From 1d16220c1ce564953551d45e2fa7689b23bd4030 Mon Sep 17 00:00:00 2001 From: Francisco Coelho Date: Tue, 30 Jan 2024 10:52:07 +0000 Subject: [PATCH] hello 2024 --- 2023-11-11--biblio.zip | Bin 0 -> 31598238 bytes NOVALINCS/00-Application_Form-FINAL.md | 53 +++++++++++++++++++++++++++++++++++++++++++++++++++++ NOVALINCS/00-Application_Form-FINAL.pdf | Bin 0 -> 44220 bytes NOVALINCS/00-Call-TEMPLATE.md | 54 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ NOVALINCS/00-Call-TEMPLATE.pdf | Bin 0 -> 81806 bytes NOVALINCS/01-Call-DRAFT.md | 12 ++++++++++++ NOVALINCS/01-Call-DRAFT.pdf | Bin 0 -> 39226 bytes NOVALINCS/01-Call-Underg_A.md | 17 +++++++++++++++++ NOVALINCS/NOVA LINCS.UE TRA Application Form | Zugzwang.docx | Bin 0 -> 12047 bytes NOVALINCS/NOVA LINCS.UE TRA Application Form | Zugzwang.gddoc | 1 + NOVALINCS/TdOS-JasonMAllen.jpg | Bin 0 -> 87283 bytes NOVALINCS/bolsas/BI/2023-01-91 | Oficio Juri.gddoc | 1 + NOVALINCS/bolsas/BI/2023-02-01 _ Ata1.pdf | Bin 0 -> 86182 bytes 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create mode 100644 text/summaries/00_DistSem.md create mode 100644 text/summaries/00_DistSem.pdf create mode 100644 text/summaries/00_ILP.md create mode 100644 text/summaries/00_ILP.pdf create mode 100644 text/summaries/00_PASP.pdf create mode 100644 text/summaries/00_PASP_credal.md create mode 100644 text/summaries/00_POTASSCO.md create mode 100644 text/summaries/00_PROB.md create mode 100644 text/summaries/00_Z3.md create mode 100644 text/summaries/00_pasp.md create mode 100644 zugzwang.code-workspace diff --git a/2023-11-11--biblio.zip b/2023-11-11--biblio.zip new file mode 100644 index 0000000..b67ef93 Binary files /dev/null and b/2023-11-11--biblio.zip differ diff --git a/NOVALINCS/00-Application_Form-FINAL.md b/NOVALINCS/00-Application_Form-FINAL.md new file mode 100644 index 0000000..1fcb5d5 --- /dev/null +++ b/NOVALINCS/00-Application_Form-FINAL.md @@ -0,0 +1,53 @@ +# Zugzwang | Logic and Artificial Intelligence + +## Team + +- Francisco Coelho +- Salvador Abreu (PI) +- Bruno Dinis (External Collaborator) + +## Research Questions + +How to extend probability annotations on an ASP program to a distribution over the possible observations? In general, the association of probabilities to some facts is not enough to uniquely define a probability of stable models. This lack of information must be carefully expressed and handled, to avoid biased results. + +Once made explicit, how to use such distribution, together with an empiric distribution from a dataset, to do general probabilistic tasks such as the estimation of a marginal or of the joint probability? Having a probability on a set of observations, including the stable models, might not cover all the sample space. Again, information and structure entailed from the ASP program must guide unbiased extensions to the sample space. + +How to use distribution measures to score ASP programs? We view ASP programs as formalized knowledge about an observable system. Some of those programs will be "better" than others, as determined by a quantitative, objective, measure, rooted on well-known functions such as the Kullback–Leibler divergence. + +This last question leads to the application of evolutionary algorithms to inductive logic programming and to a bridge with common machine learning methods, where a model is scored by a dataset. Other important questions lie behind the scope of this project. For example, how to deal with latent, unobserved, variables? + +## Novelty/Impact + +ASP has some key advantages over Prolog, most of which result from ASP being a truly declarative language and modern APS systems, such as CLASP, apply efficient optimisation techniques. Other systems, like ILASP, learn ASP programs with normal rules, choice rules, and hard and weak constraints. Moreover, ASP can use recent important advances concerning SAT solvers to ILP tasks. However, the assumption that the knowledge base (BK) is perfect, leaving no room for uncertainty, poses here a major limitation. + +One approach to overcome this restriction on logic programs is Statistical Relationship Learning (StarAI), that extends the BK with probabilities in order to setup a distribution representation. The formal setting for this approach rests on Sato's Distribution Semantics and frames systems such as Problog and PRISM. But these systems are oriented towards Prolog-like programs and semantic and leave out ASP program learning. + +One important ongoing research question is the precise semantic of an ASP program annotated with probabilities. Sato's semantic specify an unique probability distribution over Prolog-like program's atoms, but this uniqueness fails for ASP programs. Efforts to address this problem are either based on Credal sets or on selecting one probability over others, such as the P-log, the LP^MLN or the PrASP languages. + +The novelty of this research is to define the semantic of an ASP program + probability annotated facts + observations from the following process: (a) parametrise the uncertainty on stable models and annotations, (b) setup a partition of the sample space around on the stable models and (c) use observations to estimate the value of the uncertainty parameters of step (a). +If successful, it would have an impact on how logic programs express and are used to deal with real-world problems, where both uncertainty and formal KB are required ingredients. Also, when used in ILP problems the resulting models are logic programs, much simpler to understand than numeric models. + +## Expected Results / Demonstrators + +Our aim is to develop a set of software tools to apply and evaluate the theoretical results on well-known, and relevant, problems. The initial target framework is the Potassco suite, that provides a Python API to state-of-the-art grounding (CLASP) and solving (CLINGO) tools, as well as ASP-Core-2 Language support. + +The researchers will do the theoretical study and supervise one to three undergraduate, master or Ph.D. students on the implementation tasks. Theoretical work includes formalization and assessment of methods; The students will implement the tools required to experiment, explore and test those methods. The degree of complexity of the implementation tasks must be adapted to the individual competences and interests of the students. + +The theoretical work is to be reported in one or two papers with intermediate results and a final comprehensive paper for a conference. + +## Relationship of activity w/ NOVA LINCS strategic program and Research Group work plan + +This project involves two members from the NOVA LINCS "Knowledge-Based Systems" research group, from the Universidade de Évora pole and a member of the CIMA center from the Universidade de Évora. + +Part of the proposed research and software is currently being developed. + +## Resources required and justification + +Students are expected to benefit from a BIC or a BI scholarship. Depending on the candidates the respective durations can range from 3 to 12 months and the respective amounts from 486.12€ to 1144.64€ per month. + +Considering one undergraduate student with a 3 month scholarship: + +- 972.24 € = 3 months x 486.12 €/month; +- TOTAL: 972.24 € + + diff --git a/NOVALINCS/00-Application_Form-FINAL.pdf b/NOVALINCS/00-Application_Form-FINAL.pdf new file mode 100644 index 0000000..1e371bc Binary files /dev/null and b/NOVALINCS/00-Application_Form-FINAL.pdf differ diff --git a/NOVALINCS/00-Call-TEMPLATE.md b/NOVALINCS/00-Call-TEMPLATE.md new file mode 100644 index 0000000..b4b0cb0 --- /dev/null +++ b/NOVALINCS/00-Call-TEMPLATE.md @@ -0,0 +1,54 @@ +# Concurso Bolsa | TEMPLATE + +- _centro de investigação ou departamento_ + +- _identificação da tipologia de bolsa oferecida e número de vagas_ + +- _dia_ de _mês_ de 2023 + +Encontra-se aberto concurso para a atribuição de _número_ Bolsas _tipo_ no âmbito do projeto _título do projeto_, _referência_, financiado por fundos nacionais através da _…_ exemplo: FCT/MCTES e cofinanciado pelo Fundo Europeu de Desenvolvimento Regional _FEDER_ através do COMPETE – Programa Operacional Fatores de Competitividade _POFC_ _quando aplicável_, nas seguintes condições: + +- **Área Científica:** … + +- **Requisitos de admissão:** _indicar a habilitação académica necessária, bem como a experiência exigida em investigação e outros requisitos de admissão, incluindo fatores preferenciais_ +Escolher uma opção: + + - Conforme o Regulamento de Bolsas de Investigação da FCT nº950/2019 de 16 de dezembro de 2019, artigo 3º e 6º, os candidatos a BI _Bolsas de Investigação_ devem cumprir como condição para a atribuição da bolsa, a inserção efetiva em ciclos de estudos conducentes à atribuição de graus académicos ou em cursos não conferentes de grau académico. Os cursos não conferentes de grau académico correspondem aos cursos previstos na alínea e_ do nº 3 do artigo 4º do Decreto-Lei nº74/2006 de 24 de março e deverão ser desenvolvidos numa instituição de ensino superior em associação a pelo menos uma unidade de I&D, incluindo-se o plano do curso numa ou em várias áreas de investigação da unidade. + - Conforme o Regulamento de Bolsas de Investigação da FCT nº950/2019 de 16 de dezembro de 2019, artigo 5º, — As BII _Bolsas de Iniciação à Investigação_ não podem ser atribuídas a quem já tenha beneficiado de bolsas de investigação direta ou indiretamente financiadas pela FCT, atribuídas nos termos do Estatuto do Bolseiro de Investigação. + - Relativamente às BIPD _Bolsas de Investigação Pós-doutoral_, ao nível dos requisitos a verificar para atribuição das mesmas _artigo 7º do Regulamento de Bolsas de Investigação da FCT nº950/2019_ destaca-se a necessidade do grau de doutor ter sido obtido nos 3 anos anteriores à data de submissão da candidatura à bolsa e, ao nível da sua execução, o facto de apenas poder ser renovada até ao prazo máximo de 3 anos. + +- **Plano de trabalhos:** _apresentar um resumo dos trabalhos a desenvolver e dos objetivos a atingir_ + +- **Legislação e regulamentação aplicável:** A concessão da Bolsa de Investigação será realizada mediante a celebração de um contrato entre a Universidade de Évora e o bolseiro conforme minuta , nos termos do Estatuto do Bolseiro de Investigação _Lei nº40/2004 de 18 de agosto e decreto-lei nº 123/2019 de 28 de agosto_ e de acordo com a legislação e Regulamento de Bolsas de Investigação da Fundação para a Ciência e a Tecnologia, I.P em vigor, regulamento nº950/2019 de 16 de dezembro de 2019: e demais normas aplicáveis. + +- **Local de trabalho:** O trabalho será desenvolvido no(a) _denominação da unidade de investigação_ da Universidade de Évora, sob a orientação científica do Professor(a)/Doutor(a) ……… + +- **Duração da(s) bolsa(s):** A bolsa terá a duração de ..... meses, com início previsto em ..... _mês_ de .......... _ano_. O contrato de bolsa poderá ser renovado até _ex._ ao máximo de ….. meses ou até ao final da dotação orçamental do projeto de financiamento _…_. + +- **Valor do subsídio de manutenção mensal:** O montante da bolsa corresponde a €………., conforme tabela de valores das bolsas atribuídas diretamente pela FCT, I.P. no País _http://fct.pt/apoios/bolsas/valores_, sendo os pagamentos efetuados mensalmente, através de cheque ou transferência bancária. + +- **Métodos de seleção:** Os métodos de seleção a utilizar serão os seguintes: _avaliação curricular, entrevista, provas de conhecimento, ou outros_, com a respetiva valoração de _indicar os valores atribuídos a cada critério ou item avaliado e sua ponderação percentual_. + +- **Composição do Júri de Seleção:** _identificação do Presidente do Júri e dos vogais efetivos e suplentes_ + +- **Forma de publicitação/notificação dos resultados:** Os resultados finais da avaliação serão publicitados, através de lista ordenada _alfabeticamente, por nota final obtida ou outra_ afixada em local visível e público do(a) _indicar local da instituição_, sendo o candidato(a) aprovado(a) notificado através de _email, ofício ou outro_. +Nos termos de direito de audiência prévia dos interessados o projeto de Classificação Final será anunciado por qualquer meio escrito a todos os interessados. +Após comunicação da lista provisória dos resultados da avaliação, os candidatos dispõem de um período de 10 dias úteis para, querendo, se pronunciarem em sede de audiência prévia de interessados. + +- **Prazo de candidatura e forma de apresentação das candidaturas:** O concurso encontra-se aberto no período de ......................... a ......................... de 2022 e os resultados da seleção serão publicados até………………..de………………….de 2022. _O prazo de apresentação de candidaturas não deve ser inferior a 10 dias úteis_. + +- **As candidaturas devem ser formalizadas**, obrigatoriamente, através do envio de carta de candidatura acompanhada dos seguintes documentos: _ex: Curriculum Vitae, certificado de habilitações, cartas de referência ou recomendação e outros documentos comprovativos considerados relevantes_. + - Para efeitos de candidatura os comprovativos podem ser substituídos por declaração de honra do candidato, mas a não demonstração, em fase de contratualização, da posse do grau exigido à data limite da candidatura ou a não apresentação dos comprovativos de matrícula ou inscrição em ciclo de estudos ou curso não conferente de grau, para as bolsas com essa componente, implicam a anulação da avaliação do candidato. + - Os graus académicos obtidos em países estrangeiros necessitam de registo por uma Instituição Portuguesa de acordo com o Decreto-lei nº. 66/2018, de 16 de agosto e a Portaria nº. 33/2019, de 25 de janeiro. A apresentação do certificado é obrigatória para a assinatura do contrato. + Mais informação poderá ser obtida em: + + +- **As candidaturas deverão ser remetidas por e-mail para:** + - Prof. Doutor(a)…………………. + - _Centro/Dep._……………………………….. da Universidade de Évora + - e-mail: + + + +_logos do programa de financiamento quando aplicável_ + diff --git a/NOVALINCS/00-Call-TEMPLATE.pdf b/NOVALINCS/00-Call-TEMPLATE.pdf new file mode 100644 index 0000000..a8670c5 Binary files /dev/null and b/NOVALINCS/00-Call-TEMPLATE.pdf differ diff --git a/NOVALINCS/01-Call-DRAFT.md b/NOVALINCS/01-Call-DRAFT.md new file mode 100644 index 0000000..1f0f4d1 --- /dev/null +++ b/NOVALINCS/01-Call-DRAFT.md @@ -0,0 +1,12 @@ +# Concurso Bolsa | Zugzwang | DRAFT + +> A minuta das candidatura define vários campos a preencher. Quase todos são triviais mas os restantes são **críticos**. Estes necessitam de consenso entre nós. +> +> Além disso, no projeto ficou em aberto o número e tipologia de bolsas. Agora temos de concretizar essas opções: **Exatamente,** quantas bolsas de que tipo? + +## Campos Críticos + +- **Requisitos de admissão:** _indicar a habilitação académica necessária, bem como a experiência exigida em investigação e outros requisitos de admissão, incluindo fatores preferenciais_ +- **Plano de trabalhos:** _apresentar um resumo dos trabalhos a desenvolver e dos objetivos a atingir_ +- **Métodos de seleção:** Os métodos de seleção a utilizar serão os seguintes: _avaliação curricular, entrevista, provas de conhecimento, ou outros_, com a respetiva valoração de _indicar os valores atribuídos a cada critério ou item avaliado e sua ponderação percentual_. +- **As candidaturas devem ser formalizadas**, obrigatoriamente, através do envio de carta de candidatura acompanhada dos seguintes documentos: _ex: Curriculum Vitae, certificado de habilitações, cartas de referência ou recomendação e outros documentos comprovativos considerados relevantes_.<<<>>> \ No newline at end of file diff --git a/NOVALINCS/01-Call-DRAFT.pdf b/NOVALINCS/01-Call-DRAFT.pdf new file mode 100644 index 0000000..e0a5a52 Binary files /dev/null and b/NOVALINCS/01-Call-DRAFT.pdf differ diff --git a/NOVALINCS/01-Call-Underg_A.md b/NOVALINCS/01-Call-Underg_A.md new file mode 100644 index 0000000..927d500 --- /dev/null +++ b/NOVALINCS/01-Call-Underg_A.md @@ -0,0 +1,17 @@ +# Concurso para "Bolsa de Iniciação à Investigação" + +- **Requisitos de admissão:** + - Estudante inscrito numa licenciatura, que não tenha já beneficiado de bolsa de investigação direta ou indiretamente financiadas pela FCT, atribuída nos termos do Estatuto do Bolseiro de Investigação. + - Aprovação a, pelo menos, uma unidade curricular de programação, preferencialmente em `Python`. +- **Plano de trabalhos:** Implementação duma biblioteca `Python` para processar ficheiros de código _ASP_ anotados com probabilidades com as seguintes funcionalidades: + 1. Extração das anotações e associação aos factos relevantes. + 2. Comunicação com o sistema `Potassco` para obtenção dos resultados do programa _ASP_ subjacente. + 3. 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a/NOVALINCS/bolsas/BII/AliceMartins-l52768/01-candidatura/comprovativo_matricula_inscricao.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/01-candidatura/comprovativo_matricula_inscricao.pdf new file mode 100644 index 0000000..35cee79 Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/01-candidatura/comprovativo_matricula_inscricao.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/01-candidatura/dados.md b/NOVALINCS/bolsas/BII/AliceMartins-l52768/01-candidatura/dados.md new file mode 100644 index 0000000..766c43f --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/01-candidatura/dados.md @@ -0,0 +1,6 @@ +Número do Cartão de Cidadão: 14594754 8 ZX5 +Número de Contribuinte: 259379174 +Morada: + Rua Joaquim Marques Saias, Lote 19, Moradia D + 2130-299 Benavente +Ciencia-ID: 861B-FBEF-B218 diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/2023-09-01 Relatorio Atividades Bolseiros.pdf 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+\def\QUALIDADE{Professor Catedrático} + +\begin{document} + +\unidade{\textsc{\textcolor{gray}{NOVA-LINCS (Pólo de Évora)}}} + +\documento + {12/DI.SPA/23} + {Renovação de Bolsa de Iniciação à Investigação} + {31 de julho de 2023} + +Na qualidade de Coordenador do pólo de Évora do NOVA-LINCS, solicito que, em acordo com o parecer do orientador, seja renovada a bolsa de iniciação à investigação (para licenciada) com a aluna Alice Fernandes Martins, para o perı́odo de 2 de agosto a 2 de novembro de 2023, no âmbito do Financiamento Plurianual da unidade de I\&D com a referência: \texttt{UIDP/04516/2020} (NOVA LINCS: Financiamento Programático 2020–2023), cofinanciado pela Fundação para Ciência e a Tecnologia. + +\CHAU{Évora, julho de 2023,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/parecer/parecer_renovação.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/parecer/parecer_renovação.pdf new file mode 100644 index 0000000..3936907 Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/parecer/parecer_renovação.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/parecer/parecer_renovação.tex b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/parecer/parecer_renovação.tex new file mode 100644 index 0000000..099894d --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/parecer/parecer_renovação.tex @@ -0,0 +1,36 @@ +% !TeX program = xelatex +\documentclass{oficio} + +\usepackage{polyglossia} +\setmainlanguage{portuges} + +\def\ASSINATURA{\includegraphics{fcsig.pdf}} + +\begin{document} +\unidade{\textsc{\DI}} +\documento{% +NL/ZZ/BII/1/AM/1 +}{% +Renovação de BII --- Parecer}{% +31 de julho de 2023} +%\EU{} + +\qualidade{(Professor Auxiliar)} + +A aluna +% +Alice Fernandes Martins, +% +detentora de uma \emph{Bolsa de Iniciação à Investigação} no âmbito do Financiamento Plurianual da unidade I\&D Referência: UIDP/04516/2020 (NOVA LINCS: Financiamento Programático 2020-2023) cofinanciado pela Fundação para Ciência e a Tecnologia +% +\textbf{completou corretamente todas as tarefas propostas no âmbito desta bolsa}. +% + +% +Como orientador científico, +% +dado o excelente desempenho da bolseira e a necessidade, no projeto Zugzwang, de continuar o desenvolvimento destas tarefas, \textbf{recomendo a renovação da bolsa}. + +\CHAU{Com os melhores cumprimentos,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.md b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.md new file mode 100644 index 0000000..333cb92 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.md @@ -0,0 +1,20 @@ +# Plano de Trabalhos - Renovação de BII + +
+Évora, 4 de setembro de 2023 +
+ + +Considerando que foi já implementada uma biblioteca Python que proporciona as seguintes funcionalidades: +- Ler a descrição de uma rede Bayesiana num formato comum (BIF) para uma representação intermédia. +- Escrever uma especificação ASP anotada com probabilidades a partir da representação intermédia indicada no ponto anterior. + +e que, com esta biblioteca é possível aplicar métodos e ferramentas desenvolvidos no âmbito do projeto “Zugzwang | Lógica e Inteligência Artificial” a um conjunto alargado de problemas, **pretende-se continuar,** com a implementação duma biblioteca Python para processar ficheiros de código ASP anotados com probabilidades incluindo as seguintes funcionalidades: + +- Extração das anotações e associação aos factos relevantes. +- Comunicação com o sistema Potassco para obtenção dos resultados do programa ASP subjacente. +- Cálculos de probabilidades com base nos modelos estáveis obtidos no passo anterior, nas anotações extraídas no passo 1, e nos resultados teóricos deste projeto. + +
+(Francisco Coelho) +
diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.pdf new file mode 100644 index 0000000..a3e00fa Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.tex b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.tex new file mode 100644 index 0000000..5ea5c5f --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/plano-trab/plano-trabalhos.tex @@ -0,0 +1,38 @@ +% !TeX program = xelatex +\documentclass{oficio} + +\usepackage{polyglossia} +\setmainlanguage{portuges} + +\def\ASSINATURA{\includegraphics{fcsig.pdf}} + +\begin{document} +\unidade{\textsc{\DI}} +\documento{% +NL/ZZ/BII/1/AM/2 +}{% +Renovação de BII --- Plano de Trabalhos}{% +31 de julho de} +%\EU{} + +\qualidade{(Professor Auxiliar)} + +Considerando que foi já implementada uma biblioteca \texttt{Python} que proporciona as seguintes funcionalidades: +% +\begin{itemize} + \item Ler a descrição de uma rede Bayesiana num formato comum (\texttt{BIF}) para uma representação intermédia. + \item Escrever uma especificação \texttt{ASP} anotada com probabilidades a partir da representação intermédia indicada no ponto anterior. +\end{itemize} +% +e que, com esta biblioteca é possível aplicar métodos e ferramentas desenvolvidos no âmbito do projeto ``\emph{Zugzwang | Lógica e Inteligência Artificial}'' a um conjunto alargado de problemas, \textbf{pretende-se continuar,} com a implementação duma biblioteca \texttt{Python} para processar ficheiros de código ASP anotados com probabilidades incluindo as seguintes funcionalidades: + +\begin{itemize} + + \item Extração das anotações e associação aos factos relevantes. + \item Comunicação com o sistema Potassco para obtenção dos resultados do programa ASP subjacente. + \item Cálculos de probabilidades com base nos modelos estáveis obtidos no passo anterior, nas anotações extraídas no passo 1, e nos resultados teóricos deste projeto. +\end{itemize} + +\CHAU{Com os melhores cumprimentos,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf new file mode 100644 index 0000000..b6c4e5d Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/relatório/2023-09-01 Relatorio Atividades Bolseiros.xopp b/NOVALINCS/bolsas/BII/AliceMartins-l52768/02-renovação/relatório/2023-09-01 Relatorio Atividades Bolseiros.xopp new file mode 100644 index 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b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/ofício/ofício_renovação.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/ofício/ofício_renovação.tex b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/ofício/ofício_renovação.tex new file mode 100644 index 0000000..2c43559 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/ofício/ofício_renovação.tex @@ -0,0 +1,31 @@ +% !TeX program = xelatex +\documentclass{oficio} +\usepackage{polyglossia} +\setmainlanguage{portuges} +\usepackage{xcolor} + + +\def\LOGO{\LOGOECT} +\def\DESTINATARIO{\\ +Magnífica Reitora da\\ +Universidade de Évora\\ +Professora Doutora Hermínia Vilar +} +\def\EU{Salvador Pinto Abreu} +\def\ASSINATURA{\vspace{5mm}} +\def\QUALIDADE{Professor Catedrático} + +\begin{document} + +\unidade{\textsc{\textcolor{gray}{NOVA-LINCS (Pólo de Évora)}}} + +\documento + {12/DI.SPA/23} + {Renovação de Bolsa de Iniciação à Investigação} + {31 de outubro de 2023} + +Na qualidade de Coordenador do pólo de Évora do NOVA-LINCS, solicito que, em acordo com o parecer do orientador, seja renovada a bolsa de iniciação à investigação (para licenciada) com a aluna Alice Fernandes Martins, para o perı́odo de 2 de novembro de 2023 a 2 de janeiro de 2024, no âmbito do Financiamento Plurianual da unidade de I\&D com a referência: \texttt{UIDP/04516/2020} (NOVA LINCS: Financiamento Programático 2020–2023), cofinanciado pela Fundação para Ciência e a Tecnologia. + +\CHAU{Évora, outubro de 2023,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/parecer/parecer_renovação.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/parecer/parecer_renovação.pdf new file mode 100644 index 0000000..c53fa62 Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/parecer/parecer_renovação.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/parecer/parecer_renovação.tex b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/parecer/parecer_renovação.tex new file mode 100644 index 0000000..0fa4e67 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/parecer/parecer_renovação.tex @@ -0,0 +1,36 @@ +% !TeX program = xelatex +\documentclass{oficio} + +\usepackage{polyglossia} +\setmainlanguage{portuges} + +\def\ASSINATURA{\includegraphics{fcsig.pdf}} + +\begin{document} +\unidade{\textsc{\DI}} +\documento{% +NL/ZZ/BII/1/AM/3 +}{% +Renovação de BII --- Parecer} +{31 de outubro de 2023} +%\EU{} + +\qualidade{(Professor Auxiliar)} + +A aluna +% +Alice Fernandes Martins, +% +detentora de uma \emph{Bolsa de Iniciação à Investigação} no âmbito do Financiamento Plurianual da unidade I\&D Referência: UIDP/04516/2020 (NOVA LINCS: Financiamento Programático 2020-2023) cofinanciado pela Fundação para Ciência e a Tecnologia +% +\textbf{completou corretamente todas as tarefas propostas no âmbito desta bolsa}. +% + +% +Como orientador científico, +% +dado o excelente desempenho da bolseira e a necessidade, no projeto Zugzwang, de continuar o desenvolvimento destas tarefas, \textbf{recomendo a renovação da bolsa}. + +\CHAU{Com os melhores cumprimentos,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.md b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.md new file mode 100644 index 0000000..333cb92 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.md @@ -0,0 +1,20 @@ +# Plano de Trabalhos - Renovação de BII + +
+Évora, 4 de setembro de 2023 +
+ + +Considerando que foi já implementada uma biblioteca Python que proporciona as seguintes funcionalidades: +- Ler a descrição de uma rede Bayesiana num formato comum (BIF) para uma representação intermédia. +- Escrever uma especificação ASP anotada com probabilidades a partir da representação intermédia indicada no ponto anterior. + +e que, com esta biblioteca é possível aplicar métodos e ferramentas desenvolvidos no âmbito do projeto “Zugzwang | Lógica e Inteligência Artificial” a um conjunto alargado de problemas, **pretende-se continuar,** com a implementação duma biblioteca Python para processar ficheiros de código ASP anotados com probabilidades incluindo as seguintes funcionalidades: + +- Extração das anotações e associação aos factos relevantes. +- Comunicação com o sistema Potassco para obtenção dos resultados do programa ASP subjacente. +- Cálculos de probabilidades com base nos modelos estáveis obtidos no passo anterior, nas anotações extraídas no passo 1, e nos resultados teóricos deste projeto. + +
+(Francisco Coelho) +
diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.pdf new file mode 100644 index 0000000..d5f9f31 Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.tex b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.tex new file mode 100644 index 0000000..894fd92 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/plano-trab/plano-trabalhos.tex @@ -0,0 +1,38 @@ +% !TeX program = xelatex +\documentclass{oficio} + +\usepackage{polyglossia} +\setmainlanguage{portuges} + +\def\ASSINATURA{\includegraphics{fcsig.pdf}} + +\begin{document} +\unidade{\textsc{\DI}} +\documento{% +NL/ZZ/BII/1/AM/4 +}{% +Renovação de BII --- Plano de Trabalhos}{% +31 de outubro de 2023} +%\EU{} + +\qualidade{(Professor Auxiliar)} + +Considerando que foram já implementadas biblioteca \texttt{Python} que proporcionam as seguintes funcionalidades: +% +\begin{itemize} + \item Ler a descrição de uma rede Bayesiana num formato comum (\texttt{BIF}) para uma representação intermédia. + \item Escrever uma especificação \texttt{ASP} anotada com probabilidades a partir da representação intermédia indicada no ponto anterior. +\end{itemize} +% +e que, com esta biblioteca é possível aplicar métodos e ferramentas desenvolvidos no âmbito do projeto ``\emph{Zugzwang | Lógica e Inteligência Artificial}'' a um conjunto alargado de problemas, \textbf{pretende-se continuar,} com a implementação duma biblioteca \texttt{Python} para processar ficheiros de código ASP anotados com probabilidades, incluindo as seguintes funcionalidades: + +\begin{itemize} + + \item Extração das anotações e associação aos factos relevantes. + \item Comunicação com o sistema \texttt{Potassco} para obtenção dos resultados do programa \texttt{ASP} subjacente. + \item Cálculos de probabilidades com base nos modelos estáveis obtidos no passo anterior, nas anotações extraídas no passo 1, e nos resultados teóricos deste projeto. +\end{itemize} + +\CHAU{Com os melhores cumprimentos,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf new file mode 100644 index 0000000..b6c4e5d Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação (nov)/relatório/2023-09-01 Relatorio Atividades Bolseiros.xopp b/NOVALINCS/bolsas/BII/AliceMartins-l52768/03-renovação 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b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/ofício/ofício_renovação.tex new file mode 100644 index 0000000..2c43559 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/ofício/ofício_renovação.tex @@ -0,0 +1,31 @@ +% !TeX program = xelatex +\documentclass{oficio} +\usepackage{polyglossia} +\setmainlanguage{portuges} +\usepackage{xcolor} + + +\def\LOGO{\LOGOECT} +\def\DESTINATARIO{\\ +Magnífica Reitora da\\ +Universidade de Évora\\ +Professora Doutora Hermínia Vilar +} +\def\EU{Salvador Pinto Abreu} +\def\ASSINATURA{\vspace{5mm}} +\def\QUALIDADE{Professor Catedrático} + +\begin{document} + +\unidade{\textsc{\textcolor{gray}{NOVA-LINCS (Pólo de Évora)}}} + +\documento + {12/DI.SPA/23} + {Renovação de Bolsa de Iniciação à Investigação} + {31 de outubro de 2023} + +Na qualidade de Coordenador do pólo de Évora do NOVA-LINCS, solicito que, em acordo com o parecer do orientador, seja renovada a bolsa de iniciação à investigação (para licenciada) com a aluna Alice Fernandes Martins, para o perı́odo de 2 de novembro de 2023 a 2 de janeiro de 2024, no âmbito do Financiamento Plurianual da unidade de I\&D com a referência: \texttt{UIDP/04516/2020} (NOVA LINCS: Financiamento Programático 2020–2023), cofinanciado pela Fundação para Ciência e a Tecnologia. + +\CHAU{Évora, outubro de 2023,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/parecer/parecer_renovação.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/parecer/parecer_renovação.pdf new file mode 100644 index 0000000..c53fa62 Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/parecer/parecer_renovação.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/parecer/parecer_renovação.tex b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/parecer/parecer_renovação.tex new file mode 100644 index 0000000..0fa4e67 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/parecer/parecer_renovação.tex @@ -0,0 +1,36 @@ +% !TeX program = xelatex +\documentclass{oficio} + +\usepackage{polyglossia} +\setmainlanguage{portuges} + +\def\ASSINATURA{\includegraphics{fcsig.pdf}} + +\begin{document} +\unidade{\textsc{\DI}} +\documento{% +NL/ZZ/BII/1/AM/3 +}{% +Renovação de BII --- Parecer} +{31 de outubro de 2023} +%\EU{} + +\qualidade{(Professor Auxiliar)} + +A aluna +% +Alice Fernandes Martins, +% +detentora de uma \emph{Bolsa de Iniciação à Investigação} no âmbito do Financiamento Plurianual da unidade I\&D Referência: UIDP/04516/2020 (NOVA LINCS: Financiamento Programático 2020-2023) cofinanciado pela Fundação para Ciência e a Tecnologia +% +\textbf{completou corretamente todas as tarefas propostas no âmbito desta bolsa}. +% + +% +Como orientador científico, +% +dado o excelente desempenho da bolseira e a necessidade, no projeto Zugzwang, de continuar o desenvolvimento destas tarefas, \textbf{recomendo a renovação da bolsa}. + +\CHAU{Com os melhores cumprimentos,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.md b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.md new file mode 100644 index 0000000..333cb92 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.md @@ -0,0 +1,20 @@ +# Plano de Trabalhos - Renovação de BII + +
+Évora, 4 de setembro de 2023 +
+ + +Considerando que foi já implementada uma biblioteca Python que proporciona as seguintes funcionalidades: +- Ler a descrição de uma rede Bayesiana num formato comum (BIF) para uma representação intermédia. +- Escrever uma especificação ASP anotada com probabilidades a partir da representação intermédia indicada no ponto anterior. + +e que, com esta biblioteca é possível aplicar métodos e ferramentas desenvolvidos no âmbito do projeto “Zugzwang | Lógica e Inteligência Artificial” a um conjunto alargado de problemas, **pretende-se continuar,** com a implementação duma biblioteca Python para processar ficheiros de código ASP anotados com probabilidades incluindo as seguintes funcionalidades: + +- Extração das anotações e associação aos factos relevantes. +- Comunicação com o sistema Potassco para obtenção dos resultados do programa ASP subjacente. +- Cálculos de probabilidades com base nos modelos estáveis obtidos no passo anterior, nas anotações extraídas no passo 1, e nos resultados teóricos deste projeto. + +
+(Francisco Coelho) +
diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.pdf new file mode 100644 index 0000000..d5f9f31 Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.tex b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.tex new file mode 100644 index 0000000..894fd92 --- /dev/null +++ b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/plano-trab/plano-trabalhos.tex @@ -0,0 +1,38 @@ +% !TeX program = xelatex +\documentclass{oficio} + +\usepackage{polyglossia} +\setmainlanguage{portuges} + +\def\ASSINATURA{\includegraphics{fcsig.pdf}} + +\begin{document} +\unidade{\textsc{\DI}} +\documento{% +NL/ZZ/BII/1/AM/4 +}{% +Renovação de BII --- Plano de Trabalhos}{% +31 de outubro de 2023} +%\EU{} + +\qualidade{(Professor Auxiliar)} + +Considerando que foram já implementadas biblioteca \texttt{Python} que proporcionam as seguintes funcionalidades: +% +\begin{itemize} + \item Ler a descrição de uma rede Bayesiana num formato comum (\texttt{BIF}) para uma representação intermédia. + \item Escrever uma especificação \texttt{ASP} anotada com probabilidades a partir da representação intermédia indicada no ponto anterior. +\end{itemize} +% +e que, com esta biblioteca é possível aplicar métodos e ferramentas desenvolvidos no âmbito do projeto ``\emph{Zugzwang | Lógica e Inteligência Artificial}'' a um conjunto alargado de problemas, \textbf{pretende-se continuar,} com a implementação duma biblioteca \texttt{Python} para processar ficheiros de código ASP anotados com probabilidades, incluindo as seguintes funcionalidades: + +\begin{itemize} + + \item Extração das anotações e associação aos factos relevantes. + \item Comunicação com o sistema \texttt{Potassco} para obtenção dos resultados do programa \texttt{ASP} subjacente. + \item Cálculos de probabilidades com base nos modelos estáveis obtidos no passo anterior, nas anotações extraídas no passo 1, e nos resultados teóricos deste projeto. +\end{itemize} + +\CHAU{Com os melhores cumprimentos,} + +\end{document} \ No newline at end of file diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf new file mode 100644 index 0000000..b6c4e5d Binary files /dev/null and b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/relatório/2023-09-01 Relatorio Atividades Bolseiros.pdf differ diff --git a/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação (jan)/relatório/2023-09-01 Relatorio Atividades Bolseiros.xopp b/NOVALINCS/bolsas/BII/AliceMartins-l52768/04-renovação 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end of file diff --git a/NOVALINCS/demo-markdown.md b/NOVALINCS/demo-markdown.md new file mode 100644 index 0000000..7f75c7d --- /dev/null +++ b/NOVALINCS/demo-markdown.md @@ -0,0 +1,32 @@ +# Título + +> O `markdown` permite formatar um documento seguindo algumas convenções facilmente legíveis, ao contrário, por exemplo, do `LaTeX`. +> +> Este fragmento de texto é uma "citação" e pode ter vários parágrafos. +> Como no `LaTeX`, é precisa uma linha de intervalo entre parágrafos. + +A formatação de `markdown` é muito simples: usam-se dois `**` para **negrito** e um `_` para _itálico_. Para `fonte fixa` usa-se o acento agudo, "\`". + +Não há muito mais formatação: +- As listas consistem em linhas que começam por `-` e podem ter sub-listas: + - Como esta. +- É possível, em certos casos, usar os comandos `LaTeX` para fórmulas, que podem ser _inline_ $e^{i\pi} = -1$ ou _display_: +$$\alpha = \sum_{i=1}^{\infty} a_i.$$ + +As listas _numeradas_ também são visualmente definidas: +1. Mexer os ovos. +2. Derreter a manteiga na frigideira. +3. Quando a manteiga estiver a ferver, deitar os ovos na frigideira. + 1. Cuidado! Não deixe a manteiga queimar! +4. Deixar cozinhar a gosto. + +## Organização de Documentos + +Usam-se os `#` no início das linhas para indicar o nível da divisão: +- Um `#` é o _título_ do documento. +- Dois `##` é um "_capítulo_", três uma "_secção_", _etc._ + +--- + +Ah! Podem-se incluir _links_ para a [_web_](https://en.wikipedia.org/wiki/Markdown) e também imagens: ![ +Théâtre d’Opéra Spatial](TdOS-JasonMAllen.jpg) diff --git a/NOVALINCS/demo-markdown.pdf b/NOVALINCS/demo-markdown.pdf new file mode 100644 index 0000000..3cc6250 Binary files /dev/null and b/NOVALINCS/demo-markdown.pdf differ diff --git a/biblio.gdlink b/biblio.gdlink new file mode 100644 index 0000000..8d01dea --- /dev/null +++ b/biblio.gdlink @@ -0,0 +1 @@ +{"url": "https://drive.google.com/file/d/19ZBN9zT4YyLOEeMNvpA9Baqw9KKWd-dx/view?usp=drivesdk", "file_id": "19ZBN9zT4YyLOEeMNvpA9Baqw9KKWd-dx", "account_email": "mangon@gmail.com"} \ No newline at end of file diff --git a/biblio/2002 - Answer set programming and plan generation.pdf b/biblio/2002 - Answer set programming and plan generation.pdf new file mode 100644 index 0000000..91a3f2b Binary files /dev/null and b/biblio/2002 - Answer set programming and plan generation.pdf differ diff --git a/biblio/2013 - Answer Set Solving in Practice - Martin Gebser, Roland Kaminski, Benjamin Kaufmann, Torsten Schaub.pdf b/biblio/2013 - Answer Set Solving in Practice - Martin Gebser, Roland Kaminski, Benjamin Kaufmann, Torsten Schaub.pdf new file mode 100644 index 0000000..11d340f Binary files /dev/null and b/biblio/2013 - Answer Set Solving in Practice - Martin Gebser, Roland Kaminski, Benjamin Kaufmann, Torsten Schaub.pdf differ diff --git a/biblio/2014 - A new method of proving the existence of answer sets for disjunctive logic programs.pdf b/biblio/2014 - A new method of proving the existence of answer sets for disjunctive logic programs.pdf new file mode 100644 index 0000000..6fe313d Binary files /dev/null and b/biblio/2014 - A new method of proving the existence of answer sets for disjunctive logic programs.pdf differ diff --git a/biblio/2018 - Deep Learning, A critical appraisal - Gary Marcus.pdf b/biblio/2018 - Deep Learning, A critical appraisal - Gary Marcus.pdf new file mode 100644 index 0000000..f4cce60 Binary files /dev/null and b/biblio/2018 - Deep Learning, A critical appraisal - Gary Marcus.pdf differ diff --git a/biblio/2018 - Foundations of Probabilistic Logic Programming. Languages, Semantics, Inference and Learning - Fabrizio Riguzzi.pdf b/biblio/2018 - Foundations of Probabilistic Logic Programming. Languages, Semantics, Inference and Learning - Fabrizio Riguzzi.pdf new file mode 100644 index 0000000..7d52d6b Binary files /dev/null and b/biblio/2018 - Foundations of Probabilistic Logic Programming. 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+The DLV Tutorial +

+ +In this tutorial, we give an introduction to Disjunctive Datalog +(using some of the extensions of DLV). +The tutorial does not give a full description of the usage and capabilities of +DLV. +For a more complete account of these, see the +DLV +homepage and the + +DLV online user manual. + +The examples shown in this tutorial work with every recent +DLV release. +Executables of the DLV system for various platforms +can be downloaded from the + +DLV homepage. +

+ +The tutorial consists of the following sections, each of them being built +around a guiding example: + +

    +
  • The First Example : Rules and Facts
    • +
  • The Second Example : Negation and the Complete World Assumption
    • +
  • The Family Tree Example : Predicates, Variables, and Recursion
    • +
  • DLV as a Deductive Database System; Comparison Operators
    • +
  • +The Railway Crossing Example : True Negation and Negation as Finite Failure +
    • +
  • +The Broken Arm Example : Disjunctive Datalog and the Stable Model Semantics +
    • +
  • Strong Constraints
    • +
  • Graph Coloring: Guess&Check Programming
    • +
  • The Fibonacci Example: Built-in Predicates and Integer Arithmetics
    • +
  • The 8-Queens Example: Guess&Check Programming with Integers
    • +
  • A simple Physics Diagnosis example
    • +
  • A different way to implement the Physics Diagnosis example
    • +
  • The Monkey&Banana Example: Planning
    • +
+ +This page is quite long. People who are in a hurry might appreciate the +information that the tutorial is fully on this page, there will be no branches +and no links to further pages. +

+ +This tutorial is written for computer-literate people with a background +different from computer science, or students new to this area. +It was originally written for physicists at CERN, and some examples are +tailored towards this community. +

+ + +

+

+Introduction +

+ +

+Datalog is a declarative (programming) language. +This means that the programmer does not write a program that solves some +problem but instead specifies what the solution should look like, and a +Datalog inference engine (or Deductive Database System) tries +to find the the way to solve the problem and the solution itself. +This is done with rules and facts. +Facts are the input data, and rules can be used to derive more facts, and +hopefully, the solution of the given problem. +

+ +

+Disjunctive datalog is an extension of datalog in which the logical OR +expression (the disjunction) +is allowed to appear in the rules - this is not allowed in basic datalog. +

+ +

+DLV (= datalog with disjunction) is a powerful though +freely available deductive database system. +It is based on the declarative programming language datalog, +which is known for being a convenient tool for knowledge representation. +With its disjunctive extensions, it is well suited for all kinds of +nonmonotonic reasoning, including diagnosis and planning. +

+ +

+Finally, we have to mention to the more advanced reader that +DLV is relevant to two communities. Firstly, as mentioned, +it is a deductive database engine and can therefore be seen as a way to query +data from databases which is strictly more powerful than for example SQL +(everything that can be done with the core SQL language can also be done with +DLV, and more), but it is also often described as a +system for answer set programming (ASP). This is a powerful new paradigm +from the area of "Nonmonotonic Reasoning" which allows to formulate even very +complicated problems in a straightforward and highly declarative way. +One may call this paradigm even more declarative than classical logic. +Of course, every programming language to be processed by a computer has to +have both fixed syntax (i.e. a grammar that specifies what programs of this +language have to look like, and what combinations of symbols make a valid +program) and semantics (which abstractly specifies what the computer has to +do with the program by declaring how a program is to be translated into the/a +correct result). There is wide agreement (and also some excitement) that +both the syntax and semantics of the language of DLV +are very simple and intuitive. In fact, we do not know of any way to make +the language even simpler while preserving its characteristics. +

+ +

+Both the syntax and semantics of DLV will be described in +this tutorial. +

+ + +
+

+The First Example : Rules and Facts +

+ +

+Suppose we want to model that every time somebody tells us a joke, we laugh. +Furthermore, somebody now tells us a joke. +This could be done in the following way: + +

+joke.
+laugh :- joke.
+
+ +The first line is called a fact and expresses that joke +is true (a simple word such as joke appearing in a rule or fact +which has a truth value is called a proposition. A more general +name - which we will use in the following - for the constituents of rules +and facts is atom.). +The second line is called a rule. +It is read as "if joke is true, laugh must also be true". +(The sign ":-" is meant to be an arrow to the left, the logic programming +version of the implication.) +

+ +

+If the author of such a program decides it appropriate, one can also interpret +some causality into a rule and read this one as "from joke follows laugh". +This is pure matter of choice of the human, and DLV +does not care about it. +The left side of a rule is called its head, while the right side is +called its body. +

+ +

+A result of a Datalog computation is called a model. +The meaning of this is clear: it is a consistent explanation (model) +of the world, as far as the Datalog system can derive it. +If a datalog program is inconsistent, i.e., it is contradictory, there is +simply no model (we will see examples of this later). +

+ +

+Of course, since in this example joke is certainly true (this is +given by the fact), laugh is also true. +DLV now tries to find all those models of the world that correctly +and consistently explain the observations made (= the program). + +A model assigns a truth value (either true or false) to +each atom appearing in the program, +and is written as the set of atoms that are true in a certain model. +The model of the above program is {joke, laugh}. +When all atoms are false in a model, we talk about an empty model +(written as {}). Note that having an empty model is very different +from finding no model. We will see examples for this later. +

+ +

+Simple datalog programs like the one above always have exactly one model. +In general, though, DLV programs may have zero +(as mentioned) or even many models. We will see examples of such programs later. +

+ + + +
+

+The Second Example : Negation and the Complete World Assumption +

+ +

+Next, suppose we are not aware of being told a joke. In this case, the +correct datalog program looks like this: + +

+laugh :- joke.
+
+ +The program itself does not express that joke is false, but the so-called +Complete World Assumption (CWA) does. It is one of the foundations +DLV bases its computations on and says that everything about which +nothing is known is assumed to be false. +Therefore, the model for this program is {}. (This means that +there is a model but it is empty. It is also possible that for a given +program there is no model.) +We will come back to the CWA in more detail later in the section that +discusses DLV as a deductive database system. +

+ +

+Next, we elaborate a bit on this example. +First, we want to express that to be able to understand a joke, one has to +hear it and must not be stupid. To hear it, one must not be deaf and there +must be a joke. +Finally, to laugh about the joke, one must understand it. +Alternatively, stupid people might laugh without being told a joke. + +

+joke.
+hear_joke :- joke, not deaf.
+understand_joke :- hear_joke, not stupid.
+laugh :- understand_joke.
+laugh :- stupid, not joke.
+
+ +In two of the rules, we encounter negated atoms. These are true if the +atoms themselves are false. +We also encounter rules that contain more than one atom in the +body. In such a case, a body is true if each of the literals are true +(a literal is a possibly negated atom). +For example, + +
+hear_joke :- joke, not deaf.
+
+ +is read as +"if joke is true and deaf is false then +hear_joke must be true". +

+ +

+The model for this program is +{joke, hear_joke, understand_joke, laugh}. +Again, by virtue of the CWA, deaf and stupid +are assumed to be false - there are no facts making these atoms +true and no rules which can derive their truth. +Now suppose we remove joke. from the program and add +stupid. instead. Then, the resulting model would be +{stupid, laugh}. +

+ +Please note the following things: +(i) Those atoms that are not listed as elements of the models above are +not automatically rendered false. Rather, they are unknown. +(ii) Suppose the program would look like this: + +
+stupid.
+laugh :- stupid, not joke.
+
+ +The model of this program is {stupid, laugh}. If we now add the +fact joke. we get the model {stupid, joke}, from +which the atom laugh got lost. In other words, you may add +more information and lose information that could be derived before because of +that. Due to this property, +the formalism of DLV is called nonmonotonic, just +as mathematical functions which are neither monotonically increasing nor +decreasing are called nonmonotonic. + +At first sight, this may look like an ugly property of this formalism, but in +fact, it allows to do many useful things. +

+ + + +
+

+The Family Tree Example : Predicates, Variables, and Recursion +

+ +So far we have studied simple atoms as the building blocks of our rules. +In fact, atoms may be constructed to hold a number of arguments - they are +then also called predicates. +

+ +In the following program, we have two binary predicates, parent +and grandparent. (They are called binary because they both have +two arguments.) +

+ +We have to map some semantics to the two arguments of the predicates. Here, +the first argument is assumed to be the older person +(the parent or grandparent), while +the second argument refers to the younger person (the child or grandchild). +Certainly, we could do it the other way as well, but then we would have to +adjust all the rules that will follow. + +

+parent(john, james).
+parent(james, bill).
+grandparent(john, bill) :- parent(john, james), parent(james, bill).
+
+ +Of course, the model of this program is +{parent(john, james), parent(james, bill), grandparent(john,bill)}. + +

+With predicates, it is allowed to use variables, which begin with an upper-case +character, differently from the constants of the previous program that begin +with a lower-case letter. The following program has the same model as the +previous example: + +

+parent(john, james).
+parent(james, bill).
+grandparent(X, Y) :- parent(X, Z), parent(Z, Y).
+
+ +This new grandparent rule which uses variables simply models that every +parent of a parent is a grandparent. +

+ +

+Note that the facts of a program are often called the +Extensional Database (EDB), while the remaining rules are called +the Intensional Database (IDB). +With DLV, the EDB can be read either from a relational +or object-oriented database, or just simply from files, where no separation +of rules and facts is required. +

+ +

+We can now extend this example a bit to show how DLV +can be used to model knowledge as datalog rules and exploit it. + +First we add a few more facts to add more people and to express their gender: + +

+parent(william, john).
+parent(john, james).
+parent(james, bill).
+parent(sue, bill).
+parent(james, carol).
+parent(sue, carol).
+
+male(john).
+male(james).
+female(sue).
+male(bill).
+female(carol).
+
+ +Then we can add more rules that model family relationships. + +
+grandparent(X, Y) :- parent(X, Z), parent(Z, Y).
+father(X, Y) :- parent(X, Y), male(X).
+mother(X, Y) :- parent(X, Y), female(X).
+brother(X, Y) :- parent(P, X), parent(P, Y), male(X), X != Y.
+sister(X, Y)  :- parent(P, X), parent(P, Y), female(X), X != Y.
+
+ +The rules for brother and sister use X != Y to require that +X and Y are different (one cannot be his own brother). This is called a +built-in predicate, since it could be written as something like +not_equal(X, Y). DLV knows quite a few of +these built-in predicates. +For this program, DLV finds the following model +(to simplify readability, the facts already listed above were removed from the +model below; of course, they still belong there): + +
+{grandparent(william,james), grandparent(john,bill), grandparent(john,carol),
+father(john,james), father(james,bill), father(james,carol),
+mother(sue,bill), mother(sue,carol),
+brother(bill,carol), sister(carol,bill)}
+
+ +Let us now exchange the IDB rules against the following (the EDB facts remain +the same): + +
+ancestor(X, Y) :- parent(X, Y).
+ancestor(X, Y) :- parent(X, Z), ancestor(Z, Y).
+
+ +These rules are interesting, since they use recursion +to implement transitivity. +They express that, to start with, every parent is an ancestor, and, secondly, +that every parent of an ancestor is an ancestor. +Please note that the semantics used ensures that it is impossible that there +be any problems with left-recursion as they occur in languages as Prolog. +In DLV, the programmer can safely ignore such +considerations. +

+ +

+The model of this program combined with the six-entries parent +facts base above results in the following model (where the parent +facts were again removed for readability): + +

+{ancestor(william,john), ancestor(william,james), ancestor(william,bill),
+ancestor(william,carol), ancestor(john,james), ancestor(john,bill),
+ancestor(john,carol), ancestor(james,bill), ancestor(james,carol),
+ancestor(sue,bill), ancestor(sue,carol)}
+
+

+ +

+Finally, some subtle detail has to be noted which is quite useful to improve +the readability of the rules. In the case that a certain argument of a +predicate is irrelevant for a certain rule, no dummy variable has to be +inserted, but the _ can be used. For instance, suppose we want +to derive the persons from the parent facts. For this, we can write the +following rules: + +

+person(X) :- parent(X, _).
+person(X) :- parent(_, X).
+
+ +

+Finally, please avoid calling a predicate as shown in this section a +proposition. (It is fine to call them atoms.) +

+ + + +
+

+DLV as a Deductive Database System; Comparison Operators +

+ +

+When you use the CWA in one of your programs, you basically view the +DLV system as a deductive database system, since you +do not ask for what is logically right, but what you can usefully derive +from your facts base. +Following this approach, you can perform queries on the existing data (the +facts base), derive (and "store") new data using queries(=rules), +which again can be used to deduce even more data, and, using the CWA, even +ask queries as to what is not in (or derivable from) your database. +

+ +Consider the following example in SQL in the well know business domain (which +many relational database systems examples use). Emp is a relational table +containing employee information, and dept contains data on departmens of a +company in which the employees work. + +
+SELECT e.name, e.salary, d.location
+FROM   emp e, dept d
+WHERE  e.dept = d.dept_id
+AND    e.salary > 31000;
+
+ +When the relational tables are encoded as a facts base, we can rewrite the +above query into a datalog rule: + +
+emp("Jones",   30000, 35, "Accounting").
+emp("Miller",  38000, 29, "Marketing").
+emp("Koch",  2000000, 24, "IT").
+emp("Nguyen",  35000, 42, "Marketing").
+emp("Gruber",  32000, 39, "IT").
+
+dept("IT",         "Atlanta").
+dept("Marketing",  "New York").
+dept("Accounting", "Los Angeles").
+
+q1(Ename, Esalary, Dlocation) :- emp(Ename, Esalary, _, D), dept(D, Dlocation),
+   Esalary > 31000.
+
+ +

+As you can see, joins are achieved via variable binding (we use the same +variable D both in emp and in dept), selections can for example be achieved +by the comparison operators, and projections (i.e. where unwanted data columns +are excluded from a query result) can be accomplished by using _ or an +unbound variable. +

+ +

+You can use DLV to ask all the queries that are possible in the +core SQL language. Furthermore, (as you will see when the full expressive power +of DLV is unveiled later in this tutorial) you can also encode +many useful queries that cannot be expressed in SQL. +

+ +

+This example used another feature of DLV that has not been +introduced yet: comparison operators. DLV supports the operators +<, >, >=, <=, and = for integers, floating point values, and strings. +This is an extension that is not part of basic datalog, but it is convenient +and also compatible with the philosophy of datalog, as you can think of an +expression X > Y as a predicate +greater_than(X,Y) for which the facts base of all the greater-than +relationships between constant symbols in your program are automatically +generated. +Therefore, we call these comparison operators built-in predicates. +

+ +

+Note that you could also rewrite q1 to use the operator = for the +join. The rule below obtains the same result as the one shown earlier: + +

+q1(Ename, Esalary, Dlocation) :- emp(Ename, Esalary, _, D1),
+                                 dept(D2, Dlocation), D1 = D2,
+                                 Esalary > 31000.
+
+ +

+Download example program. +

+ +

+ + + +
+

+The Railway Crossing Example : True Negation and Negation as Finite Failure +

+ +

+DLV supports two kinds of negation. + +Here, we emphasize the difference between explicitly expressing the falseness +of an atom and having it done by the Complete World Assumption. + +The following program uses the CWA. It has the model +{cross} because train_approaching is assumed to be false (as it +being true is not stated anywhere). +This kind of negation is called +negation as (finite) failure or naf. +

+cross :- not train_approaching.
+
+ +The next program uses so-called true or classical negation. +Since -train_approaching is not known to be true, the following +program has only an empty model. + +
+cross :- -train_approaching.
+
+ +The difference between the two kinds of negation is quite important: +In the first example, we cross the railroad track if we have no information +on any trains approaching, which is quite dangerous, +while in the second example, we only cross if we know for +sure that no train comes. +In particular, the left side of the previous rule will only be true if + +
+-train_approaching.
+
+ +is in the facts base of the program. +

+ +

+True negation is stronger than negation as finite failure. If something is +true via true negation, it is always also true if negated by negation as +finite failure. +For example, the program +

+ +
+cross :- not train_approaching.
+-train_approaching.
+
+ +has the model {cross, -train_approaching}. + +

+Using True Negation also allows to build programs that are contradictory and +have no models. Consider the following example: + +

+cross.
+-cross.
+
+ +Certainly, this program cannot have a model. +This is very different from a program that has an empty model, which would just +mean that the program represents a possible situation but that all of its atoms +are assumed to be false. +

+ + + +
+

+The Broken Arm Example : Disjunctive Datalog and the Stable Model Semantics +

+ +Suppose you have met a friend recently and you know that he +had one of his arms broken, but you don't know which one. +Now you didn't receive a greeting card for your birthday and +wonder if you should be angry on him or if he just cannot +write because of his broken arm. +Finally, you know that he writes with his right hand. +The following DLV program computes the two possible +explanations for the observations you made. + +
+left_arm_broken v right_arm_broken.
+can_write :- left_arm_broken.
+be_angry :- can_write.
+
+ +The first rule is called a disjunctive rule; The v is read as "or" +and the whole rule is read as +"For sure, either the left or the right arm is broken." +As we can see here, a disjunctive rule may (but does not have to) have an +empty body (= lack a body). +It is still called a rule, since it is certainly not a fact. +(It is unknown if the left or the right arm is broken.) +

+ +Being able to process incomplete information (i.e. being unsure if the +left or the right arm is broken) is one of the great strengths of +DLV. + +The resulting models of this query are +{left_arm_broken, can_write, be_angry} and +{right_arm_broken}. +

+ +In fact, the disjunction left_arm_broken v right_arm_broken. +also allows both left_arm_broken and right_arm_broken +to be true at the same time. Still, DLV does not output +the model {left_arm_broken, right_arm_broken, can_write, be_angry} +due to the computing paradigm that it uses to cope with uncertainty, and which +is called the Stable Model Semantics. +Under this semantics, a model is not stable if there is a smaller model which +is a subset of it (which is the case for both stable models shown +above with respect to the "big" model containing left_arm_broken +and right_arm_broken). +While this might seem complicated, it is a very powerful feature of +DLV which is very useful for all kinds of reasoning. +We will come back to this later in this tutorial. +(For the moment, we want to emphasize that this one "big" model which +is not stable would be obviously wrong in this application.) +

+ + +Note that the same uncertainty can also be expressed by the following program: + +

+left_arm_broken :- not right_arm_broken.
+right_arm_broken :- not left_arm_broken.
+can_write :- left_arm_broken.
+be_angry :- can_write.
+
+ +

+This program results in the same pair of models. The method used here is called +Unstratified Negation and is considered less elegant than the first +method. Also, there are certain interesting reasoning problems that +DLV can solve and which can only be expressed with true +disjunction but not with unstratified negation. +

+ +

+Finally, please note that rule bodies may either contain positive (nonnegated) +atoms, atoms negated by true negation, and atoms negated by negation as +failure, while rule heads may only contain positive atoms and true negation, +but no negation as failure. In other words, a rule such as + +

+not a :- b.   % INVALID !!!
+
+ +is not valid! (The % sign in a DLV program +starts a comment which goes to the right to the end of the line.) +

+ + + +
+

+Strong Constraints +

+ +

+DLV also supports integrity constraints +(strong constraints). +A constraint is a rule with an empty head. If its body is true (which is of +course the case exactly if all the literals in the body are true at the same +time), a model is made inconsistent and is removed. + +For example, in the family tree example which was presented earlier, we +can easily write an integrity constraint to assure that the facts base does not +erroneously contain contradicting facts saying that a person is male and +female at the same time. + +

+:- male(X), female(X).
+
+ +This kind of constraints is called strong constraints because there +is also a different kind (weak constraints) supported by +DLV which is not addressed in this tutorial. +This other kind of constraints is very useful to solve optimization problems. +

+ + + +
+

+Graph Coloring: Guess&Check Programming +

+ +

+Graph 3-colorability is a hard (NP-complete) problem. +It is the problem of deciding if there exists a coloring of a map of +countries corresponding to the given graph using no more than three colors in +which no two neighbour countries (nodes connected by an arc) have the same +color. +It is known that every map can be colored given these constraints if four +colors are available. +

+ +
+ + +
+node(minnesota).
+node(wisconsin).
+node(illinois).
+node(iowa).
+node(indiana).
+node(michigan).
+node(ohio).
+
+arc(minnesota, wisconsin).
+arc(illinois, iowa).
+arc(illinois, michigan).
+arc(illinois, wisconsin).
+arc(illinois, indiana).
+arc(indiana, ohio).
+arc(michigan, indiana).
+arc(michigan, ohio).
+arc(michigan, wisconsin).
+arc(minnesota, iowa).
+arc(wisconsin, iowa).
+arc(minnesota, michigan).
+
+
+ +This problem can now be solved with a very simple datalog program, in which +we first guess a coloring by using a disjunctive rule and then check it by +adding a (strong) constraint which deletes all those colorings that do not +satisfy our requirements (that there may be no arc between two nodes of +equal color): + +
+% guess coloring
+col(Country, red) v col(Country, green) v col(Country, blue) :- node(Country).
+
+% check coloring
+:- arc(Country1, Country2), col(Country1, CommonColor), col(Country2, CommonColor).
+
+ +This problem instance has 6 solutions (stable models), therefore, it is +3-colorable. Below, one solution is shown, in which the facts base has again +be removed for better readability: + +
+{col(minnesota,green), col(wisconsin,red), col(illinois,green),
+ col(iowa,blue), col(indiana,red), col(michigan,blue), col(ohio,green)}
+
+ +

+This method (guess&check programming) allows to encode a large number of +complicated problems in an intuitive way. DLV can then +use such an encoding to solve the problems surprisingly efficiently. +

+ +

+Download example program. +

+ +

+As an exercise, you can use DLV to prove that a +map of Germany, Belgium, Luxembourg and France +is not 3-colorable. +

+ + + +
+

+The Fibonacci Example: Built-in Predicates and Integer Arithmetics +

+ +Note that this section introduces some features of DLV +which are not part of standard datalog. +

+ +In the following example, the Fibonacci function is defined, +which is relevant in areas as disparate as Chaos Theory and Botanics. +Its starts with the following values: +1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ... +(Apart from the first two values, each value is defined as the sum of the +previous two.) + +

+true.
+fibonacci(1, 1) :- true.
+fibonacci(1, 2) :- true.
+fibonacci(F, Index) :- +(F1, F2, F),
+                       fibonacci(F1, Index1),
+                       fibonacci(F2, Index2),
+                       #succ(Index1, Index2),
+                       #succ(Index2, Index).
+
+ +This program uses the built-in predicates + (which adds or +subtracts integer numbers) and #succ (the successor function). +Note that for better readability, it is also correct to write +F = F1 + F2 instead of +(F1, F2, F) and +Index2 = Index1 + 1 instead of +#succ(Index1, Index2). +Still, these simple equations always map to the built-in predicates and +may not be extended any further. (It is not allowed to write +A = B + C + D, this has to be split into two parts.) +

+ +

+The second topic that has to be discussed at this point is why the fact +true. was introduced. +The reason for this is the strong separation that is made between EDB and +IDB predicates. Since fibonacci is used on the left-hand side +of a rule, it is in the IDB. IDB predicates cannot be used in facts (because +then they would have to be in the EDB). +Because of that, a fact is introduced and rules are built that are always true +and are therefore equivalent to facts. + +Note that this distinction between IDB and EDB predicates is not necessary +anymore in the most recent versions of DLV. Therefore, +you can now declare fibonacci(1, 1) and +fibonacci(1, 2) simply as facts. +

+ +

+Whenever integer arithmetics are used, the range of possible values has to +be restricted, since DLV requires the space of possible +solutions to be finite. This is done by invoking DLV +with the option -N. (For a full description of +DLV usage, refer to the + +DLV manual.) + +For example, invoking DLV with + +

+dl -N=100 fibonacci.dl
+
+ +results in the model + +
+{true, fibonacci(1,1), fibonacci(1,2), fibonacci(2,3), fibonacci(3,4),
+ fibonacci(5,5), fibonacci(8,6), fibonacci(13,7), fibonacci(21,8),
+ fibonacci(34,9), fibonacci(55,10), fibonacci(89,11)}
+
+ +These are all the Fibonacci numbers not greater than 100. +

+ +

+Download example program. +

+ + + + + +
+Click +here +for some interesting material on Fibonacci numbers. + + + +
+

+The 8-Queens Example: Guess&Check Programming with Integers +

+ +The 8 queens problem asks for a solution in which 8 queens are placed on a +8 x 8 chess board without threatening eachother. A queen threatens another +if it is in the same row, column, or on a diagonal. + +
+% guess horizontal position for each row
+q(X,1) v q(X,2) v q(X,3) v q(X,4) v q(X,5) v q(X,6) v q(X, 7) v q(X,8) :- #int(X), X > 0.
+
+% check
+
+% assert that each column may only contain (at most) one queen
+:- q(X1,Y), q(X2,Y), X1 <> X2.
+
+% assert that no two queens are in a diagonal from top left to bottom right
+:- q(X1,Y1), q(X2,Y2), X2=X1+N, Y2=Y1+N, N > 0.
+
+% assert that no two queens are in a diagonal from top right to bottom left 
+:- q(X1,Y1), q(X2,Y2), X2=X1+N, Y1=Y2+N, N > 0.
+
+ +To run this program with DLV, type the following: + +
+dl -n=1 -N=8 8queens.dl
+
+ +This will return a result like + +
+{q(1,3), q(2,7), q(3,2), q(4,8), q(5,5), q(6,1), q(7,4), q(8,6)}
+
+ +To get all 92 correct solutions, type + +
+dl -N=8 8queens.dl
+
+ +

+Download example program. +

+ + + +
+

+A simple Physics Diagnosis example +

+ +We will now show how to use DLV to do diagnosis. +We choose a physics application domain, a simplified version of ECAL +pre-calibration. + + +
+ +As shown in the picture, a test beam is directed onto a scintillating crystal +whose light emission is measured by an avalanche photodiode (APD). +The measurement is then read with some readout electronics. +Alternatively to the beam reading, the APD can receive a test pulse signal, +which allows to check the correct functioning of the APD independently from +the crystal. +The following program allows to automatically diagnose malfunctioning parts: + +
+ok(testpulse_reading).
+ok(beam_reading).
+
+good(crystal) v bad(crystal).
+good(apd) v bad(apd).
+
+:- good(X), bad(X).
+
+good(crystal) :- ok(beam_reading).
+good(apd) :-     ok(beam_reading).
+bad(apd) v bad(crystal) :- not ok(beam_reading).
+good(apd) :-     ok(testpulse_reading).
+bad(apd)  :- not ok(testpulse_reading).
+
+ +The program starts with two facts expressing our observations. +Here, both the testpulse reading and the beam reading were found to be correct; +below, we will evaluate the program with different observations. +The following two rules tell the system that crystals and APDs are either +working or broken. After this follows a constraint that assures that they +cannot be both at the same time. + +Finally, there are five rules that are a collection of expert knowledge. +They model the knowledge about the domain and show quite clearly why the +test pulse is available as a separate input to the APDs: it allows to +find out if the APD works correctly without having to make any assumptions +about the crystal. +If the reaout of the beam on the other hand were not correct, one could +not be sure if the responsible part is the crystal or the APD. +

+ +Here, the unique result is the model {good(crystal), good(apd)}. +Suppose we exchange the two EDB facts (the first two lines of this program) +to ok(testpulse_reading). then the result changes to +{good(apd), bad(crystal)}. +The whole set of different cases is shown in the following table: +

+ + + + + + + + + + + + + + + + + + + + + + + + + + +
EDBModel(s)
{ok(testpulse_reading). ok(beam_reading).} {good(crystal), good(apd)}
{ok(testpulse_reading).} {good(apd), bad(crystal)}
{ok(beam_reading).} no model
{} {bad(apd), good(crystal)},
+{bad(apd), bad(crystal)}
+

+ +The case that the facts base is {ok(beam_reading).} +is also interesting: +According to our program, if ok(beam_reading) is true, +ok(testpulse_reading) also has to be true. +Therefore, there is no consistent model in this case. +In other words, according to our program, such observations cannot be made. +

+ + + +

+

+A different way to implement the Physics Diagnosis example +

+ +The way to do diagnosis that was presented in the previous section has two +drawbacks: +It requires that more knowledge than necessary has to be coded in the program, +and resulting from this, the program does not really do anything original. +Also, it it hard to extend. +Here, we show a different (better) way to do diagnosis in the same +application domain. +We represent the system as a graph of its units: + +
+connected(beam, crystal).
+connected(crystal, apd).
+connected(testpulse_injector, apd).
+connected(apd, readout).
+
+good_path(X,Y) :- not bad(X), not bad(Y), connected(X, Y).
+good_path(X,Z) :- good_path(X,Y), good_path(Y, Z).
+
+bad(crystal) v bad(apd).
+
+testpulse_readout_ok :- good_path(testpulse_injector, readout).
+beam_readout_ok :- good_path(beam, readout).
+
+ +In this example program, we have left away all the possible observations, +which we implement as constraints, as shown in the following table: +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Observations (Constraints)Model(s) (good_path predicates omitted)
{} +{bad(crystal), testpulse_readout_ok}, +
+{bad(apd)} +
{:- testpulse_readout_ok.} {bad(apd)}
{:- beam_readout_ok.} +{bad(crystal), testpulse_readout_ok}, +
+{bad(apd)} +
{:- beam_readout_ok.
:- testpulse_readout_ok.} 		{bad(apd)}
{:- not testpulse_readout_ok.} +{bad(crystal), testpulse_readout_ok} +
{:- not beam_readout_ok.} no model
{:- not beam_readout_ok.
+:- not testpulse_readout_ok.} 		no model
+

+ +Download example program. +

+ + + +

+

+The Monkey&Banana Example: Planning +

+ +

+The following example shall give an idea of how DLV +can be used to do planning. +

+ +

+Please note that there is a DLV planning frontend +that uses a convenient special-purpose planning language and which is not +described in this tutorial. Instead, we use plain disjunctive datalog for +solving planning problems here. +If you are interested in this frontend, please refer to the +DLV +homepage for further information. +

+ +

+Consider the following classic planning problem. +A monkey is in a room with a chair and a banana which is fixed to the +ceiling. +The monkey cannot reach the banana unless it stands on the chair; it is simply +too high up. The chair is now at a position different from the place +where the banana is hung up, and the monkey itself initially is at again +a different place. +

+ +Since the program is quite long compared to the earlier examples, it will +be explained step by step. + +

+walk(Time) v move_chair(Time) v ascend(Time) v idle(Time) :- #int(Time).
+
+ +At each discrete point in time, the monkey performs one of the following +for actions: it walks, it moves the chair (while doing this, it also moves +through the room), it climbs up the chair, or it does nothing. +#int is again a built-in predicate which is true exactly if its +argument is an integer value. + +
+monkey_motion(T) :- walk(T).
+monkey_motion(T) :- move_chair(T).
+
+stands_on_chair(T2) :- ascend(T), T2 = T + 1.
+:- stands_on_chair(T), ascend(T).
+:- stands_on_chair(T), monkey_motion(T).
+stands_on_chair(T2) :- stands_on_chair(T), T2 = T + 1.
+
+ +After climbing up the chair, it is on it. If is is already on it, it cannot +climb up any further. While on the chair, it cannot walk around. +If it was on the chair earlier, it will be there in the future. + +
+chair_at_place(X, T2) :- chair_at_place(X, T1), T2 = T1 + 1, not move_chair(T1).
+chair_at_place(Pos, T2) :- move_chair(T1), T2 = T1 + 1,
+   monkey_at_place(Pos, T2).
+
+ +If the chair is not moved, it will stay at the same place. +If the monkey moves the chair, it changes its position. + +
+monkey_at_place(monkey_starting_point, T) v
+monkey_at_place(chair_starting_point, T) v
+monkey_at_place(below_banana, T) :- #int(T).
+
+ +The monkey is somewhere in the room. (For simplicity, only three positions are +possible.) + +
+:- monkey_at_place(Pos1, T2), monkey_at_place(Pos2, T1),
+   T2 = T1 + 1, Pos1 != Pos2, not monkey_motion(T1).
+
+:- monkey_at_place(Pos, T2), monkey_at_place(Pos, T1), T2 = T1 + 1,
+   monkey_motion(T1).
+
+:- ascend(T), monkey_at_place(Pos1, T), chair_at_place(Pos2, T), Pos1 != Pos2.
+
+:- move_chair(T), monkey_at_place(Pos1, T), chair_at_place(Pos2, T),
+   Pos1 != Pos2.
+
+ +The monkey cannot change its position without moving. +The monkey cannot stay at the same place if it moves. +It cannot climb up the chair if it is somewhere else. +It cannot move the chair if it is somewhere else. + +
+monkey_at_place(monkey_starting_point, 0) :- true.
+chair_at_place(chair_starting_point, 0) :- true.
+true.
+
+ +Initially, the monkey and the chair are at different positions. + +
+can_reach_banana :- stands_on_chair(T), chair_at_place(below_banana, T).
+eats_banana :- can_reach_banana.
+happy :- eats_banana.
+
+:- not happy.
+
+ +The monkey can only reach the banana if it stands on the chair and the +chair is below the banana. +If it can reach the banana, it will eat it, and this will make it happy. +Our goal is to make the monkey happy. + +
+step(N, walk, Destination) :- walk(N), monkey_at_place(Destination, N2),
+                              N2 = N + 1.
+step(N, move_chair, Destination) :- move_chair(N),
+                                    monkey_at_place(Destination, N2),
+                                    N2 = N + 1.
+step(N, ascend, " ") :- ascend(N).
+
+ + +The step rules collect all the things we can derive from the situation and +build a consistent plan. (There is no step rule for the action idle +since we are not interested in it.) +

+ +This program again uses integer arithmetics; to find a plan, the maximum +integer variable has to be set to at least 3: + +

+dl -N=3 banana.dl
+
+ +This results in the following model (If N is set to a value greater than 3, +DLV will find other plans that make the monkey happy.) + +
+{chair_at_place(chair_starting_point,0),
+monkey_at_place(monkey_starting_point,0),
+monkey_at_place(chair_starting_point,1),
+monkey_at_place(below_banana,2),
+monkey_at_place(below_banana,3),
+walk(0), move_chair(1), ascend(2), idle(3),
+chair_at_place(chair_starting_point,1),
+chair_at_place(below_banana,2),
+chair_at_place(below_banana,3),
+monkey_motion(0), monkey_motion(1),
+step(0,walk,chair_starting_point),
+step(1,move_chair,below_banana),
+step(2,ascend," "),
+stands_on_chair(3), can_reach_banana, eats_banana, happy}
+
+ +Download example program. +

+ + + +

+ + + + + + + diff --git a/biblio/ASP-DLV_tutorial.md b/biblio/ASP-DLV_tutorial.md new file mode 100644 index 0000000..b152ce6 --- /dev/null +++ b/biblio/ASP-DLV_tutorial.md @@ -0,0 +1,923 @@ +## The **`DLV`** Tutorial + +In this tutorial, we give an introduction to *Disjunctive Datalog* +(using some of the extensions of **`DLV`**). The tutorial does not give +a full description of the usage and capabilities of **`DLV`**. For a +more complete account of these, see the [**`DLV`** +homepage](http://www.dlvsystem.com/) and the [**`DLV`** online user +manual](http://www.dlvsystem.com/man/). The examples shown in this +tutorial work with every recent **`DLV`** release. Executables of the +**`DLV`** system for various platforms can be downloaded from the +[**`DLV`** homepage](http://www.dlvsystem.com/). + +The tutorial consists of the following sections, each of them being +built around a guiding example: + +- The First Example : Rules and Facts +- The Second Example : Negation and the Complete World Assumption +- The Family Tree Example : Predicates, Variables, and Recursion +- `DLV` as a Deductive Database System; Comparison Operators +- The Railway Crossing Example : True Negation and Negation as Finite + Failure +- The Broken Arm Example : Disjunctive Datalog and the Stable Model + Semantics +- Strong Constraints +- Graph Coloring: Guess&Check Programming +- The Fibonacci Example: Built-in Predicates and Integer Arithmetics +- The 8-Queens Example: Guess&Check Programming with Integers +- A simple Physics Diagnosis example +- A different way to implement the Physics Diagnosis example +- The Monkey&Banana Example: Planning + +This page is quite long. People who are in a hurry might appreciate the +information that the tutorial is fully on this page, there will be no +branches and no links to further pages. + +**This tutorial is written for computer-literate people with a +background different from computer science, or students new to this +area. It was originally written for physicists at CERN, and some +examples are tailored towards this community.** + +------------------------------------------------------------------------ + +### Introduction + +Datalog is a *declarative* (programming) language. This means that the +programmer does not write a program that solves some problem but instead +specifies what the solution should look like, and a Datalog inference +engine (or *Deductive Database System*) tries to find the the way to +solve the problem and the solution itself. This is done with *rules* and +*facts*. Facts are the input data, and rules can be used to derive more +facts, and hopefully, the solution of the given problem. + +Disjunctive datalog is an extension of datalog in which the logical OR +expression (the disjunction) is allowed to appear in the rules - this is +not allowed in basic datalog. + +**`DLV`** (= datalog with disjunction) is a powerful though freely +available deductive database system. It is based on the declarative +programming language *datalog*, which is known for being a convenient +tool for knowledge representation. With its disjunctive extensions, it +is well suited for all kinds of nonmonotonic reasoning, including +diagnosis and planning. + +Finally, we have to mention to the more advanced reader that **`DLV`** +is relevant to two communities. Firstly, as mentioned, it is a deductive +database engine and can therefore be seen as a way to query data from +databases which is strictly more powerful than for example SQL +(everything that can be done with the core SQL language can also be done +with **`DLV`**, and more), but it is also often described as a system +for answer set programming (ASP). This is a powerful new paradigm from +the area of \"Nonmonotonic Reasoning\" which allows to formulate even +very complicated problems in a straightforward and highly declarative +way. One may call this paradigm even more declarative than classical +logic. Of course, every programming language to be processed by a +computer has to have both fixed syntax (i.e. a grammar that specifies +what programs of this language have to look like, and what combinations +of symbols make a valid program) and semantics (which abstractly +specifies what the computer has to do with the program by declaring how +a program is to be translated into the/a correct result). There is wide +agreement (and also some excitement) that both the syntax and semantics +of the language of **`DLV`** are very simple and intuitive. In fact, we +do not know of any way to make the language even simpler while +preserving its characteristics. + +Both the syntax and semantics of **`DLV`** will be described in this +tutorial. + +------------------------------------------------------------------------ + +### The First Example : Rules and Facts + +Suppose we want to model that every time somebody tells us a joke, we +laugh. Furthermore, somebody now tells us a joke. This could be done in +the following way: + +> joke. +> laugh :- joke. + +The first line is called a fact and expresses that `joke` is true (a +simple word such as `joke` appearing in a rule or fact which has a truth +value is called a *proposition*. A more general name - which we will use +in the following - for the constituents of rules and facts is *atom*.). +The second line is called a rule. It is read as \"if joke is true, laugh +must also be true\". (The sign \":-\" is meant to be an arrow to the +left, the logic programming version of the implication.) + +If the author of such a program decides it appropriate, one can also +interpret some causality into a rule and read this one as \"from joke +follows laugh\". This is pure matter of choice of the human, and +**`DLV`** does not care about it. The left side of a rule is called its +*head*, while the right side is called its *body*. + +A result of a Datalog computation is called a *model*. The meaning of +this is clear: it is a consistent explanation (model) of the world, as +far as the Datalog system can derive it. If a datalog program is +inconsistent, i.e., it is contradictory, there is simply no model (we +will see examples of this later). + +Of course, since in this example `joke` is certainly true (this is given +by the fact), `laugh` is also true. `DLV` now tries to find all those +models of the world that correctly and consistently explain the +observations made (= the program). A model assigns a truth value (either +*true* or *false*) to each atom appearing in the program, and is written +as the set of atoms that are true in a certain model. The model of the +above program is `{joke, laugh}`. When all atoms are false in a model, +we talk about an empty model (written as `{}`). Note that having an +empty model is very different from finding no model. We will see +examples for this later. + +Simple datalog programs like the one above always have exactly one +model. In general, though, **`DLV`** programs may have zero (as +mentioned) or even many models. We will see examples of such programs +later. + +------------------------------------------------------------------------ + +### The Second Example : Negation and the Complete World Assumption + +Next, suppose we are not aware of being told a joke. In this case, the +correct datalog program looks like this: + +> laugh :- joke. + +The program itself does not express that joke is false, but the +so-called *Complete World Assumption (CWA)* does. It is one of the +foundations `DLV` bases its computations on and says that everything +about which nothing is known is assumed to be false. Therefore, the +model for this program is `{}`. (This means that there is a model but it +is empty. It is also possible that for a given program there is no +model.) We will come back to the CWA in more detail later in the section +that discusses `DLV` as a deductive database system. + +Next, we elaborate a bit on this example. First, we want to express that +to be able to understand a joke, one has to hear it and must not be +stupid. To hear it, one must not be deaf and there must be a joke. +Finally, to laugh about the joke, one must understand it. Alternatively, +stupid people might laugh without being told a joke. + +> joke. +> hear_joke :- joke, not deaf. +> understand_joke :- hear_joke, not stupid. +> laugh :- understand_joke. +> laugh :- stupid, not joke. + +In two of the rules, we encounter negated atoms. These are true if the +atoms themselves are false. We also encounter rules that contain more +than one atom in the body. In such a case, a body is true if each of the +literals are true (a literal is a possibly negated atom). For example, + +> hear_joke :- joke, not deaf. + +is read as \"if `joke` is true and `deaf` is false then `hear_joke` must +be true\". + +The model for this program is +`{joke, hear_joke, understand_joke, laugh}`. Again, by virtue of the +CWA, `deaf` and `stupid` are assumed to be false - there are no facts +making these atoms true and no rules which can derive their truth. Now +suppose we remove `joke.` from the program and add `stupid.` instead. +Then, the resulting model would be `{stupid, laugh}`. + +Please note the following things: (i) Those atoms that are not listed as +elements of the models above are *not* automatically rendered false. +Rather, they are unknown. (ii) Suppose the program would look like this: + +> stupid. +> laugh :- stupid, not joke. + +The model of this program is `{stupid, laugh}`. If we now add the fact +`joke.` we get the model `{stupid, joke}`, from which the atom `laugh` +got lost. In other words, you may add more information and lose +information that could be derived before because of that. Due to this +property, the formalism of **`DLV`** is called *nonmonotonic*, just as +mathematical functions which are neither monotonically increasing nor +decreasing are called nonmonotonic. At first sight, this may look like +an ugly property of this formalism, but in fact, it allows to do many +useful things. + +------------------------------------------------------------------------ + +### The Family Tree Example : Predicates, Variables, and Recursion + +So far we have studied simple atoms as the building blocks of our rules. +In fact, atoms may be constructed to hold a number of arguments - they +are then also called *predicates*. + +In the following program, we have two binary predicates, `parent` and +`grandparent`. (They are called binary because they both have two +arguments.) + +We have to map some semantics to the two arguments of the predicates. +Here, the first argument is assumed to be the older person (the parent +or grandparent), while the second argument refers to the younger person +(the child or grandchild). Certainly, we could do it the other way as +well, but then we would have to adjust all the rules that will follow. + +> parent(john, james). +> parent(james, bill). +> grandparent(john, bill) :- parent(john, james), parent(james, bill). + +Of course, the model of this program is +`{parent(john, james), parent(james, bill), grandparent(john,bill)}`. + +With predicates, it is allowed to use variables, which begin with an +upper-case character, differently from the constants of the previous +program that begin with a lower-case letter. The following program has +the same model as the previous example: + +> parent(john, james). +> parent(james, bill). +> grandparent(X, Y) :- parent(X, Z), parent(Z, Y). + +This new grandparent rule which uses variables simply models that every +parent of a parent is a grandparent. + +Note that the facts of a program are often called the *Extensional +Database (EDB)*, while the remaining rules are called the *Intensional +Database (IDB)*. With **`DLV`**, the EDB can be read either from a +relational or object-oriented database, or just simply from files, where +no separation of rules and facts is required. + +We can now extend this example a bit to show how **`DLV`** can be used +to model knowledge as datalog rules and exploit it. First we add a few +more facts to add more people and to express their gender: + +> parent(william, john). +> parent(john, james). +> parent(james, bill). +> parent(sue, bill). +> parent(james, carol). +> parent(sue, carol). +> +> male(john). +> male(james). +> female(sue). +> male(bill). +> female(carol). + +Then we can add more rules that model family relationships. + +> grandparent(X, Y) :- parent(X, Z), parent(Z, Y). +> father(X, Y) :- parent(X, Y), male(X). +> mother(X, Y) :- parent(X, Y), female(X). +> brother(X, Y) :- parent(P, X), parent(P, Y), male(X), X != Y. +> sister(X, Y) :- parent(P, X), parent(P, Y), female(X), X != Y. + +The rules for brother and sister use `X != Y` to require that X and Y +are different (one cannot be his own brother). This is called a built-in +predicate, since it could be written as something like +`not_equal(X, Y)`. **`DLV`** knows quite a few of these built-in +predicates. For this program, **`DLV`** finds the following model (to +simplify readability, the facts already listed above were removed from +the model below; of course, they still belong there): + +> {grandparent(william,james), grandparent(john,bill), grandparent(john,carol), +> father(john,james), father(james,bill), father(james,carol), +> mother(sue,bill), mother(sue,carol), +> brother(bill,carol), sister(carol,bill)} + +Let us now exchange the IDB rules against the following (the EDB facts +remain the same): + +> ancestor(X, Y) :- parent(X, Y). +> ancestor(X, Y) :- parent(X, Z), ancestor(Z, Y). + +These rules are interesting, since they use recursion to implement +transitivity. They express that, to start with, every parent is an +ancestor, and, secondly, that every parent of an ancestor is an +ancestor. Please note that the semantics used ensures that it is +impossible that there be any problems with left-recursion as they occur +in languages as Prolog. In **`DLV`**, the programmer can safely ignore +such considerations. + +The model of this program combined with the six-entries `parent` facts +base above results in the following model (where the `parent` facts were +again removed for readability): + +> {ancestor(william,john), ancestor(william,james), ancestor(william,bill), +> ancestor(william,carol), ancestor(john,james), ancestor(john,bill), +> ancestor(john,carol), ancestor(james,bill), ancestor(james,carol), +> ancestor(sue,bill), ancestor(sue,carol)} + +Finally, some subtle detail has to be noted which is quite useful to +improve the readability of the rules. In the case that a certain +argument of a predicate is irrelevant for a certain rule, no dummy +variable has to be inserted, but the `_` can be used. For instance, +suppose we want to derive the persons from the parent facts. For this, +we can write the following rules: + +> person(X) :- parent(X, _). +> person(X) :- parent(_, X). + +Finally, please *avoid* calling a predicate as shown in this section a +proposition. (It is fine to call them atoms.) + +------------------------------------------------------------------------ + +### `DLV` as a Deductive Database System; Comparison Operators + +When you use the CWA in one of your programs, you basically view the +`DLV` system as a *deductive* database system, since you do not ask for +what is logically right, but what you can usefully derive from your +facts base. Following this approach, you can perform queries on the +existing data (the facts base), derive (and \"store\") new data using +queries(=rules), which again can be used to deduce even more data, and, +using the CWA, even ask queries as to what is *not* in (or derivable +from) your database. + +Consider the following example in SQL in the well know business domain +(which many relational database systems examples use). Emp is a +relational table containing employee information, and dept contains data +on departmens of a company in which the employees work. + +> SELECT e.name, e.salary, d.location +> FROM emp e, dept d +> WHERE e.dept = d.dept_id +> AND e.salary > 31000; + +When the relational tables are encoded as a facts base, we can rewrite +the above query into a datalog rule: + +> emp("Jones", 30000, 35, "Accounting"). +> emp("Miller", 38000, 29, "Marketing"). +> emp("Koch", 2000000, 24, "IT"). +> emp("Nguyen", 35000, 42, "Marketing"). +> emp("Gruber", 32000, 39, "IT"). +> +> dept("IT", "Atlanta"). +> dept("Marketing", "New York"). +> dept("Accounting", "Los Angeles"). +> +> q1(Ename, Esalary, Dlocation) :- emp(Ename, Esalary, _, D), dept(D, Dlocation), +> Esalary > 31000. + +As you can see, joins are achieved via variable binding (we use the same +variable D both in emp and in dept), selections can for example be +achieved by the comparison operators, and projections (i.e. where +unwanted data columns are excluded from a query result) can be +accomplished by using \_ or an unbound variable. + +You can use `DLV` to ask all the queries that are possible in the core +SQL language. Furthermore, (as you will see when the full expressive +power of `DLV` is unveiled later in this tutorial) you can also encode +many useful queries that cannot be expressed in SQL. + +This example used another feature of `DLV` that has not been introduced +yet: comparison operators. `DLV` supports the operators \<, \>, \>=, +\<=, and = for integers, floating point values, and strings. This is an +extension that is not part of basic datalog, but it is convenient and +also compatible with the philosophy of datalog, as you can think of an +expression X \> Y as a predicate `greater_than(X,Y)` for which the facts +base of all the greater-than relationships between constant symbols in +your program are automatically generated. Therefore, we call these +comparison operators *built-in predicates*. + +Note that you could also rewrite `q1` to use the operator = for the +join. The rule below obtains the same result as the one shown earlier: + +> q1(Ename, Esalary, Dlocation) :- emp(Ename, Esalary, _, D1), +> dept(D2, Dlocation), D1 = D2, +> Esalary > 31000. + +[Download example +program.](http://www.dlvsystem.com/tutorial/examples/emp.dl) + +------------------------------------------------------------------------ + +### The Railway Crossing Example : True Negation and Negation as Finite Failure + +**`DLV`** supports *two* kinds of negation. Here, we emphasize the +difference between explicitly expressing the falseness of an atom and +having it done by the *Complete World Assumption*. The following program +uses the CWA. It has the model `{cross}` because train_approaching is +assumed to be false (as it being true is not stated anywhere). This kind +of negation is called *negation as (finite) failure* or *naf*. + +> cross :- not train_approaching. + +The next program uses so-called *true* or *classical negation*. Since +`-train_approaching` is not known to be true, the following program has +only an empty model. + +> cross :- -train_approaching. + +The difference between the two kinds of negation is quite important: In +the first example, we cross the railroad track if we have no information +on any trains approaching, which is quite dangerous, while in the second +example, we only cross if we know for sure that no train comes. In +particular, the left side of the previous rule will only be true if + +> -train_approaching. + +is in the facts base of the program. + +True negation is stronger than negation as finite failure. If something +is true via true negation, it is always also true if negated by negation +as finite failure. For example, the program + +> cross :- not train_approaching. +> -train_approaching. + +has the model `{cross, -train_approaching}`. + +Using True Negation also allows to build programs that are contradictory +and have no models. Consider the following example: + +> cross. +> -cross. + +Certainly, this program cannot have a model. This is very different from +a program that has an empty model, which would just mean that the +program represents a possible situation but that all of its atoms are +assumed to be false. + +------------------------------------------------------------------------ + +### The Broken Arm Example : Disjunctive Datalog and the Stable Model Semantics + +Suppose you have met a friend recently and you know that he had one of +his arms broken, but you don\'t know which one. Now you didn\'t receive +a greeting card for your birthday and wonder if you should be angry on +him or if he just cannot write because of his broken arm. Finally, you +know that he writes with his right hand. The following **DLV** program +computes the two possible explanations for the observations you made. + +> left_arm_broken v right_arm_broken. +> can_write :- left_arm_broken. +> be_angry :- can_write. + +The first rule is called a disjunctive rule; The v is read as \"or\" and +the whole rule is read as \"For sure, either the left or the right arm +is broken.\" As we can see here, a disjunctive rule may (but does not +have to) have an empty body (= lack a body). It is still called a rule, +since it is certainly not a fact. (It is unknown if the left or the +right arm is broken.) + +Being able to process incomplete information (i.e. being unsure if the +left or the right arm is broken) is one of the great strengths of +**`DLV`**. The resulting models of this query are +`{left_arm_broken, can_write, be_angry}` and `{right_arm_broken}`. + +In fact, the disjunction `left_arm_broken v right_arm_broken.` also +allows both `left_arm_broken` and `right_arm_broken` to be true at the +same time. Still, **`DLV`** does not output the model +`{left_arm_broken, right_arm_broken, can_write, be_angry}` due to the +computing paradigm that it uses to cope with uncertainty, and which is +called the *Stable Model Semantics*. Under this semantics, a model is +not stable if there is a smaller model which is a subset of it (which is +the case for both stable models shown above with respect to the \"big\" +model containing `left_arm_broken` and `right_arm_broken`). While this +might seem complicated, it is a very powerful feature of **`DLV`** which +is very useful for all kinds of reasoning. We will come back to this +later in this tutorial. (For the moment, we want to emphasize that this +one \"big\" model which is not stable would be obviously wrong in this +application.) + +Note that the same uncertainty can also be expressed by the following +program: + +> left_arm_broken :- not right_arm_broken. +> right_arm_broken :- not left_arm_broken. +> can_write :- left_arm_broken. +> be_angry :- can_write. + +This program results in the same pair of models. The method used here is +called *Unstratified Negation* and is considered less elegant than the +first method. Also, there are certain interesting reasoning problems +that **`DLV`** can solve and which can only be expressed with true +disjunction but not with unstratified negation. + +Finally, please note that rule bodies may either contain positive +(nonnegated) atoms, atoms negated by true negation, and atoms negated by +negation as failure, while rule heads may only contain positive atoms +and true negation, but no negation as failure. In other words, a rule +such as + +> not a :- b. % INVALID !!! + +is *not* valid! (The % sign in a **`DLV`** program starts a comment +which goes to the right to the end of the line.) + +------------------------------------------------------------------------ + +### Strong Constraints + +**`DLV`** also supports integrity constraints (strong constraints). A +constraint is a rule with an empty head. If its body is true (which is +of course the case exactly if all the literals in the body are true at +the same time), a model is made inconsistent and is removed. For +example, in the family tree example which was presented earlier, we can +easily write an integrity constraint to assure that the facts base does +not erroneously contain contradicting facts saying that a person is male +and female at the same time. + +> :- male(X), female(X). + +This kind of constraints is called *strong constraints* because there is +also a different kind (*weak constraints*) supported by **`DLV`** which +is not addressed in this tutorial. This other kind of constraints is +very useful to solve optimization problems. + +------------------------------------------------------------------------ + +### Graph Coloring: Guess&Check Programming + +Graph 3-colorability is a hard (NP-complete) problem. It is the problem +of deciding if there exists a coloring of a map of countries +corresponding to the given graph using no more than three colors in +which no two neighbour countries (nodes connected by an arc) have the +same color. It is known that every map can be colored given these +constraints if four colors are available. + ++-----------------------------------+-----------------------------------+ +| ![](http://www.dlv | > node(minnesota). | +| system.com/tutorial/midwest2.gif) | > node(wisconsin). | +| | > node(illinois). | +| | > node(iowa). | +| | > node(indiana). | +| | > node(michigan). | +| | > node(ohio). | +| | > | +| | > arc(minnesota, wisconsin). | +| | > arc(illinois, iowa). | +| | > arc(illinois, michigan). | +| | > arc(illinois, wisconsin). | +| | > arc(illinois, indiana). | +| | > arc(indiana, ohio). | +| | > arc(michigan, indiana). | +| | > arc(michigan, ohio). | +| | > arc(michigan, wisconsin). | +| | > arc(minnesota, iowa). | +| | > arc(wisconsin, iowa). | +| | > arc(minnesota, michigan). | ++-----------------------------------+-----------------------------------+ + +This problem can now be solved with a very simple datalog program, in +which we first guess a coloring by using a disjunctive rule and then +check it by adding a (strong) constraint which deletes all those +colorings that do not satisfy our requirements (that there may be no arc +between two nodes of equal color): + +> % guess coloring +> col(Country, red) v col(Country, green) v col(Country, blue) :- node(Country). +> +> % check coloring +> :- arc(Country1, Country2), col(Country1, CommonColor), col(Country2, CommonColor). + +This problem instance has 6 solutions (stable models), therefore, it is +3-colorable. Below, one solution is shown, in which the facts base has +again be removed for better readability: + +> {col(minnesota,green), col(wisconsin,red), col(illinois,green), +> col(iowa,blue), col(indiana,red), col(michigan,blue), col(ohio,green)} + +This method (guess&check programming) allows to encode a large number of +complicated problems in an intuitive way. **`DLV`** can then use such an +encoding to solve the problems surprisingly efficiently. + +[Download example +program.](http://www.dlvsystem.com/tutorial/examples/3col.dl) + +As an exercise, you can use **`DLV`** to prove that a [map of Germany, +Belgium, Luxembourg and +France](http://www.dlvsystem.com/tutorial/benelux.jpg) is not +3-colorable. + +------------------------------------------------------------------------ + +### The Fibonacci Example: Built-in Predicates and Integer Arithmetics + +Note that this section introduces some features of **`DLV`** which are +not part of standard datalog. + +In the following example, the Fibonacci function is defined, which is +relevant in areas as disparate as Chaos Theory and Botanics. Its starts +with the following values: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, +233, 377, \... (Apart from the first two values, each value is defined +as the sum of the previous two.) + +> true. +> fibonacci(1, 1) :- true. +> fibonacci(1, 2) :- true. +> fibonacci(F, Index) :- +(F1, F2, F), +> fibonacci(F1, Index1), +> fibonacci(F2, Index2), +> #succ(Index1, Index2), +> #succ(Index2, Index). + +This program uses the built-in predicates `+` (which adds or subtracts +integer numbers) and `#succ` (the successor function). Note that for +better readability, it is also correct to write `F = F1 + F2` instead of +`+(F1, F2, F)` and `Index2 = Index1 + 1` instead of +`#succ(Index1, Index2)`. Still, these simple equations always map to the +built-in predicates and may not be extended any further. (It is not +allowed to write `A = B + C + D`, this has to be split into two parts.) + +The second topic that has to be discussed at this point is why the fact +`true.` was introduced. The reason for this is the strong separation +that is made between EDB and IDB predicates. Since `fibonacci` is used +on the left-hand side of a rule, it is in the IDB. IDB predicates cannot +be used in facts (because then they would have to be in the EDB). +Because of that, a fact is introduced and rules are built that are +always true and are therefore equivalent to facts. Note that this +distinction between IDB and EDB predicates is not necessary anymore in +the most recent versions of **`DLV`**. Therefore, you can now declare +`fibonacci(1, 1)` and `fibonacci(1, 2)` simply as facts. + +Whenever integer arithmetics are used, the range of possible values has +to be restricted, since **`DLV`** requires the space of possible +solutions to be finite. This is done by invoking **`DLV`** with the +option `-N`. (For a full description of **`DLV`** usage, refer to the +[**`DLV`** manual](http://www.dlvsystem.com/man/).) For example, +invoking **`DLV`** with + +> dl -N=100 fibonacci.dl + +results in the model + +> {true, fibonacci(1,1), fibonacci(1,2), fibonacci(2,3), fibonacci(3,4), +> fibonacci(5,5), fibonacci(8,6), fibonacci(13,7), fibonacci(21,8), +> fibonacci(34,9), fibonacci(55,10), fibonacci(89,11)} + +These are all the Fibonacci numbers not greater than 100. + +[Download example +program.](http://www.dlvsystem.com/tutorial/examples/fibonacci.dl) + +![](http://www.dlvsystem.com/tutorial/fibSpiralANIM.gif) +![](http://www.dlvsystem.com/tutorial/fibspiral2.GIF) +![](http://www.dlvsystem.com/tutorial/shell.gif){width="84" +height="120"}\ +Click +[here](http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html) +for some interesting material on Fibonacci numbers. + +------------------------------------------------------------------------ + +### The 8-Queens Example: Guess&Check Programming with Integers + +The 8 queens problem asks for a solution in which 8 queens are placed on +a 8 x 8 chess board without threatening eachother. A queen threatens +another if it is in the same row, column, or on a diagonal. + +> % guess horizontal position for each row +> q(X,1) v q(X,2) v q(X,3) v q(X,4) v q(X,5) v q(X,6) v q(X, 7) v q(X,8) :- #int(X), X > 0. +> +> % check +> +> % assert that each column may only contain (at most) one queen +> :- q(X1,Y), q(X2,Y), X1 <> X2. +> +> % assert that no two queens are in a diagonal from top left to bottom right +> :- q(X1,Y1), q(X2,Y2), X2=X1+N, Y2=Y1+N, N > 0. +> +> % assert that no two queens are in a diagonal from top right to bottom left +> :- q(X1,Y1), q(X2,Y2), X2=X1+N, Y1=Y2+N, N > 0. + +To run this program with **`DLV`**, type the following: + +> dl -n=1 -N=8 8queens.dl + +This will return a result like + +> {q(1,3), q(2,7), q(3,2), q(4,8), q(5,5), q(6,1), q(7,4), q(8,6)} + +To get all 92 correct solutions, type + +> dl -N=8 8queens.dl + +[Download example +program.](http://www.dlvsystem.com/tutorial/examples/8queens.dl) + +------------------------------------------------------------------------ + +### A simple Physics Diagnosis example + +We will now show how to use **`DLV`** to do diagnosis. We choose a +physics application domain, a simplified version of ECAL +pre-calibration. ![](http://www.dlvsystem.com/tutorial/h4.gif)\ +As shown in the picture, a test beam is directed onto a scintillating +crystal whose light emission is measured by an avalanche photodiode +(APD). The measurement is then read with some readout electronics. +Alternatively to the beam reading, the APD can receive a test pulse +signal, which allows to check the correct functioning of the APD +independently from the crystal. The following program allows to +automatically diagnose malfunctioning parts: + +> ok(testpulse_reading). +> ok(beam_reading). +> +> good(crystal) v bad(crystal). +> good(apd) v bad(apd). +> +> :- good(X), bad(X). +> +> good(crystal) :- ok(beam_reading). +> good(apd) :- ok(beam_reading). +> bad(apd) v bad(crystal) :- not ok(beam_reading). +> good(apd) :- ok(testpulse_reading). +> bad(apd) :- not ok(testpulse_reading). + +The program starts with two facts expressing our observations. Here, +both the testpulse reading and the beam reading were found to be +correct; below, we will evaluate the program with different +observations. The following two rules tell the system that crystals and +APDs are either working or broken. After this follows a constraint that +assures that they cannot be both at the same time. Finally, there are +five rules that are a collection of expert knowledge. They model the +knowledge about the domain and show quite clearly why the test pulse is +available as a separate input to the APDs: it allows to find out if the +APD works correctly without having to make any assumptions about the +crystal. If the reaout of the beam on the other hand were not correct, +one could not be sure if the responsible part is the crystal or the APD. + +Here, the unique result is the model `{good(crystal), good(apd)}`. +Suppose we exchange the two EDB facts (the first two lines of this +program) to `ok(testpulse_reading).` then the result changes to +`{good(apd), bad(crystal)}`. The whole set of different cases is shown +in the following table: + + ---------------------------------------------- --------------------------------------------------------- + **EDB** **Model(s)** + `{ok(testpulse_reading). ok(beam_reading).}` `{good(crystal), good(apd)}` + `{ok(testpulse_reading).}` `{good(apd), bad(crystal)}` + `{ok(beam_reading).}` no model + `{}` `{bad(apd), good(crystal)}, {bad(apd), bad(crystal)} ` + ---------------------------------------------- --------------------------------------------------------- + +The case that the facts base is `{ok(beam_reading).}` is also +interesting: According to our program, if `ok(beam_reading)` is true, +`ok(testpulse_reading)` also has to be true. Therefore, there is no +consistent model in this case. In other words, according to our program, +such observations cannot be made. + +------------------------------------------------------------------------ + +### A different way to implement the Physics Diagnosis example + +The way to do diagnosis that was presented in the previous section has +two drawbacks: It requires that more knowledge than necessary has to be +coded in the program, and resulting from this, the program does not +really do anything original. Also, it it hard to extend. Here, we show a +different (better) way to do diagnosis in the same application domain. +We represent the system as a graph of its units: + +> connected(beam, crystal). +> connected(crystal, apd). +> connected(testpulse_injector, apd). +> connected(apd, readout). +> +> good_path(X,Y) :- not bad(X), not bad(Y), connected(X, Y). +> good_path(X,Z) :- good_path(X,Y), good_path(Y, Z). +> +> bad(crystal) v bad(apd). +> +> testpulse_readout_ok :- good_path(testpulse_injector, readout). +> beam_readout_ok :- good_path(beam, readout). + +In this example program, we have left away all the possible +observations, which we implement as constraints, as shown in the +following table: + + ----------------------------------------------------------- ------------------------------------------------------ + **Observations (Constraints)** **Model(s) (good_path predicates omitted)** + `{}` ` {bad(crystal), testpulse_readout_ok}, {bad(apd)}` + `{:- testpulse_readout_ok.}` `{bad(apd)}` + `{:- beam_readout_ok.}` ` {bad(crystal), testpulse_readout_ok}, {bad(apd)}` + `{:- beam_readout_ok. :- testpulse_readout_ok.}` `{bad(apd)}` + `{:- not testpulse_readout_ok.}` ` {bad(crystal), testpulse_readout_ok}` + `{:- not beam_readout_ok.}` no model + `{:- not beam_readout_ok. :- not testpulse_readout_ok.}` no model + ----------------------------------------------------------- ------------------------------------------------------ + +[Download example +program.](http://www.dlvsystem.com/tutorial/examples/diagnosis.dl) + +------------------------------------------------------------------------ + +### The Monkey&Banana Example: Planning + +The following example shall give an idea of how **`DLV`** can be used to +do planning. + +Please note that there is a `DLV` planning frontend that uses a +convenient special-purpose planning language and which is not described +in this tutorial. Instead, we use plain disjunctive datalog for solving +planning problems here. If you are interested in this frontend, please +refer to the [**`DLV`** homepage](http://www.dlvsystem.com) for further +information. + +Consider the following classic planning problem. A monkey is in a room +with a chair and a banana which is fixed to the ceiling. The monkey +cannot reach the banana unless it stands on the chair; it is simply too +high up. The chair is now at a position different from the place where +the banana is hung up, and the monkey itself initially is at again a +different place. + +Since the program is quite long compared to the earlier examples, it +will be explained step by step. + +> walk(Time) v move_chair(Time) v ascend(Time) v idle(Time) :- #int(Time). + +At each discrete point in time, the monkey performs one of the following +for actions: it walks, it moves the chair (while doing this, it also +moves through the room), it climbs up the chair, or it does nothing. +#int is again a built-in predicate which is true exactly if its argument +is an integer value. + +> monkey_motion(T) :- walk(T). +> monkey_motion(T) :- move_chair(T). +> +> stands_on_chair(T2) :- ascend(T), T2 = T + 1. +> :- stands_on_chair(T), ascend(T). +> :- stands_on_chair(T), monkey_motion(T). +> stands_on_chair(T2) :- stands_on_chair(T), T2 = T + 1. + +After climbing up the chair, it is on it. If is is already on it, it +cannot climb up any further. While on the chair, it cannot walk around. +If it was on the chair earlier, it will be there in the future. + +> chair_at_place(X, T2) :- chair_at_place(X, T1), T2 = T1 + 1, not move_chair(T1). +> chair_at_place(Pos, T2) :- move_chair(T1), T2 = T1 + 1, +> monkey_at_place(Pos, T2). + +If the chair is not moved, it will stay at the same place. If the monkey +moves the chair, it changes its position. + +> monkey_at_place(monkey_starting_point, T) v +> monkey_at_place(chair_starting_point, T) v +> monkey_at_place(below_banana, T) :- #int(T). + +The monkey is somewhere in the room. (For simplicity, only three +positions are possible.) + +> :- monkey_at_place(Pos1, T2), monkey_at_place(Pos2, T1), +> T2 = T1 + 1, Pos1 != Pos2, not monkey_motion(T1). +> +> :- monkey_at_place(Pos, T2), monkey_at_place(Pos, T1), T2 = T1 + 1, +> monkey_motion(T1). +> +> :- ascend(T), monkey_at_place(Pos1, T), chair_at_place(Pos2, T), Pos1 != Pos2. +> +> :- move_chair(T), monkey_at_place(Pos1, T), chair_at_place(Pos2, T), +> Pos1 != Pos2. + +The monkey cannot change its position without moving. The monkey cannot +stay at the same place if it moves. It cannot climb up the chair if it +is somewhere else. It cannot move the chair if it is somewhere else. + +> monkey_at_place(monkey_starting_point, 0) :- true. +> chair_at_place(chair_starting_point, 0) :- true. +> true. + +Initially, the monkey and the chair are at different positions. + +> can_reach_banana :- stands_on_chair(T), chair_at_place(below_banana, T). +> eats_banana :- can_reach_banana. +> happy :- eats_banana. +> +> :- not happy. + +The monkey can only reach the banana if it stands on the chair and the +chair is below the banana. If it can reach the banana, it will eat it, +and this will make it happy. Our goal is to make the monkey happy. + +> step(N, walk, Destination) :- walk(N), monkey_at_place(Destination, N2), +> N2 = N + 1. +> step(N, move_chair, Destination) :- move_chair(N), +> monkey_at_place(Destination, N2), +> N2 = N + 1. +> step(N, ascend, " ") :- ascend(N). + +The step rules collect all the things we can derive from the situation +and build a consistent plan. (There is no step rule for the action idle +since we are not interested in it.) + +This program again uses integer arithmetics; to find a plan, the maximum +integer variable has to be set to at least 3: + +> dl -N=3 banana.dl + +This results in the following model (If N is set to a value greater than +3, **`DLV`** will find other plans that make the monkey happy.) + +> {chair_at_place(chair_starting_point,0), +> monkey_at_place(monkey_starting_point,0), +> monkey_at_place(chair_starting_point,1), +> monkey_at_place(below_banana,2), +> monkey_at_place(below_banana,3), +> walk(0), move_chair(1), ascend(2), idle(3), +> chair_at_place(chair_starting_point,1), +> chair_at_place(below_banana,2), +> chair_at_place(below_banana,3), +> monkey_motion(0), monkey_motion(1), +> step(0,walk,chair_starting_point), +> step(1,move_chair,below_banana), +> step(2,ascend," "), +> stands_on_chair(3), can_reach_banana, eats_banana, happy} + +[Download example 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b/code/.vscode/launch.json @@ -0,0 +1,17 @@ +{ + // Use IntelliSense to learn about possible attributes. + // Hover to view descriptions of existing attributes. + // For more information, visit: https://go.microsoft.com/fwlink/?linkid=830387 + "version": "0.2.0", + "configurations": [ + { + "type": "julia", + "request": "launch", + "name": "Run active Julia file", + "program": "${file}", + "stopOnEntry": false, + "cwd": "${workspaceFolder}", + "juliaEnv": "${command:activeJuliaEnvironment}" + } + ] +} \ No newline at end of file diff --git a/code/asp/drafts/alarm.lp b/code/asp/drafts/alarm.lp new file mode 100644 index 0000000..ebb5f05 --- /dev/null +++ b/code/asp/drafts/alarm.lp @@ -0,0 +1,5 @@ +burglary;-burglary. % 0.1:: +earthquake;-earthquake. % 0.01:: + +alarm:- burglary. +alarm;-alarm :- earthquake. % 0.8:: \ No newline at end of file diff --git a/code/asp/drafts/assp_pag43.lp b/code/asp/drafts/assp_pag43.lp new file mode 100644 index 0000000..3267d55 --- /dev/null +++ b/code/asp/drafts/assp_pag43.lp @@ -0,0 +1,2 @@ +q. +p :- q, not r. \ No newline at end of file diff --git a/code/asp/drafts/assp_pag46.lp b/code/asp/drafts/assp_pag46.lp new file mode 100644 index 0000000..1bc10fe --- /dev/null +++ b/code/asp/drafts/assp_pag46.lp @@ -0,0 +1 @@ +p :- -p. \ No newline at end of file diff --git a/code/asp/drafts/bachelor.lp b/code/asp/drafts/bachelor.lp new file mode 100644 index 0000000..30f593e --- /dev/null +++ b/code/asp/drafts/bachelor.lp @@ -0,0 +1,9 @@ +person(joey). + +married(X); -married(X) :- person(X). + +male(X); female(X) :- person(X). + +bachelor(X) :- male(X), not married(X). + +#show bachelor/1. \ No newline at end of file diff --git a/code/asp/drafts/bachelor.pl b/code/asp/drafts/bachelor.pl new file mode 100644 index 0000000..7a3fd9d --- /dev/null +++ b/code/asp/drafts/bachelor.pl @@ -0,0 +1,6 @@ +person(joey). +% male(X); female(X) :- person(X). +male(X) :- person(X). +female(X) :- person(X). + +bachelor(X) :- male(X), \+ married(X). \ No newline at end of file diff --git a/code/asp/drafts/barber.lp b/code/asp/drafts/barber.lp new file mode 100644 index 0000000..2682de2 --- /dev/null +++ b/code/asp/drafts/barber.lp @@ -0,0 +1,8 @@ +shaves(X, Y) :- barber(X), villager(Y), not shaves(Y, Y). + +villager(a). +barber(b). + +villager(b) ; -villager(b). + +query :- not shaves(b, b). \ No newline at end of file diff --git a/code/asp/drafts/choices_01.lp b/code/asp/drafts/choices_01.lp new file mode 100644 index 0000000..189c5f3 --- /dev/null +++ b/code/asp/drafts/choices_01.lp @@ -0,0 +1,11 @@ +{ a ; b }. + +% Answer: 1 +% +% Answer: 2 +% b +% Answer: 3 +% a +% Answer: 4 +% a b +% SATISFIABLE \ No newline at end of file diff --git a/code/asp/drafts/choices_02.lp b/code/asp/drafts/choices_02.lp new file mode 100644 index 0000000..c742984 --- /dev/null +++ b/code/asp/drafts/choices_02.lp @@ -0,0 +1,9 @@ +a ; b. + +% Answer: 1 +% b +% Answer: 2 +% a +% SATISFIABLE +% +% In Answer 2 we (also) have not b. \ No newline at end of file diff --git a/code/asp/drafts/choices_03.lp b/code/asp/drafts/choices_03.lp new file mode 100644 index 0000000..616525b --- /dev/null +++ b/code/asp/drafts/choices_03.lp @@ -0,0 +1,10 @@ +{ a ; b }. +:- a, not b. + +% Answer: 1 +% +% Answer: 2 +% b +% Answer: 3 +% a b +% SATISFIABLE \ No newline at end of file diff --git a/code/asp/drafts/choices_04.lp b/code/asp/drafts/choices_04.lp new file mode 100644 index 0000000..44e5e93 --- /dev/null +++ b/code/asp/drafts/choices_04.lp @@ -0,0 +1,5 @@ +a ; b. +:- a, not b. +% Answer: 1 +% b +% SATISFIABLE \ No newline at end of file diff --git a/code/asp/drafts/choices_05.lp b/code/asp/drafts/choices_05.lp new file mode 100644 index 0000000..f991810 --- /dev/null +++ b/code/asp/drafts/choices_05.lp @@ -0,0 +1,8 @@ +{ a ; -a }. +:- a, not -a. + +% Answer: 1 +% +% Answer: 2 +% -a +% SATISFIABLE diff --git a/code/asp/drafts/choices_06.lp b/code/asp/drafts/choices_06.lp new file mode 100644 index 0000000..deafa47 --- /dev/null +++ b/code/asp/drafts/choices_06.lp @@ -0,0 +1,5 @@ +a::0.7. +b ; c :- a. +% Answer: 1 +% -a +% SATISFIABLE \ No newline at end of file diff --git a/code/asp/drafts/color.lp b/code/asp/drafts/color.lp new file mode 100644 index 0000000..625b4db --- /dev/null +++ b/code/asp/drafts/color.lp @@ -0,0 +1,15 @@ +node(1..6). + +edge(1,2). edge(2,4). edge(3,1). +edge(4,1). edge(5,3). edge(6,2). +edge(1,3). edge(2,5). edge(3,4). +edge(4,2). edge(5,4). edge(6,3). +edge(1,4). edge(2,6). edge(3,5). +edge(5,6). edge(6,5). + +col(r). col(b). col(g). + +1 { color(X,C) : col(C) } 1 :- node(X). +:- edge(X,Y), color(X,C), color(Y,C). + +#show color/2. \ No newline at end of file diff --git a/code/asp/drafts/credal_01.lp b/code/asp/drafts/credal_01.lp new file mode 100644 index 0000000..5cafa70 --- /dev/null +++ b/code/asp/drafts/credal_01.lp @@ -0,0 +1,4 @@ +a ; -a. + +s :- not w, not a. +w :- not s. \ No newline at end of file diff --git a/code/asp/drafts/cyclic.lp b/code/asp/drafts/cyclic.lp new file mode 100644 index 0000000..20cb323 --- /dev/null +++ b/code/asp/drafts/cyclic.lp @@ -0,0 +1,9 @@ +% p :- not p. % UNSAT +% not p :- p. % SAT: {} +% p :- -p. % SAT: {} +% -p :- p. % SAT: {} +% not p :- -p. % SAT: {} +% -p :- not p. % SAT: -p +a :- b. +b :- a. +a :- not a. \ No newline at end of file diff --git a/code/asp/drafts/cyclic.pl b/code/asp/drafts/cyclic.pl new file mode 100644 index 0000000..20c8813 --- /dev/null +++ b/code/asp/drafts/cyclic.pl @@ -0,0 +1,14 @@ +a :- b. +a :- \+ a. +b :- a. +% > swipl cyclic.pl +% ERROR: /home/fc/sci/projetos/plp/code/asp/cyclic.pl:2: +% ERROR: Stack limit (1.0Gb) exceeded +% ERROR: Stack sizes: local: 1.0Gb, global: 27Kb, trail: 1Kb +% ERROR: Stack depth: 7,455,777, last-call: 0%, Choice points: 7,455,758 +% ERROR: Probable infinite recursion (cycle): +% ERROR: [7,455,777] user:p +% ERROR: [7,455,776] user:p +% Warning: /home/fc/sci/projetos/plp/code/asp/cyclic.pl:2: +% Warning: Goal (directive) failed: user:p +% \ No newline at end of file diff --git a/code/asp/drafts/disj.lp b/code/asp/drafts/disj.lp new file mode 100644 index 0000000..6d62252 --- /dev/null +++ b/code/asp/drafts/disj.lp @@ -0,0 +1,3 @@ +% prob(a) = 0.3 +a ; -a. +b ; c :- a. \ No newline at end of file diff --git a/code/asp/drafts/double_negation.lp b/code/asp/drafts/double_negation.lp new file mode 100644 index 0000000..850c811 --- /dev/null +++ b/code/asp/drafts/double_negation.lp @@ -0,0 +1,42 @@ +%not not a. +%% UNSATISFIABLE +%%% ie no models. + +% a. +%% Answer: 1 +%% a +%% SATISFIABLE +%%% ie there is (only) one (stable) model: {a} + +% -a. +%% Answer: 1 +%% -a +%% SATISFIABLE + +% --a. +%% *** ERROR: (clingo): parsing failed +%%% WTF? + +% not a. +%% Answer: 1 +%% +%% SATISFIABLE +%%% ie there is (only) one (stable) model: {} +%%% +%%% this program states that there is no information. In particular, there is no information about a. +%%% Therefore there are no provable atoms. Hence the empty set is a stable model. + +% not -a. +%% Answer: 1 +%% +%% SATISFIABLE + +% b. +% a;c. +% not a :- b. +%% Answer: 1 +%% b -a +%% SATISFIABLE + +a. +b :- not a. \ No newline at end of file diff --git a/code/asp/drafts/ex03.lp b/code/asp/drafts/ex03.lp new file mode 100644 index 0000000..0300b60 --- /dev/null +++ b/code/asp/drafts/ex03.lp @@ -0,0 +1,3 @@ +p(a) :- not p(b). +p(b) :- not p(a). +:- p(b). \ No newline at end of file diff --git a/code/asp/drafts/gcE01.lp b/code/asp/drafts/gcE01.lp new file mode 100644 index 0000000..6b49a6e --- /dev/null +++ b/code/asp/drafts/gcE01.lp @@ -0,0 +1,9 @@ +col(r ; g ; b). + +% var C is local in this rule. +% More specificlly, it is bound to the (lhs) cardinality contraint. +% Also, it varies over all instantiations of col(C). +1 { color(X, C) : col(C) } 1 :- node(X). +:- edge(X, Y), color(X, C), color(Y, C). + +#show color/2. \ No newline at end of file diff --git a/code/asp/drafts/gcI01.lp b/code/asp/drafts/gcI01.lp new file mode 100644 index 0000000..05a2c5a --- /dev/null +++ b/code/asp/drafts/gcI01.lp @@ -0,0 +1,8 @@ +node(1 .. 6). + +edge(1 ,2). edge(2 ,4). edge(3 ,1). +edge(4 ,1). edge(5 ,3). edge(6 ,2). +edge(1 ,3). edge(2 ,5). edge(3 ,4). +edge(4 ,2). edge(5 ,4). +edge(6 ,3). edge(1 ,4). edge(2 ,6). +edge(3 ,5). edge(5 ,6). edge(6 ,5). \ No newline at end of file diff --git a/code/asp/drafts/hanoiE.lp b/code/asp/drafts/hanoiE.lp new file mode 100644 index 0000000..453c7dd --- /dev/null +++ b/code/asp/drafts/hanoiE.lp @@ -0,0 +1,21 @@ +% Generating +1 { move(D, P, T) : disk(D), peg(P) } 1 :- T = 1 .. moves. + +% Defining +move(D, T) :- move(D, _, T). % when target peg is irrelevant +on(D, P, 0) :- init_on(D, P). % init +on(D, P, T) :- move(D, P, T). % effect: move +on(D, P, T + 1) :- on(D, P, T), not move(D, T + 1), T < moves. % inertia +blocked(D - 1, P, T + 1) :- on(D, P, T), T < moves. % D on P, T => P blocked to D - 1 +blocked(D - 1, P, T) :- blocked(D, P, T), disk(D). % P blocked to D => P blocked D - 1 + +% Testing +:- move(D, P, T), blocked(D - 1, P, T). +:- move(D, T), on(D, P, T - 1), blocked(D, P, T). +:- not 1 { on(D, P, T) } 1, disk(D), T = 1 .. moves. + +:- goal_on(D, P), not on(D, P, moves). + +% Displaying +% #hide. +#show move/3. \ No newline at end of file diff --git a/code/asp/drafts/hanoiI.lp b/code/asp/drafts/hanoiI.lp new file mode 100644 index 0000000..ace48dc --- /dev/null +++ b/code/asp/drafts/hanoiI.lp @@ -0,0 +1,11 @@ +#const moves = 15. + +peg(a ; b ; c). +disk(1..4). +init_on(1..4, a). +goal_on(1..4, c). + +%* + Implicit relative locations on a peg. + In a peg, disk are placed top-down from the largest to the smallest. +*% diff --git a/code/asp/drafts/heads_tails.lp b/code/asp/drafts/heads_tails.lp new file mode 100644 index 0000000..b2e30b1 --- /dev/null +++ b/code/asp/drafts/heads_tails.lp @@ -0,0 +1,4 @@ +heads; -heads. + +tails :- -heads. +-tails :- heads. \ No newline at end of file diff --git a/code/asp/drafts/monty_hall.lp b/code/asp/drafts/monty_hall.lp new file mode 100644 index 0000000..a1c9f74 --- /dev/null +++ b/code/asp/drafts/monty_hall.lp @@ -0,0 +1,20 @@ +prize(1, (1,3) ) ; prize(2, (1,3)) ; prize(3, (1, 3)). + +selected(1). + +open_door(2, (1,2)) ; open_door(3, (1,2)) :- prize(1, _). +open_door(2, (1,1)) :- prize(3, _). +open_door(3, (1,1)) :- prize(2, _). + +win_keep :- + prize(1, _), + open_door(_, _). +win_switch :- + prize(2, _), + open_door(3, _). +win_switch :- + prize(3, _), + open_door(2, _). + +% #show win_keep/1. +% #show win_switch/1. \ No newline at end of file diff --git a/code/asp/drafts/no_model_cyclic.lp b/code/asp/drafts/no_model_cyclic.lp new file mode 100644 index 0000000..729189e --- /dev/null +++ b/code/asp/drafts/no_model_cyclic.lp @@ -0,0 +1 @@ +a :- not a. \ No newline at end of file diff --git a/code/asp/drafts/no_model_wa.lp b/code/asp/drafts/no_model_wa.lp new file mode 100644 index 0000000..4508a01 --- /dev/null +++ b/code/asp/drafts/no_model_wa.lp @@ -0,0 +1,8 @@ +a. +b. +-ok :- a, b. +ok :- not -ok. +% +% WHAT ABOUT +% a :- not a. +% diff --git a/code/asp/drafts/nonstrat.lp b/code/asp/drafts/nonstrat.lp new file mode 100644 index 0000000..a13ff51 --- /dev/null +++ b/code/asp/drafts/nonstrat.lp @@ -0,0 +1,7 @@ +-a ; a. % c1 +b :- not c, not a. % c2 +c :- not b. % c3 + +prob_r(a, 3, 10). +prob_r(-A, C - B, C) :- prob_r(A, B, C). + diff --git a/code/asp/drafts/onabove.lp b/code/asp/drafts/onabove.lp new file mode 100644 index 0000000..8a809eb --- /dev/null +++ b/code/asp/drafts/onabove.lp @@ -0,0 +1,5 @@ +on(a,b). +on(b,c). + +above(X,Y) :- on(X,Y). +above(X,Y) :- on(X,Z), above(Z,Y). \ No newline at end of file diff --git a/code/asp/drafts/p01.lp b/code/asp/drafts/p01.lp new file mode 100644 index 0000000..4c558ef --- /dev/null +++ b/code/asp/drafts/p01.lp @@ -0,0 +1,4 @@ +a. %::0.3 => a;-a +b;-b. +c :- not b, not d. +d :- a, not c. \ No newline at end of file diff --git a/code/asp/drafts/p18.lp b/code/asp/drafts/p18.lp new file mode 100644 index 0000000..8075624 --- /dev/null +++ b/code/asp/drafts/p18.lp @@ -0,0 +1,3 @@ +p(a, b). +p(b, c). +p(X, Y) :- p(X, Z), p(Z, Y). \ No newline at end of file diff --git a/code/asp/drafts/p19.lp b/code/asp/drafts/p19.lp new file mode 100644 index 0000000..b1a1019 --- /dev/null +++ b/code/asp/drafts/p19.lp @@ -0,0 +1,2 @@ +q(f(a)). +p(X) :- q(X). \ No newline at end of file diff --git a/code/asp/drafts/p20.lp b/code/asp/drafts/p20.lp new file mode 100644 index 0000000..3ea3934 --- /dev/null +++ b/code/asp/drafts/p20.lp @@ -0,0 +1,2 @@ +q(f(a)). +p(X) :- -q(X). \ No newline at end of file diff --git a/code/asp/drafts/pasp01.lp b/code/asp/drafts/pasp01.lp new file mode 100644 index 0000000..18687dd --- /dev/null +++ b/code/asp/drafts/pasp01.lp @@ -0,0 +1,3 @@ +b ; -b. + +h1 ; h2 :- b. \ No newline at end of file diff --git a/code/asp/drafts/pasp_general.lp b/code/asp/drafts/pasp_general.lp new file mode 100644 index 0000000..ef0571a --- /dev/null +++ b/code/asp/drafts/pasp_general.lp @@ -0,0 +1,49 @@ +% 0.6 :: edge(1, 2). +% 0.1 :: edge(1, 3). +% 0.4 :: edge(2, 5). +% 0.3 :: edge(2, 6). +% 0.3 :: edge(3, 4). +% 0.8 :: edge(4, 5). +% 0.2 :: edge(5, 6). +% +% ======================================= +% PARTICULAR INSTANCE +% ======================================= +% +edge(1, 2). +edge(1, 3). +edge(2, 5). +edge(2, 6). +edge(3, 4). +edge(4, 5). +edge(5, 6). +% +% == HARD COLORING +% +red(1). +green(4). +green(6). +% +% ======================================= +% GENERAL DESCRIPTION +% ======================================= +% +edge(X, Y) :- edge(Y, X). +% +node(X) :- edge(X, _) ; edge(_, X). +% +red(X); green(X); blue(X) :- node (X). +% +% == CONSTRAINTS +% +:- edge(X, Y), red(X), red(Y). +:- edge(X, Y), green(X), green(Y). +:- edge(X, Y), blue(X), blue(Y). +% +% ======================================= +% SOLVER DIRECTIVES +% ======================================= +% +#show red/1. +#show green/1. +#show blue/1. \ No newline at end of file diff --git a/code/asp/drafts/paths.lp b/code/asp/drafts/paths.lp new file mode 100644 index 0000000..baa83d4 --- /dev/null +++ b/code/asp/drafts/paths.lp @@ -0,0 +1,15 @@ +obs(a, c, 9, 1). +obs(c, b, 7, 3). +obs(d, c, 3, 2). +obs(d, b, 9, 1). + +edge(A,B); -edge(A,B) :- obs(A, B, _, _). + +prob(X,Y, P, P + N) :- edge(X,Y), obs(X,Y, P, N). +prob(X,Y, N, P + N) :- -edge(X,Y), obs(X,Y, P, N). + +path(X,Y, 1) :- edge(X,Y). +path(X,Y, 2) :- edge(X,Z), path(Z, Y, _). + +#show prob/4. +#show path/3. \ No newline at end of file diff --git a/code/asp/drafts/pdist.lp b/code/asp/drafts/pdist.lp new file mode 100644 index 0000000..a62615b --- /dev/null +++ b/code/asp/drafts/pdist.lp @@ -0,0 +1,2 @@ +a; -a. +b; c :- a. \ No newline at end of file diff --git a/code/asp/drafts/pqr.lp b/code/asp/drafts/pqr.lp new file mode 100644 index 0000000..8acdf51 --- /dev/null +++ b/code/asp/drafts/pqr.lp @@ -0,0 +1,6 @@ +p(1). +p(2). +p(3). + +q(3) :- not r(3). +r(X) :- p(X), not q(X). \ No newline at end of file diff --git a/code/asp/drafts/pqueens.lp b/code/asp/drafts/pqueens.lp new file mode 100644 index 0000000..c87e92b --- /dev/null +++ b/code/asp/drafts/pqueens.lp @@ -0,0 +1,20 @@ +% Column and row domains. +col(1 .. n). +row(1 .. n). +% One queen on some row per colum. +1 { queen(I, J) : row(J) } 1 :- col(I). +% One queen on some columns per row. +1 { queen(I, J) : col(I) } 1 :- row(J). +% Number descending diagonals. +diag1(I, J, I - J + n) :- col(I), row(J). +% Number ascending diagonals. +diag2(I, J, I + J - 1) :- col(I), row(J). +% +% Negative Restrictions +:- D = 1 .. 2 * n - 1, not { queen(I, J) : diag1(I, J, D) } 1. +:- D = 1 .. 2 * n - 1, not { queen(I, J) : diag2(I, J, D) } 1. +% +% Output this predicate. +#show queen/2. +%#show diag1/3. +%#show diag2/3. \ No newline at end of file diff --git a/code/asp/drafts/prob01.lp b/code/asp/drafts/prob01.lp new file mode 100644 index 0000000..7ee0efb --- /dev/null +++ b/code/asp/drafts/prob01.lp @@ -0,0 +1,5 @@ +gripe;-gripe. +febre;-febre :- gripe. + +#show febre/0. +#show gripe/0. \ No newline at end of file diff --git a/code/asp/drafts/prob02.lp b/code/asp/drafts/prob02.lp new file mode 100644 index 0000000..597fa4e --- /dev/null +++ b/code/asp/drafts/prob02.lp @@ -0,0 +1,2 @@ +cross :- -train. +train;-train. \ No newline at end of file diff --git a/code/asp/drafts/py_01.lp b/code/asp/drafts/py_01.lp new file mode 100644 index 0000000..9b69a25 --- /dev/null +++ b/code/asp/drafts/py_01.lp @@ -0,0 +1,22 @@ +#script (python) + +import clingo + +N = clingo.Number + +def gcd(a, b): + if a.number == 0: + return b + else: + na = a.number + nb = b.number + nc = N(nb % na) + return gcd(nc, a) + +#end. + +p(210, 213). +p(1364, 385). +gcd(X, Y, @gcd(X,Y)) :- p(X, Y). + +#show gcd/3. \ No newline at end of file diff --git a/code/asp/drafts/py_02.lp b/code/asp/drafts/py_02.lp new file mode 100644 index 0000000..f0fd009 --- /dev/null +++ b/code/asp/drafts/py_02.lp @@ -0,0 +1,16 @@ +#script (python) + +def main(prg): + p = [] + p.append(("base", [])) + p.append(("a", [1, 3])) + prg.ground(p) + prg.solve() + +#end. + +a. +#program a(s,t). +b(s,t). +#program base. +c. \ No newline at end of file diff --git a/code/asp/drafts/roads.lp b/code/asp/drafts/roads.lp new file mode 100644 index 0000000..0d5cda3 --- /dev/null +++ b/code/asp/drafts/roads.lp @@ -0,0 +1,14 @@ +road(berlin, potsdam). +road(potsdam, werder). +road(werder, brandenburg). +road(X, Y) :- road(Y, X). + +blocked(werder, brandenburg). + +route(X, Y) :- road(X, Y), not blocked(X, Y). +route(X, Y) :- route(X, Z), route(Z, Y). + +% prob: 0.5 +drive(X) :- route(berlin, X). + +#show drive/1. \ No newline at end of file diff --git a/code/asp/drafts/s.lp b/code/asp/drafts/s.lp new file mode 100644 index 0000000..17add7e --- /dev/null +++ b/code/asp/drafts/s.lp @@ -0,0 +1,7 @@ +p(1). +p(2). +p(3). + +q(3) :- not r(3). + +r(X) :- p(X), not q(X). diff --git a/code/asp/drafts/toh.lp b/code/asp/drafts/toh.lp new file mode 100644 index 0000000..c5805b2 --- /dev/null +++ b/code/asp/drafts/toh.lp @@ -0,0 +1,29 @@ +#const moves=15. + +% instance +peg(a;b;c). +disk(1..4). +init_on(1..4, a). +goal_on(1..4, c).% generating +1 { move(D, P, T) : disk(D) : peg(P) } 1 :- T = 1..moves. + +% defining +move(D, T) :- move(D, _, T). + +on(D, P, 0) :- init_on(D, P). +on(D, P, T) :- move(D, P, T). +on(D, P, T + 1) :- on(D, P, T), not move D(T + 1), T < moves. + +blocked(D - 1, P, T + 1) :- on(D, P, T), T < moves. +blocked(D - 1, P, T) :- blocked(D, P, T), disk(D). + +% testing +:- move(D, P, T), blocked(D - 1, P, T). +:- move(D, T), on(D, P, T - 1), blocked(D, P, T). +:- not 1 { on(D, P, T) } 1, disk(D), T = 1..moves. + +:- goal_on(D, P), not on(D, P, moves). + +% displaying +#hide +#show move/3 diff --git a/code/asp/drafts/tohE.lp b/code/asp/drafts/tohE.lp new file mode 100644 index 0000000..df4a5a0 --- /dev/null +++ b/code/asp/drafts/tohE.lp @@ -0,0 +1,26 @@ +% generating + +% Only one move per "move" +% Also, restrict the "Disk" and "Peg" +1 { move(D, P, T) : disk(D) , peg(P) } 1 :- T = 1 .. moves. + +% defining +move(D, T) :- move(D, _, T). + +on(D, P, 0) :- init_on(D, P). +on(D, P, T) :- move(D, P, T). +on(D, P, T + 1) :- on(D, P, T), not move(D, T + 1), T < moves. + +blocked(D - 1, P, T + 1) :- on(D, P, T), T < moves. +blocked(D - 1, P, T) :- blocked(D, P, T), disk(D). + +% testing +:- move(D, P, T), blocked(D - 1, P, T). +:- move(D, T), on(D, P, T - 1), blocked(D, P, T). +:- not 1 { on(D, P, T) } 1, disk(D), T = 1..moves. + +:- goal_on(D, P), not on(D, P, moves). + +% displaying +% #hide +#show move/3. diff --git a/code/asp/drafts/tohI.lp b/code/asp/drafts/tohI.lp new file mode 100644 index 0000000..918d2de --- /dev/null +++ b/code/asp/drafts/tohI.lp @@ -0,0 +1,7 @@ +#const moves = 15. + +% instance +peg(a ; b ; c). +disk(1 .. 4). +init_on(1 .. 4, a). +goal_on(1 .. 4, c). \ No newline at end of file diff --git a/code/asp/drafts/tutorial_01.pl b/code/asp/drafts/tutorial_01.pl new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/code/asp/drafts/tutorial_01.pl diff --git a/code/asp/drafts/tutotial_01.lp b/code/asp/drafts/tutotial_01.lp new file mode 100644 index 0000000..a73152d --- /dev/null +++ b/code/asp/drafts/tutotial_01.lp @@ -0,0 +1,5 @@ +p(1;2). +q(1;2). +r(X,Y) :- p(X), q(Y), X != Y. + +#show r/2. \ No newline at end of file diff --git a/code/asp/drafts/unsound.lp b/code/asp/drafts/unsound.lp new file mode 100644 index 0000000..196625e --- /dev/null +++ b/code/asp/drafts/unsound.lp @@ -0,0 +1,4 @@ +sleep :- not work, not insomnia. +work :- not sleep. + +insomnia ; -insomnia. diff --git a/code/asp/fplp/example_05-1.lp b/code/asp/fplp/example_05-1.lp new file mode 100644 index 0000000..e8c989f --- /dev/null +++ b/code/asp/fplp/example_05-1.lp @@ -0,0 +1,6 @@ +b :- not a. +c :- not b. +c :- a. + +prob(frac(1,100)) :- a. +prob(frac(99,100)) :- not a. diff --git a/code/asp/fplp/example_05-2.lp b/code/asp/fplp/example_05-2.lp new file mode 100644 index 0000000..a980849 --- /dev/null +++ b/code/asp/fplp/example_05-2.lp @@ -0,0 +1,5 @@ +b :- not a. +c :- not b. +c :- a, not p. +p :- not q. +q :- not p, b. \ No newline at end of file diff --git a/code/asp/plp/alarm.plp b/code/asp/plp/alarm.plp new file mode 100644 index 0000000..947e590 --- /dev/null +++ b/code/asp/plp/alarm.plp @@ -0,0 +1,22 @@ +{burglary}. +:~not burglary.["2.19722"@0,0,()] + +{earthquake}. +:~not earthquake.["1.38629"@0,1,()] + +{hears_alarm(mary)}. +{hears_alarm(john)}. +:~not hears_alarm(mary).["-0.84729"@0,2,(mary,)] +:~not hears_alarm(john).["-0.84729"@0,2,(john,)] + +alarm :- burglary. +alarm :- earthquake. +calls(X) :- alarm, hears_alarm(X). + +person(mary). +person(john). + +% Evidence that alarm has (been observed) to be true +:- not alarm. + +&query(burglary). \ No newline at end of file diff --git a/code/asp/plp/coins.plp b/code/asp/plp/coins.plp new file mode 100644 index 0000000..88cbb15 --- /dev/null +++ b/code/asp/plp/coins.plp @@ -0,0 +1,7 @@ +{ heads(C) } :- C=1..2. +:~ not heads(C), C=1..2. [0@0,0,(C,)] + +two_heads :- heads(1), heads(2). +&query(heads(1)). + +:- two_heads. \ No newline at end of file diff --git a/code/asp/plp/cplint.plp b/code/asp/plp/cplint.plp new file mode 100644 index 0000000..0a440c4 --- /dev/null +++ b/code/asp/plp/cplint.plp @@ -0,0 +1,15 @@ +heads :- &problog("0.4"). +col(1,red) :- &log("0.3"). +col(1,blue) :- &log("0.7"). +col(2,red) :- &log("0.2"). +col(2,green) :- &log("0.3"). +col(2,blue) :- &log("0.5"). + +:- not { col(C,_) } = 1, C=1..2. + +win :- heads, col(_,red). +win :- col(1,C), col(2,C). + +% &evidence(col(2,green),true). +&query(win). +&query(win, col(2,green)). \ No newline at end of file diff --git a/code/asp/plp/sbfe.plp b/code/asp/plp/sbfe.plp new file mode 100644 index 0000000..2ab9e69 --- /dev/null +++ b/code/asp/plp/sbfe.plp @@ -0,0 +1,7 @@ +{ a }. +:~ not a. ["0.3"@0,0,()] + +b ; c :- a. +:- a. + +&query(b). \ No newline at end of file diff --git a/code/asplang/ASPLang.jl b/code/asplang/ASPLang.jl new file mode 100644 index 0000000..3c686eb --- /dev/null +++ b/code/asplang/ASPLang.jl @@ -0,0 +1,356 @@ +module ASPLang + +export parse_asp, debug_asp, + literal_grammar, term_grammar, aggregate_grammar, asp_grammar, + Node, Variable, StringConstant, SymbolicConstant, AnonymousVariable, NumberConstant, Variable, + Functor, FunctionTerm, + Negated, Inverted, Product, Sum, + LT, LE, EQ, NE, GE, GT, + LTAtom, LEAtom, EQAtom, NEAtom, GEAtom, GTAtom, + Atom, ClassicalLiteral, NafLiteral, + Level, Contribution, WeightAtLevel, + CountAggregate, SumAggregate, MaxAggregate, MinAggregate, + Aggregate, Elements, InclusionCondition, + Constraint, LeftBound, RightBound, + Choice, RestrictedChoice, Choices, + Disjunction, Body, Rule, Restriction, WeakRestriction, Annotation, + Query, Program + +macro terminal(lang) + return quote + struct $lang <: Node end + end +end + +macro variable(lang) + return quote + struct $lang <: Node + args + end + end +end + +using ParserCombinator +#= + DATA STRUCTURES + =# + +abstract type Node end +Base.:(==)(n1::Node, n2::Node) = (typeof(n1) == typeof(n2)) && (n1.args == n2.args) + +@variable Variable +@variable StringConstant +@variable SymbolicConstant +@variable AnonymousVariable +@variable NumberConstant + +@variable Functor +@variable FunctionTerm +@variable Negated +@variable Inverted +@variable Product +@variable Sum + +@terminal LT +@terminal LE +@terminal EQ +@terminal NE +@terminal GE +@terminal GT +@variable LTAtom +@variable LEAtom +@variable EQAtom +@variable NEAtom +@variable GEAtom +@variable GTAtom + +@variable Atom +@variable ClassicalLiteral + +@variable NafLiteral + +@variable Level +@variable Weight +@variable Contribution +@variable WeightAtLevel + +@variable CountAggregate +@variable SumAggregate +@variable MaxAggregate +@variable MinAggregate + +@variable Aggregate +@variable Elements +@variable InclusionCondition +@variable Constraint +@variable LeftBound +@variable RightBound + +@variable RestrictedChoice +@variable Choice +@variable Choices +@variable Disjunction +@variable Annotation +@variable Body + +@variable Rule +@variable Restriction +@variable WeakRestriction +@variable Query +@variable Program + +#= +------------------------------------------------------ +GRAMMARS +------------------------------------------------------ +=# +@with_names begin +spc = Drop(Star(Space())) +@with_pre spc begin +@with_post spc begin +# +# +# TERMS +# +# +#---------------------------# +# variable # +#---------------------------# +variable = p"[A-Z]([a-zA-Z0-9_]*)" > Variable +#---------------------------# +# string constant # +#---------------------------# +string_constant = E"\"" + p"[^\"]*" + E"\"" > StringConstant +#---------------------------# +# symbolic constant # +#---------------------------# +symbolic_constant = p"[a-z]([a-zA-Z0-9_]*)" > SymbolicConstant +#---------------------------# +# anonymous variable # +#---------------------------# +anonymous_variable = e"_" > AnonymousVariable +#---------------------------# +# number constant # +#---------------------------# +number = (e"0" | p"[1-9]([0-9]*)") > NumberConstant +neg_number = E"-" + number > Negated +#---------------------------# +# variable term # +#---------------------------# +variable_term = variable | anonymous_variable +#---------------------------# +# ground term # +#---------------------------# +ground_term = symbolic_constant | string_constant | number # | neg_number +#---------------------------# +# basic term # +#---------------------------# +basic_term = ground_term | variable_term +basic_terms = StarList(basic_term, E",") +#---------------------------# +# arith op # +#---------------------------# +# arith_op = e"+" | e"-" | e"*" | e"/" +#---------------------------# +# functor # +#---------------------------# +functor = p"[a-z]([a-zA-Z0-9_]*)" > Functor +#---------------------------# +# term # +#---------------------------# +term = Delayed() +#---------------------------# +# terms # +#---------------------------# +terms = StarList(term, E",") +#---------------------------# +# function term # +#---------------------------# +function_term = functor + E"(" + terms + E")" |> FunctionTerm +#---------------------------# +# value # +#---------------------------# +value = ground_term | variable_term | (E"(" + term + E")") +#---------------------------# +# negated # +#---------------------------# +negated = Delayed() +negated.matcher = function_term | value | (spc + E"-" + negated > Negated) +#---------------------------# +# multiplicative factor # +#---------------------------# +mul_factor = E"*" + negated +#---------------------------# +# inverted factor # +#---------------------------# +inv_factor = E"/" + negated > Inverted +#---------------------------# +# product # +#---------------------------# +product = negated | (negated + Plus(mul_factor | inv_factor) |> Product) +#---------------------------# +# positive parcel # +#---------------------------# +pos_parcel = E"+" + product +#---------------------------# +# negated parcel # +#---------------------------# +neg_parcel = E"-" + product > Negated +#---------------------------# +# sum # +#---------------------------# +sum = product | (product + Plus(pos_parcel | neg_parcel) |> Sum) +#---------------------------# +# term # +#---------------------------# +term.matcher = value | function_term | negated | product | sum +term_grammar = term + Eos() +# +# +# LITERALS +# +# +#---------------------------# +# bin op # +#---------------------------# +bin_lt = (E"<" > LT) +bin_le = (E"<=" > LE) +bin_eq = (E"=" > EQ) +bin_ne = (E"!=" > NE) +bin_gt = (E">" > GT) +bin_ge = (E">=" > GE) +bin_op = bin_lt | bin_le | bin_eq | bin_ne | bin_gt | bin_ge +#---------------------------# +# builtin atom # +#---------------------------# +builtin_atom = + (term + bin_lt + term |> LTAtom) | # less than + (term + bin_le + term |> LEAtom) | # less equal + (term + bin_eq + term |> EQAtom) | # equal + (term + bin_ne + term |> NEAtom) | # not equal + (term + bin_gt + term |> GTAtom) | # greater than + (term + bin_ge + term |> GEAtom) # greater equal +#---------------------------# +# classical literal # +#---------------------------# +atom = symbolic_constant + Opt(E"(" + terms + E")") # |> Atom + +negated_atom = Delayed() +negated_atom.matcher = atom | (spc + E"-" + negated_atom > Negated) + +classical_literal = atom | negated_atom +#---------------------------# +# naf literal # +#---------------------------# +naf_literal = builtin_atom | classical_literal | (E"not" + classical_literal > NafLiteral) +naf_literals = StarList(naf_literal, E",") +# +literal_grammar = naf_literal + Eos() +# +# +# AGGREGATES +# +# +#---------------------------# +# weight at level # +#---------------------------# +weight_at_level = (term > Weight) + Opt(E"@" + term > Level) + Opt(E"," + terms |> Contribution) |> WeightAtLevel +#---------------------------# +# aggregates # +#---------------------------# +aggregate_element = (basic_terms |> Elements) + Opt(E":" + naf_literals |> InclusionCondition) +aggregate_elements = StarList(aggregate_element, E";") +# +count_aggregate = Opt(term + bin_op |> LeftBound) + + E"#count" + spc + E"{" + (aggregate_elements |> Constraint) + E"}" + + Opt(bin_op + term |> RightBound) |> CountAggregate +# +sum_aggregate = Opt(term + bin_op |> LeftBound) + + E"#sum" + spc + E"{" + (aggregate_elements |> Constraint) + E"}" + + Opt(bin_op + term |> RightBound) |> SumAggregate +# +min_aggregate = Opt(term + bin_op |> LeftBound) + + E"#min" + spc + E"{" + (aggregate_elements |> Constraint) + E"}" + + Opt(bin_op + term |> RightBound) |> MinAggregate +# +max_aggregate = Opt(term + bin_op |> LeftBound) + + E"#max" + spc + E"{" + (aggregate_elements |> Constraint) + E"}" + + Opt(bin_op + term |> RightBound) |> MaxAggregate +# +aggregate = count_aggregate | sum_aggregate | min_aggregate | max_aggregate +aggregate_grammar = aggregate + Eos() +# +# +# PROGRAMS +# +# +#---------------------------# +# choice # +#---------------------------# +choice_element = classical_literal | (classical_literal + E":" + naf_literals |> RestrictedChoice) +choice_elements = StarList(choice_element, E";") +choice = Opt(term + bin_op |> LeftBound) + + E"{" + (choice_elements |> Choices) + E"}" + + Opt(bin_op + term |> RightBound) |> Choice +#---------------------------# +# disjunction # +#---------------------------# +disjunction = PlusList(classical_literal, E"|") |> Disjunction +#---------------------------# +# body # +#---------------------------# +body_element = naf_literal | aggregate | (E"not" + aggregate > Negated) +body = PlusList(body_element, E",") |> Body +#---------------------------# +# head # +#---------------------------# +head = disjunction | choice +#---------------------------# +# annotation # +#---------------------------# +annotation = E"::" + term > Annotation +#---------------------------# +# statement # +#---------------------------# +statement = + # strong restriction + (E":-" + body + E"." > Restriction) | + # rule + (head + Opt(annotation) + Opt(E":-" + body) + E"." |> Rule) | + # weak restriction + (E":~" + Opt(body) + E"." + spc + E"[" + weight_at_level + E"]" |> WeakRestriction) +statements = StarList(statement, E"") +#---------------------------# +# query # +#---------------------------# +query = classical_literal + E"?" > Query +#---------------------------# +# program # +#---------------------------# +program = statements + Opt(query) |> Program +#---------------------------# +# asp # +#---------------------------# +asp_grammar = program + Eos() +end +end +end + +parse_asp(source::String, grammar=asp_grammar) = parse_one(source, grammar)[1] # |> simplify + +function debug_asp(source::String, grammar=asp_grammar, show_trace=false) + debug, task = make(Debug, source, grammar; delegate=NoCache) + try + once(task) + expr = parse_asp(source, grammar) + println("---\n\"$source\"\n\tparses to\n$(expr)\n---") + catch #_::ParserException + println("Error parsing after \"$(source[1:debug.max_iter - 1])\" and before \"$(source[debug.max_iter:end])\".\n") + + if show_trace + parse_dbg(source, Trace(grammar=grammar)) + end + end +end + +end \ No newline at end of file diff --git a/code/asplang/asp-core-2_syntax.ebnf b/code/asplang/asp-core-2_syntax.ebnf new file mode 100644 index 0000000..1b46024 --- /dev/null +++ b/code/asplang/asp-core-2_syntax.ebnf @@ -0,0 +1,54 @@ +Id ::= [a-z] [A-Za-z0-9_]* +Variable ::= [A-Z] [A-Za-z0-9_]* +StringConstant ::= \"([^\"] | \\\")\" +SymbolicConstant ::= [a-z] [A-Za-z0-9_]* +AnonymousVariable ::= "_" +Number ::= 0 | [1-9][0-9]* +VariableTerm ::= Variable | AnonymousVariable +GroundTerm ::= SymbolicConstant | StringConstant | "-"? Number +BasicTerm ::= GroundTerm | VariableTerm +BasicTerms ::= BasicTerm ("," BasicTerm)* +ArithOp ::= "+" | "-" | "*" | "/" + +Term ::= Id ("(" Terms? ")")? + | Number + | StringConstant + | Variable + | AnonymousVariable + | "(" Term ")" + | "-" Term + | Term ArithOp Term +Terms ::= Term ("," Term)* + +BinOp ::= "<" | "<=" | "=" | "!=" | ">" | ">=" + +BuiltinAtom ::= Term BinOp Term +ClassicalLiteral ::= "-"? SymbolicConstant ("(" Terms? ")")? + +NafLiteral ::= "not"? ClassicalLiteral | BuiltinAtom +NafLiterals ::= NafLiteral ("," NafLiteral)* + +WeightAtLevel ::= Term ("@" Term)? ("," Terms)? + +AggregateFunction ::= "#count" | "#sum" | "#max" | "#min" +AggregateElement ::= BasicTerms? (":" NafLiterals?)? +AggregateElements ::= AggregateElement (";" AggregateElement)* +Aggregate ::= (Term BinOp)? AggregateFunction "{" AggregateElements? "}" (BinOp Term)? + +ChoiceElement ::= ClassicalLiteral (":" NafLiterals?)? +ChoiceElements ::= ChoiceElement (";" ChoiceElement)* +Choice ::= (Term BinOp)? "{" ChoiceElements? "}" (BinOp Term)? + +Disjunction ::= ClassicalLiteral ("|" ClassicalLiteral)* + +Body ::= (Body ",")? (NafLiteral | "not"? Aggregate) +Head ::= Disjunction | Choice + +Statement ::= ":-" Body? "." + | Head (":-" Body?)? "." + | ":~" Body? "." "[" WeightAtLevel "]" + +Query ::= ClassicalAtom "?" +Statements ::= Statement+ + +Program ::= Statements? Query? diff --git a/code/asplang/draftASP.jl b/code/asplang/draftASP.jl new file mode 100644 index 0000000..e2304c3 --- /dev/null +++ b/code/asplang/draftASP.jl @@ -0,0 +1,8 @@ +using ParserCombinator + +symbolic_constant = p"[a-z]([a-zA-Z0-9_]*)" +grammar = symbolic_constant + Eos() + +source = "A" +expr = parse_one(source, grammar)[1] +println("Source $source => $expr") \ No newline at end of file diff --git a/code/asplang/drafts.jl b/code/asplang/drafts.jl new file mode 100644 index 0000000..3622b0f --- /dev/null +++ b/code/asplang/drafts.jl @@ -0,0 +1,58 @@ +include("ASPLang.jl") + +using .ASPLang + +function test_parse(source, grammar=asp_grammar) + expr = parse_asp(source, grammar) + println("$source => $(expr).\n") + print(pstring(expr)) +end + +# test_parse("\"a\"", term_grammar) +# test_parse(" - - 2", term_grammar) +# test_parse("f(0) + (A - 1) * a / 3 + f( 1, B, 2 + 3, g(0) )", term_grammar) + +# test_parse("a", literal_grammar) +# test_parse("-a(3)", literal_grammar) +# test_parse("not a(3)", literal_grammar) +# test_parse("not -a(3)", literal_grammar) +# test_parse("- a(3)", literal_grammar) +# test_parse(" - - a(3)", literal_grammar) + +#test_parse("1 - f(e) < #count { a, b : c, -d ; e, f : not e, f } < 2", aggregate_grammar) + +# test_parse("a | b :: p(3) :- c, d, e(1).") +# test_parse(":~ a. [ 3@0, b, -1 ]") +# test_parse("a :: 3 / 10. +# b | c :- a. +# b?") + +src = read("sbf.plp", String) +test_parse(src) + +# test_parse("(2 + A) * a", term_grammar) + +# test_parse(" 1 < { a : not a ; b ; c } <= 3 :- c.") + +# Program(Any[ +# Rule(Any[ +# Choice(Any[ +# LeftBound(Any[ +# NumberConstant("1"), LT() +# ]), +# Choices(Any[ +# RestrictedChoice(Any[ +# SymbolicConstant("a"), NafLiteral(SymbolicConstant("a")) +# ]), +# SymbolicConstant("b"), +# SymbolicConstant("c") +# ]), +# RightBound(Any[ +# LE(), NumberConstant("3") +# ]) +# ]), +# Body(Any[ +# SymbolicConstant("c") +# ]) +# ]) +# ]) \ No newline at end of file diff --git a/code/asplang/sbf.plp b/code/asplang/sbf.plp new file mode 100644 index 0000000..ffa07cb --- /dev/null +++ b/code/asplang/sbf.plp @@ -0,0 +1,2 @@ +a::"0.3". +b | c :- a. \ No newline at end of file diff --git a/code/asplang/testASPLang.jl b/code/asplang/testASPLang.jl new file mode 100644 index 0000000..10e133c --- /dev/null +++ b/code/asplang/testASPLang.jl @@ -0,0 +1,45 @@ +using Test + +include("ASPLang.jl") +using .ASPLang + +@testset "invalid term" begin + @test_throws "" parse_asp("", term_grammar) + @test_throws "" parse_asp("1+", term_grammar) + @test_throws "" parse_asp("()", term_grammar) + @test_throws "" parse_asp("(", term_grammar) + @test_throws "" parse_asp(")", term_grammar) + @test_throws "" parse_asp("[]", term_grammar) + @test_throws "" parse_asp("[", term_grammar) + @test_throws "" parse_asp("]", term_grammar) + @test_throws "" parse_asp("{}", term_grammar) + @test_throws "" parse_asp("{", term_grammar) + @test_throws "" parse_asp("}", term_grammar) + @test_throws "" parse_asp("1+()", term_grammar) + @test_throws "" parse_asp("1<2", term_grammar) + @test_throws "" parse_asp("1a", term_grammar) + @test_throws "" parse_asp("1+(a+", term_grammar) + @test_throws "" parse_asp("1+(1+))", term_grammar) + @test_throws "" parse_asp("(1+(1)()", term_grammar) +end + +@testset "variable" begin + @test parse_asp("A", term_grammar) == Variable("A") + @test parse_asp("A_a1b_2", term_grammar) == Variable("A_a1b_2") + @test parse_asp("a", term_grammar) != Variable("a") + @test_throws "" parse_asp("", term_grammar) != Variable("") +end + +@testset "string constant" begin + @test parse_asp("\"\"", term_grammar) == StringConstant("") + @test parse_asp("\"A\"", term_grammar) == StringConstant("A") + @test parse_asp("\"a\"", term_grammar) == StringConstant("a") + @test parse_asp("\"1\"", term_grammar) == StringConstant("1") + @test parse_asp("\"A_a1b_2\"", term_grammar) == StringConstant("A_a1b_2") +end + +@testset "symbolic constant" begin + @test parse_asp("a", term_grammar) == SymbolicConstant("a") + @test parse_asp("a_A1B_2", term_grammar) == SymbolicConstant("a_A1B_2") + @test parse_asp("A", term_grammar) != SymbolicConstant("A") +end \ No newline at end of file diff --git a/code/asplang/tests_ASPlang.jl b/code/asplang/tests_ASPlang.jl new file mode 100644 index 0000000..df49163 --- /dev/null +++ b/code/asplang/tests_ASPlang.jl @@ -0,0 +1,132 @@ +using Test + +include("ASPLang.jl") +using .ASPLang + +@testset "malformed" begin + # + # + # + @test_throws "" parse_term("1a") + @test_throws "" parse_term("1+") + @test_throws "" parse_term("+") + @test_throws "" parse_term("(") + @test_throws "" parse_term(")") + # + # Spaces + # + @test_throws "" parse_term("1 1") + @test_throws "" parse_term("+ 1") +end +@testset "constants" begin + # + # NumberConstant + # + @test parse_term("1") == NumberConstant("1") + @test parse_term("11") == NumberConstant("11") + @test parse_term("+1") == NumberConstant("+1") + @test parse_term("+++1") == NumberConstant("+++1") + @test parse_term("-1") == Negated(NumberConstant("1")) + @test parse_term("- 11") == Negated(NumberConstant("11")) + @test parse_term("---1") == Negated(Negated(Negated(NumberConstant("1")))) + # + # SymbolicConstant + # + @test parse_term("a") == SymbolicConstant("a") + @test parse_term("aA") == SymbolicConstant("aA") + @test parse_term("a1") == SymbolicConstant("a1") + @test parse_term("a_1") == SymbolicConstant("a_1") + # + # StringConstant + # + @test parse_term("\"a\"") == StringConstant("a") + @test parse_term("\"aA\"") == StringConstant("aA") + @test parse_term("\"a_1\"") == StringConstant("a_1") + @test parse_term("\"a1\"") == StringConstant("a1") + @test parse_term("\"a 1\"") == StringConstant("a 1") + @test parse_term("\"A\"") == StringConstant("A") + @test parse_term("\"AA\"") == StringConstant("AA") + @test parse_term("\"A_1\"") == StringConstant("A_1") + @test parse_term("\"A1\"") == StringConstant("A1") + @test parse_term("\"A 1\"") == StringConstant("A 1") + @test parse_term("\"0\"") == StringConstant("0") + @test parse_term("\"0A\"") == StringConstant("0A") + @test parse_term("\"0_1\"") == StringConstant("0_1") + @test parse_term("\"01\"") == StringConstant("01") + @test parse_term("\"0 1\"") == StringConstant("0 1") +end + +@testset "variables" begin + # + # Variable + # + @test parse_term("Ax") == Variable("Ax") + @test parse_term("A1") == Variable("A1") + @test parse_term("A_x") == Variable("A_x") + @test parse_term("_x") == Variable("_x") + @test parse_term("_1") == Variable("_1") + @test parse_term("__x") == Variable("__x") +end + +@testset "basic arithmetic terms" begin + # + # Single Operations + # + @test parse_term("1+1") == Sum(Any[NumberConstant("1"), NumberConstant("1")]) + @test parse_term("1-1") == Sum(Any[NumberConstant("1"), Negated(NumberConstant("1"))]) + @test parse_term("1*1") == Product(Any[NumberConstant("1"), NumberConstant("1")]) + @test parse_term("1/1") == Product(Any[NumberConstant("1"), Inverted(NumberConstant("1"))]) + @test parse_term("1^1") == Power(Any[NumberConstant("1"), NumberConstant("1")]) + # + # Simple Parenthesis + # + @test parse_term("(1)") == NumberConstant("1") + @test parse_term("(1+1)") == Sum(Any[NumberConstant("1"), NumberConstant("1")]) + @test parse_term("(1-1)") == Sum(Any[NumberConstant("1"), Negated(NumberConstant("1"))]) + @test parse_term("(1*1)") == Product(Any[NumberConstant("1"), NumberConstant("1")]) + @test parse_term("(1/1)") == Product(Any[NumberConstant("1"), Inverted(NumberConstant("1"))]) + @test parse_term("(1^1)") == Power(Any[NumberConstant("1"), NumberConstant("1")]) +end + +@testset "arithmetic associative rules" begin + @test parse_term("1 + 2 + 3") == Sum(Any[NumberConstant("1"), NumberConstant("2"), NumberConstant("3")]) + @test parse_term("1 + (2 + 3)") == Sum(Any[NumberConstant("1"), Sum(Any[NumberConstant("2"), NumberConstant("3")])]) + @test parse_term("(1 + 2) + 3") == Sum(Any[Sum(Any[NumberConstant("1"), NumberConstant("2")]), NumberConstant("3")]) + # + # Default Associative Rules + # + @test parse_term("2 + 3 * 4") == Sum(Any[ + NumberConstant("2"), Product(Any[NumberConstant("3"), NumberConstant("4")])]) + @test parse_term("2 * 3 + 4") == Sum(Any[ + Product(Any[NumberConstant("2"), NumberConstant("3")]), NumberConstant("4")]) +end + +@testset "functional terms" begin + @test parse_term("1 + 2 + 3") == Sum(Any[NumberConstant("1"), NumberConstant("2"), NumberConstant("3")]) + @test parse_term("1 + (2 + 3)") == Sum(Any[NumberConstant("1"), Sum(Any[NumberConstant("2"), NumberConstant("3")])]) + @test parse_term("(1 + 2) + 3") == Sum(Any[Sum(Any[NumberConstant("1"), NumberConstant("2")]), NumberConstant("3")]) + # + # Functional Terms + # + @test parse_term("f()") == FunctionalTerm(Any[Functor("f")]) + @test parse_term("f(1)") == FunctionalTerm(Any[Functor("f"), NumberConstant("1")]) + @test parse_term("f(1,2)") == FunctionalTerm(Any[Functor("f"), NumberConstant("1"), NumberConstant("2")]) + @test parse_term("f(1,g(2))") == FunctionalTerm(Any[Functor("f"), NumberConstant("1"), FunctionalTerm(Any[Functor("g"), NumberConstant("2")])]) +end + +@testset "predicate atoms" begin + # + # Predicate Atoms + # + @test parse_atom("p") == PredicateName("p") + @test parse_atom("p()") == PredicateAtom(Any[PredicateName("p")]) + @test parse_atom("p(1)") == PredicateAtom(Any[PredicateName("p"), NumberConstant("1")]) + @test parse_atom("p(1,2)") == PredicateAtom(Any[PredicateName("p"), NumberConstant("1"), NumberConstant("2")]) + @test parse_atom("p(1,f(2))") == PredicateAtom(Any[PredicateName("p"), NumberConstant("1"), FunctionalTerm(Any[Functor("f"), NumberConstant("2")])]) +end + +@testset "negated atoms" begin + @test parse_atom("-p") == NegatedAtom(PredicateName("p")) + @test parse_atom("- p") == NegatedAtom(PredicateName("p")) + @test parse_atom("-- p") == NegatedAtom(NegatedAtom(PredicateName("p"))) +end \ No newline at end of file diff --git a/code/biflang/BIFLang.jl b/code/biflang/BIFLang.jl new file mode 100644 index 0000000..bf13667 --- /dev/null +++ b/code/biflang/BIFLang.jl @@ -0,0 +1,157 @@ +module BIFLang + +export parse_bif, debug_bif, fmt, + NetworkKW, + VariableKW, + ProbabilityKW, + PropertyKW, + VariabletypeKW, + DiscreteKW, + DefaultvalueKW, + TablevaluesKW, + Property, + Properties, + Variable, + Parents, + Values, + Key, + ProbabilityDeclaration, + VariableDeclaration, + Distribution, + DefaultEntry, + Entry, + Discrete, + Network, + Unit + +using ParserCombinator + +abstract type Node end; + +Base.:(==)(a::Node, b::Node) = (typeof(a) == typeof(b)) && (a.args == b.args) +function Base.print(io::IO, n::Node) + show_type = n |> typeof |> string + Base.print(io, show_type) + if isa(n.args, AbstractArray) && length(n.args) > 0 + show_args = join(n.args .|> ni -> "$ni", ",") + Base.print(io, "($show_args)") + end +end + +macro grammar(lang) + return quote + struct $lang <: Node args end; + end +end + + +abstract type Keyword end; +Base.:(==)(a::Keyword, b::Keyword) = (typeof(a) == typeof(b)) +fmt(n::Keyword) = "$(n |> typeof |> string)" +macro keyword(lang) + return quote + struct $lang <: Keyword end; + end +end + +@keyword NetworkKW +@keyword VariableKW +@keyword ProbabilityKW +@keyword PropertyKW +@keyword VariabletypeKW +@keyword DiscreteKW +@keyword DefaultvalueKW +@keyword TablevaluesKW + +@grammar Property +@grammar Properties +@grammar Variable +@grammar Parents +@grammar Values +@grammar Key + +@grammar ProbabilityDeclaration +@grammar VariableDeclaration + +@grammar Distribution +@grammar DefaultEntry +@grammar Entry + +@grammar Discrete +@grammar Network +@grammar Compilation +@grammar Unit + +@with_names begin +spc = Drop(Star(Space())) +@with_pre spc begin +@with_post spc begin + +word = p"[a-zA-Z]([a-zA-Z0-9_-])*" + +decimal = PInt64() +floating_point = PFloat64() + +network_kw = E"network" > NetworkKW +variable_kw = E"variable" > VariableKW +probability_kw = E"probability" > ProbabilityKW +property_kw = E"property" > PropertyKW +variabletype_kw = E"type" > VariabletypeKW +discrete_kw = E"discrete" > DiscreteKW +defaultvalue_kw = E"default" > DefaultvalueKW +tablevalues_kw = E"table" > TablevaluesKW + +property = Drop(property_kw) + spc + p"[^;]+" + E";" |> Property + +floatingpoint_list = PlusList(floating_point, E",") +probability_table = Drop(tablevalues_kw) + floatingpoint_list + spc + E";" |> Distribution +probability_defaultentry = Drop(defaultvalue_kw) + floatingpoint_list + spc + E";" |> DefaultEntry + +probability_value = word +probability_values = E"(" + spc + PlusList(probability_value, spc + E","+spc) + spc + E")" +probability_entry = (probability_values |> Key) + (floatingpoint_list |> Distribution) + spc + E";" |> Entry + +probability_content = E"{" + spc + + Star( property | probability_defaultentry | probability_entry | probability_table ) + spc + + E"}" + +probability_variable = word |> Variable +probability_variables = E"(" + spc + probability_variable + Opt(E"|" + PlusList(probability_variable, E",") |> Parents) + spc + E")" +probability_declaration = Drop(probability_kw) + probability_variables + probability_content |> ProbabilityDeclaration + +variable_values = PlusList(probability_value, spc + E"," + spc) + +variable_discrete = Drop(variabletype_kw + discrete_kw) + + E"[" + decimal + E"]" + spc + E"{" + spc + (variable_values |> Values) + spc + E"}" |> Discrete +variable_content = E"{" + Star((property | variable_discrete) + spc + E";" + spc) + E"}" +variable_declaration= Drop(variable_kw) + probability_variable + variable_content |> VariableDeclaration + +network_content = E"{" + spc + Star(property) + spc + E"}" |> Properties +network_declaration = Drop(network_kw) + word + network_content |> Network + +compilation_unit = network_declaration + Star(variable_declaration | probability_declaration) |> Unit + +bif_grammar = compilation_unit + Eos() +end +end +end + + +parse_bif(source::String) = parse_one(source, bif_grammar)[1] + +function debug_bif(source::String, show_trace=false) + debug, task = make(Debug, source, bif_grammar; delegate=NoCache) + try + once(task) + expr = parse_bif(source) + println("---\n\"$source\"\n\tparses to\n$(expr)\n---") + catch #_::ParserException + p = debug.max_iter + println("Error parsing after \"$(source[max(1,p-20):p - 1])\" and before \"$(source[p:p+20])\".\n") + + if show_trace + parse_dbg(source, Trace(grammar=bif_grammar)) + end + end +end +end \ No newline at end of file diff --git a/code/biflang/asia2.bif b/code/biflang/asia2.bif new file mode 100644 index 0000000..73ecc01 --- /dev/null +++ b/code/biflang/asia2.bif @@ -0,0 +1,63 @@ +network asia2 { + property what: ever; + property goes: well; + property with_me; +} +variable asia { + type discrete [ 2 ] { yes, no }; +} +variable tub { + type discrete [ 2 ] { yes, no }; +} +variable smoke { + type discrete [ 2 ] { yes, no }; +} +variable lung { + type discrete [ 2 ] { yes, no }; +} +variable bronc { + type discrete [ 2 ] { yes, no }; +} +variable either { + type discrete [ 2 ] { yes, no }; +} +variable xray { + type discrete [ 2 ] { yes, no }; +} +variable dysp { + type discrete [ 2 ] { yes, no }; +} +probability ( asia ) { + table 0.01, 0.99; +} +probability ( tub | asia ) { + (yes) 0.05, 0.95; + (no) 0.01, 0.99; +} +probability ( smoke ) { + table 0.5, 0.5; +} +probability ( lung | smoke ) { + (yes) 0.1, 0.9; + (no) 0.01, 0.99; +} +probability ( bronc | smoke ) { + (yes) 0.6, 0.4; + (no) 0.3, 0.7; +} +probability ( either | lung, tub ) { + (yes, yes) 1.0, 0.0; + (no, yes) 1.0, 0.0; + (yes, no) 1.0, 0.0; + (no, no) 0.0, 1.0; +} +probability ( xray | either ) { + (yes) 0.98, 0.02; + (no) 0.05, 0.95; +} +probability ( dysp | bronc, either ) { + (yes, yes) 0.9, 0.1; + (no, yes) 0.7, 0.3; + (yes, no) 0.8, 0.2; + (no, no) 0.1, 0.9; +} diff --git a/code/biflang/bif.ebnf b/code/biflang/bif.ebnf new file mode 100644 index 0000000..4aee7bd --- /dev/null +++ b/code/biflang/bif.ebnf @@ -0,0 +1,63 @@ +Digit ::= [0-9] +PDigit::= [1-0] +Letter ::= ["a"-"z" "A"-"Z" "_" "-"] +Word ::= Letter (Letter | Digit)* + +Decimal ::= PDigit Digit* +Exponent ::= ("e" | "E") ("+" | "-")? Digit+ +FloatingPoint ::= Digit+ "." Digit* Exponent? + | "." Digit+ Exponent? + | Digit+ Exponent + + +NetworkKW ::= "network" +VariableKW ::= "variable" +ProbabilityKW ::= "probability" +PropertyKW ::= "property" +VariableTypeKW ::= "type" +DiscreteKW ::= "discrete" +DefaultValueKW ::= "default" +TableValuesKW ::= "table" + +PropertyString ::= PropertyKW (~";")* ";" + +Property ::= +FloatingPointToken ::= + +FloatingPointList ::= FloatingPointToken+ + +ProbabilityTable ::= FloatingPointList ";" + +ProbabilityDefaultEntry ::= FloatingPointList ";" + +ProbabilityVariableValue ::= Word + +ProbabilityValuesList ::= "(" ProbabilityVariableValue+ ")" + +ProbabilityEntry ::= ProbabilityValuesList FloatingPointList ";" + +ProbabilityContent ::= "{" ( Property | ProbabilityDefaultEntry | ProbabilityEntry | ProbabilityTable )* "}" + +ProbabilityVariableName ::= + +ProbabilityVariablesList ::= "(" ProbabilityVariableName+ ")" + +ProbabilityDeclaration ::= ProbabilityKW ProbabilityVariablesList ProbabilityContent + + +VariableValuesList ::= ProbabilityVariableValue+ + +VariableDiscrete ::= VariableTypeKW DiscreteKW "[" Decimal "]" + "{" VariableValuesList "}" ";" + +VariableContent ::= "{" (Property | VariableDiscrete)* "}" + +VariableDeclaration ::= VariableKW ProbabilityVariableName VariableContent + + +NetworkContent ::= "{" ( Property )* "}" +NetworkDeclaration ::= NetworkKW Word NetworkContent + +CompilationUnit ::= NetworkDeclaration (VariableDeclaration | ProbabilityDeclaration)* EOS + + diff --git a/code/biflang/bif.txt b/code/biflang/bif.txt new file mode 100644 index 0000000..ecc5a8b --- /dev/null +++ b/code/biflang/bif.txt @@ -0,0 +1,562 @@ + The Interchange Format for Bayesian Networks + + +-------------------------------------------------------------------------------- + +Introduction +This page describes a proposed format for interchange of Bayesian +networks between researchers in the field. The Interchange Format is a +vehicle for interoperability of belief network tools and aims at +facilitating comparison and discussion of research results. The +Interchange Format resembles the Bayesian Network Interchange Format +proposal, referred to as BNIF. Simplifications were made such that a +parser for the Interchange Format can be easily built and prototyped. +The goal of the current implementation is to agree on a format that +can represent networks with discrete variables. Several extensions +will be entertained for next releases of the format. + +This document gives a brief overview of the basics of the proposed +Interchange Format. Open issues are emphasized in bold font. + + +-------------------------------------------------------------------------------- + +Basics +The Interchange Format uses only ASCII symbols and expects one stream +to contain a single network (a stream is either a file, a socket, +etc). It seems reasonable to find a common file extension for input +files, which could also be used as an acronym for the Interchange +Format. At this point there is no agreement on this. How about bif +(for Bayesian Interchange Format) or dsc (as suggested in the BNIF)? +White spaces, tabs and newlines are ignored; the C/C++ style of +comments (//... and multi-line /*...*/ comment sequences) are +adopted. Two other characters are also ignored when they occur between +tokens: "," and "|". These characters can be used to separate +variables in the definition of a probability distribution. + +The basic unit of information is a block: a piece of text which starts +with a keyword and ends with the end of an attribute list (to be +explained later). Arbitrary characters are allowed between +blocks. This allows the user to insert arbitrarily long comments +outside the blocks, and reserve the //, /* */, comments to be placed +inside blocks. It also allows user-specific blocks and commands to be +placed outside the standard blocks. + +Other than blocks, the Interchange Format refers to three entities: +words, non-negative integers and non-negative reals. + +A word is a contiguous sequence of characters, with the restriction +that the first character be a letter. Characters are letters plus +numbers plus the underline symbol (_) plus the dash symbol (-). + +A non-negative integer is a sequence of numeric characters which is +not followed by a decimal point or an exponent. The first character of +a non-negative integer has to be larger than zero. A non-negative real +is a sequence of numeric characters, containing a decimal point or an +exponent or both. Notice that there is no overlap between non-negative +integer and reals; for example the sequence "88" is an integer but not +a real, and the sequence "88.0" is a real but not an integer. Finally +"088" is not an integer nor a real. These conventions roughly follow +the Java language and are also used by many C compilers (should these +conventions be changed?). + +Blocks +A block is a unit of information. The general format of a block is: + block-type block-name { + attribute-name attribute-value; + attribute-name attribute-value; + attribute-name attribute-value; + } + +with as many attributes as necessary. The closing semicolon is +mandatory after each attribute. +There are three possible blocks: network, variable and probability +blocks. + +A network block defines the name of the network and lists the +properties. Example: + network Robot-Planning { + property version 1.1; + property author Nobody; + } + +Variable blocks define the variables in a network. These blocks used +to be called node blocks in the BNIF; it seems that variable conveys +more of a statistical meaning while node just refers to a graphical +concept. Opinions? Example: + variable Leg { + type discrete[2] { long, short }; + property temporary yes; + } + +Probability blocks specify the (conditional) probability tables (CPTs) +for these variables, and hence the topology of the network. The block +indicates the variables of the probability distribution right after +the keyword probability. Example: + probability ( Leg | Arm ) { + table 0.1 0.9 0.9 0.1; + } + +The blocks occur in the following order: +A network declaration block (one, must be first). +A series of variable declaration blocks and probability definition +blocks, possibly inter-mixed. +Attributes +Several attributes are defined at this point: property, type, table, +default and entry attributes (the entry attribute is not associated +with any keyword). +The attribute property can appear in all types of blocks. A property +is just a string of arbitrary text to be associated with a +block. Examples of properties: + + property size 12; + property name "Trial number ten"; + +Any text is valid between the keyword property and the ending +semicolon. The idea is to store information that is specific to a +particular system or network in the properties. Any number of property +attributes can appear in a block. +The type attribute is specific to variable blocks. type lists the +values of a discrete variable: + +type discrete[ number-of-values ] { list-of-values }; + +The number-of-values token is a non-negative integer which indicates +how many different values this variable may assume (the size of the +list-of-values). The list-of-values is a sequence of words, each one +the name of a variable value. At this point continuous variables are +not supported (should they be?). +There are attributes that are specific to probability blocks (these +attributes are discussed in the next section): + +table lists a sequence of non-negative real numbers. +default lists a sequence of non-negative real numbers. +the entry attribute, which is not associated with any keyword. +Probability Blocks +Probability blocks are used to define the actual network topology and +conditional probability tables (CPTs). There are two kinds of +probability blocks: +Blocks for standard nodes; that is, nodes for which we have to define +the probabilities for each discrete parent instantiation. +Blocks for noisy functions, like noisy OR, noisy AND, noisy adder, +etc. +An example of a standard probability block is: +probability(GasGauge | Gas, BatteryPower) { + (yes, high) 0.999 0.001; + (yes, low) 0.850 0.150; + (yes, medium) 0.000 1.000; + (no, high) 0.000 1.000; + (no, low) 0.000 1.000; + (no, medium) 0.000 1.000; +} + +As explained before, the symbols "|" and "," are ignored between +tokens so they do not affect the list of variables given after the +keyword probability. The variables however must be enclosed by +parenthesis. +The example above uses the entry attribute, which is different from +the other attributes in that it has no keyword. It simply starts with +an opening parenthesis, and has a list of values for all the +conditioning variables. After the closing parenthesis, a list of +probability values for the first variable is given (note the user must +provide numbers that add to one but this is not mandatory). + +The probability vectors can be listed in any order, since the names in +parentheses uniquely identify the parent instantiation. + +In addition to the entry attribute, the Interchange Format supports +the concept of a default entry. So the above CPT could have been +specified equivalently as: + +probability(GasGauge | Gas, BatteryPower) { + default 0.000 1.000; + (yes, low) 0.850 0.150; + (no, medium) 0.000 1.000; +} + +Note that each number is a separate token, so we can use "," and "|" +between numbers; these symbols are ignored. +Another way to define a probability distribution is through the table +attribute. The body of such attribute is a sequence of non-negative +real numbers, in the counting order of the declared variables (if all +variables were binary, we would say binary counting with least +significant digit in the right). So, for the example above, we could +simply say: + +probability(GasGauge | Gas, BatteryPower) { + table 0.999 0.850 0.0 0.0 0.0 0.0 0.001 0.15 1.0 1.0 1.0 + 1.0; +} + +There are some subtle rules that regulate these declarations. + +If multiple default declarations exist, only the last one is valid. +If multiple table declarations exist, only the last one is valid. +A table can contain more elements than the necessary to specify a +distribution; the excess elements are discarded. +A table can contain less elements than the necessary to specify a +distribution, which is then padded with zeros. +Specified entries override conflicting default and table +declarations. + +Noisy functions are characterized by the property that the probability +vectors for each entry can be derived from the probability vectors of +the parent instantiations. This proposal has not settled yet in a +general format for noisy functions. + +Currently the Interchange Format adopts the suggestion from the BNIF, +noting that all that is necessary to reconstruct a noisy-or/max/sum is +the name of the function and the probability that the child is true +given that each parent singly is true. For example if there are four +parents, each taking values 0 and 1, we need the rows for the +instantiations 1 0 0 0, 0 1 0 0, 0 0 1 0, 0 0 0 1. + +From the BNIF proposal: + +Noisy functions are characterized by the property that the probability +vectors for each combination of conditional variables can be derived +from the probability vectors of the leak parent instantiation and the +parent instantiations in which one and only one parent assumes a value +different from its leak value. Conceptually, the leak parent +instantiation represents the situation in which none of the parents is +causing the child node to be in a abnormal state, and hence the +probability vector associated with the leak instantiation models +influences on the child node that are not explicitly accounted for the +parents. + +Currently, we suggest the use of a property "function" to insert +information about the particular noisy function. For example: + + probability(GasGauge | Gas, BatteryPower) + { + property function max; + (0, 0): 0.999, 0.001; // leak term + (0, 1): 0.850, 0.150; + (0, 2): 0.000, 1.000; + (1, 0): 0.000, 1.000; + } + +Other formats for noisy-functions will be considered for +implementation. + + +-------------------------------------------------------------------------------- + +Examples +Three files are available as examples: +dog-problem.bif, a very simple network based on the discussion at +Charniak, E., Bayesian Networks without Tears, AI Magazine, 1991. +elimbel2.bif, a simple network based on the second example in the +Elimbel system. +car-starts.bif, a somewhat large network contributed by Sreekanth +Nagarajan, based on the automobile belief network that David Heckerman +and Jack Breese presented in the March, 1995 issue of Communications +of the ACM. +Here is a portion of the car-starts.bif network that was originally +given in the BNIF distribution, adapted for the current proposal: +network Internal-Network{ //18 variables and 18 probability +distributions +} + +variable Alternator{ //2 values + type discrete[2] { Ok Faulted }; + property position = (47, 42) ; +} + +variable FanBelt{ //3 values + type discrete[3] { Ok Slipping Broken }; + property position = (154, 42) ; +} + +probability ( BatteryPower Charge BatteryState ) { //3 variable(s) and +8 values + table 1 0 0 0 0 1 1 1 ; +} + +probability ( GasInTank ) { //1 variable(s) and 2 values + table 0.5 0.5 ; +} + + + +-------------------------------------------------------------------------------- + +Playing with the standard +A number of tools have been produced so that users can test and +experiment with the Interchange Format. +A parser for the Interchange Format written in Java. +A syntax checker for the Interchange Format. +Support for the Interchange Format in the Bayesian Networks Editor by +Sreekanth Nagarajan and Bruce D'Ambrosio. +Support for the Interchange Format in the JavaBayes system. +A parser for the Interchange Format +A parser for the Interchange Format was generated using the Jack +parser generator. The parser is generated as a Java program and should +run in any platform which has a Java virtual machine. The complete +specification of the parser in Jack's specification language is given +later. The parser scans an input stream, which can be a file, a socket +or a string, and produces either a ParseError object (signifying a +syntax error occurred) or a BayesNet object. The BayesNet object +contains variables and probability distributions that are distributed +as a Java package called BayesianNetworks. The complete distribution +for the parser is available at +ftp://ftp.cs.cmu.edu/afs/cs/project/lri-3/ftp/outgoing/JavaBayes/InterchangeFormat.tar. +If this address does not work for your setup, then try: + ftp ftp.cs.cmu.edu (login as anonymous) + cd /afs/cs/project/lri-3/ftp/outgoing/JavaBayes/ + binary + get InterchangeFormat.tar + +You have to use the cd command only once; you cannot cd into +intermediate directories. Now use the tar utility to create the +distribution from the InterchangeFormat.tar file. The distribution +should have: +A README file with basic information, a Makefile which allows you to +build the Java classes for the parser in a Unix system. +The directory Examples, with car-starts.bif, dog-problem.bif and +elimbel2.bif. +The directory InterchangeFormat, with the parser source code. The +parser description is in the file InterchangeFormat.jack; all other +files in this directory are generated automatically through the parser +generator. +The directory BayesianNetworks, with the classes and methods the +parser uses to build a BayesNet object. The BayesianNetworks package +defined by these classes is also used in the JavaBayes system; more +information about the particular data structures used in the package +can be found in the JavaBayes web site. +The directory ParseTest, with files Test.html and Test.java, +containing a syntax checker program for the Interchange Format. +The directory Classes (with the subdirectory BayesianNetworks), which +contains all the bytecode files for the Java interpreter. You can +generate these files by running the Java compiler in the source files +provided with the distribution. +A syntax checker for the Interchange Format +The Test.java file that comes with the parser distribution (previous +section) is a syntax checker for the Interchange Format. It can be +used as an application and as an applet. +The bytecodes for the Test program should be in the Classes directory +in the parser distribution. To run it as an application, go to that +directory and run java Test. A frame will be created; you will be +asked to insert the name of a file (with the path to it, if +necessary). Then click the Load button, and the Test program will run +the parser in the file. A BayesNet object will be constructed and +displayed, and some basic checks will be made. If you see the word +"Result" popping out followed by some numbers, the file was +successfully read. Check the console to see the BayesNet object +printed (if you are running an applet, open the Java console window +from your browser). If there were problems with the file, a brief +error message will be displayed in the Test program frame; check the +console to get a more descriptive message. + +To run Test as an applet, you will need a small piece of HTML to call +the program. The file Test.html contains a minimal HTML document which +calls the Test program. + +The Test applet is displayed below. If you have a Java compatible +browser you should see the Test program frame. Type the name of one of +the files (at this point only server side files are allowed; you can +choose between the examples dog-problem.bif, elimbel2.bif and +car-starts.bif. Type one of these names and press the Load button. You +will see messages indicating the files are fetched and read; to +observe the full messages with the BayesNet objects, open the Java +console in your browser. + + + + +Support for the Interchange Format in other systems +Support for the Interchange Format is available in the Bayesian +Networks Editor by Sreekanth Nagarajan and Bruce D'Ambrosio. This +system gives the user a graphical interface for construction of +Bayesian networks and performs inferences through a server connection +(the inference engine is maintained at Oregon State University). For +more information on this system, consult its web site. The current +version can be used either as an applet (in this case the user can +load/save files in the server side) or as an application (in this case +the user can load/save local files in the Interchange Format). +Support for the Interchange Format is also provided in the JavaBayes +system, by Fabio Cozman. This system uses the same graphical interface +by Sreekanth Nagarajan and a Java based inference engine. It runs as +an applet (without load/save operations) and as an application (with +load/save operations). The distribution can generate and read files in +the Interchange Format. + +A more formal description +A more formal description of the proposed Interchange Format is given +here. The notation used by the Jack parser generator is used here. +In the description below, the patterns used by the lexer to define +tokens are very similar to regular expressions used by the Unix regexp +facility. Non-terminals have a parenthesis pair "()" after their +names; terminals are usually capitalized. Some structures that may +appear in expansions are: + + ( e )? : An optional occurrence of e + e1 | e2 | e3 | ... : A choice of e1, e2, e3, etc. + ( e )+ : One or more occurrences of e + ( e )* : Zero or more occurrences of e + ["a"-"z"] matches all lower case letters + ~["\n","\r"] matches any character except the new line characters + + +-------------------------------------------------------------------------------- + +The following patterns are ignored when they appear between tokens: + +" " +"\t" +"\n" +"\r" +"//" (~["\n","\r"])* ("\n"|"\r\n") +"/*" (~["*"])* "*" (~["/"] (~["*"])* "*")* "/" +"," +"|" + + +-------------------------------------------------------------------------------- + +The definition of a word is: + +WORD: LETTER (LETTER | DIGIT)* +LETTER: ["a"-"z","A"-"Z","_","-"] +DIGIT: ["0"-"9"] + + +-------------------------------------------------------------------------------- + +The definition of a non-negative integer number is: + +DECIMAL_LITERAL: ["1"-"9"] (["0"-"9"])* + + + +-------------------------------------------------------------------------------- +The definition of a non-negative real number is: +FLOATING_POINT_LITERAL: (["0"-"9"])+ "." (["0"-"9"])* (EXPONENT)? + | "." (["0"-"9"])+ (EXPONENT)? + | (["0"-"9"])+ (EXPONENT)? +#EXPONENT: ["e","E"] (["+","-"])? (["0"-"9"])+ + + +-------------------------------------------------------------------------------- + +The following words are keywords: + +NETWORK: "network" +VARIABLE: "variable" +PROBABILITY: "probability" +PROPERTY: "property" +VARIABLETYPE: "type" +DISCRETE: "discrete" +DEFAULTVALUE: "default" +TABLEVALUES: "table" + + +-------------------------------------------------------------------------------- + +A property is defined as: + +PROPERTYSTRING: PROPERTY (~[";"])* ";" + + +-------------------------------------------------------------------------------- + +The productions of the grammar are: + +CompilationUnit() : + NetworkDeclaration() + ( VariableDeclaration() | ProbabilityDeclaration() )* + EOF + +NetworkDeclaration() : + NETWORK WORD NetworkContent() + +NetworkContent() : + "{" ( Property() )* "}" + +VariableDeclaration() : + VARIABLE ProbabilityVariableName() VariableContent() + +VariableContent(String name) : + "{" ( Property() | VariableDiscrete() )* "}" + +VariableDiscrete() : + VARIABLETYPE DISCRETE + "[" DECIMAL_LITERAL "]" "{" VariableValuesList() "}" ";" + +void VariableValuesList() : + ProbabilityVariableValue() + ( ProbabilityVariableValue() )* + +ProbabilityVariableValue() : WORD + +ProbabilityDeclaration() : + PROBABILITY ProbabilityVariablesList() ProbabilityContent() + +ProbabilityVariablesList() : + "(" ProbabilityVariableName() ( ProbabilityVariableName() )* ")" + +ProbabilityVariableName() : + +ProbabilityContent() + "{" ( Property() | ProbabilityDefaultEntry() | ProbabilityEntry() + | + ProbabilityTable() )* "}" + +ProbabilityEntry() : + ProbabilityValuesList() FloatingPointList() ";" + +ProbabilityValuesList() : + "(" ProbabilityVariableValue() ( ProbabilityVariableValue() )* + ")" + +ProbabilityDefaultEntry() : + FloatingPointList() ";" + +ProbabilityTable() : + FloatingPointList() ";" + +FloatingPointList() : + FloatingPointToken() ( FloatingPointToken() )* + +FloatingPointToken() : + +Property() : + + + +-------------------------------------------------------------------------------- + +Wish List +Here are some of the comments and thoughts related to this +proposal. This is mostly for easy reference; if you're not interested +in the discussion, please skip this section. + +(Wray Buntine) I'd suggest you have the ability to define vector +constants or probability table constants, and the noisy-or should just +be viewed as some random distribution named "noisy-or" applied to a +random vector that just happens to be probabilities. +(Wray Buntine) For conditional probability tables, don't-cares, i.e. + (yes, no, *): 0.2, 0.8 + (no, *, yes): 0.45, 0.55 + (yes, yes, no): 0.34, 0.66 + default: 0.5, 0.5 + +(Fabio Cozman) We could suppress the keyword "default" and just say: + (yes, no, *): 0.2, 0.8 + (no, *, yes): 0.45, 0.55 + (yes, yes, no): 0.34, 0.66 + (*,*,*): 0.5, 0.5 + +(Sreekanth Nagarajan) Can the type info be associated with the +variable declaration itself? For eg, + variable continuous Flow { + } + + +-------------------------------------------------------------------------------- + +-------------------------------------------------------------------------------- + +-------------------------------------------------------------------------------- + +This page is maintained by Fabio Cozman [Send Mail?] diff --git a/code/biflang/draft_biflang.jl b/code/biflang/draft_biflang.jl new file mode 100644 index 0000000..3209664 --- /dev/null +++ b/code/biflang/draft_biflang.jl @@ -0,0 +1,110 @@ +include("BIFLang.jl") + +using .BIFLang + +src = read("asia2.bif", String) + +# src = "network unknown { } +# variable asia { +# type discrete [ 2 ] { yes, no }; +# } +# probability ( either | lung, tub ) { +# (yes, yes) 1.0, 0.0; +# (no, yes) 1.0, 0.0; +# (yes, no) 1.0, 0.0; +# (no, no) 0.0, 1.0; +# }" +expr = parse_bif(src) +# println(expr) + +children(s::Any) = [] +function children(node::BIFLang.Node) + node.args +end + +function descendants(node::BIFLang.Node) + unvisited = filter(n -> isa(n, BIFLang.Node), children(node)) + visited = [] + while length(unvisited) > 0 + h = popfirst!(unvisited) + push!(visited, h) + fresh = filter(n -> isa(n, BIFLang.Node), children(h)) + union!(unvisited, setdiff(fresh, visited)) + end + return visited +end + +is_leaf(x::Any) = true +is_leaf(x::BIFLang.Node) = false +is_leaf(x::BIFLang.Property) = true +is_leaf(x::BIFLang.Variable) = true +is_leaf(x::BIFLang.Values) = true +is_leaf(x::BIFLang.Key) = true +is_leaf(x::BIFLang.Distribution) = true +branches(x::Any) = [] +function branches(node::BIFLang.Node) + if is_leaf(node) + return [[node.args]] + else + cs = children(node) + return [ [ci, bi...] for ci in cs for bi in branches(ci) ] + end +end + +# println.(filter(n -> isa(n, Variable), children(expr))) +# println.(filter(n -> isa(n, Values), descendants(expr))) +# println.(filter(n -> 2 ∈ n.args, descendants(expr))) +for bs in branches(expr) + println.(bs) + println() +end +#= + +Unit( + Network(asia2,Properties( + Property(what: ever), + Property(goes: well), + Property(with_me))), + VariableDeclaration(Variable(asia), + Discrete(2,Values(yes,no))), + VariableDeclaration(Variable(tub), + Discrete(2,Values(yes,no))), + VariableDeclaration(Variable(smoke), + Discrete(2,Values(yes,no))), + VariableDeclaration(Variable(lung), + Discrete(2,Values(yes,no))), + VariableDeclaration(Variable(bronc), + Discrete(2,Values(yes,no))), + VariableDeclaration(Variable(either), + Discrete(2,Values(yes,no))), + VariableDeclaration(Variable(xray), + Discrete(2,Values(yes,no))), + VariableDeclaration(Variable(dysp), + Discrete(2,Values(yes,no))), + ProbabilityDeclaration(Variable(asia), + Distribution(0.01,0.99)), + ProbabilityDeclaration(Variable(tub),Parents(Variable(asia)), + Entry(Key(yes),Distribution(0.05,0.95)), + Entry(Key(no),Distribution(0.01,0.99))), + ProbabilityDeclaration(Variable(smoke), + Distribution(0.5,0.5)), + ProbabilityDeclaration(Variable(lung),Parents(Variable(smoke)), + Entry(Key(yes),Distribution(0.1,0.9)), + Entry(Key(no),Distribution(0.01,0.99))), + ProbabilityDeclaration(Variable(bronc),Parents(Variable(smoke)), + Entry(Key(yes),Distribution(0.6,0.4)), + Entry(Key(no),Distribution(0.3,0.7))), + ProbabilityDeclaration(Variable(either),Parents(Variable(lung),Variable(tub)), + Entry(Key(yes,yes),Distribution(1.0,0.0)), + Entry(Key(no,yes),Distribution(1.0,0.0)), + Entry(Key(yes,no),Distribution(1.0,0.0)), + Entry(Key(no,no),Distribution(0.0,1.0))), + ProbabilityDeclaration(Variable(xray),Parents(Variable(either)), + Entry(Key(yes),Distribution(0.98,0.02)), + Entry(Key(no),Distribution(0.05,0.95))), + ProbabilityDeclaration(Variable(dysp),Parents(Variable(bronc),Variable(either)), + Entry(Key(yes,yes),Distribution(0.9,0.1)), + Entry(Key(no,yes),Distribution(0.7,0.3)), + Entry(Key(yes,no),Distribution(0.8,0.2)), + Entry(Key(no,no),Distribution(0.1,0.9)))) +=# \ No newline at end of file diff --git a/code/biflang/notes.md b/code/biflang/notes.md new file mode 100644 index 0000000..f766ddd --- /dev/null +++ b/code/biflang/notes.md @@ -0,0 +1,142 @@ + + +A more formal description +A more formal description of the proposed Interchange Format is given +here. The notation used by the Jack parser generator is used here. +In the description below, the patterns used by the lexer to define +tokens are very similar to regular expressions used by the Unix regexp +facility. Non-terminals have a parenthesis pair "()" after their +names; terminals are usually capitalized. Some structures that may +appear in expansions are: + + ( e )? : An optional occurrence of e + e1 | e2 | e3 | ... : A choice of e1, e2, e3, etc. + ( e )+ : One or more occurrences of e + ( e )* : Zero or more occurrences of e + ["a"-"z"] matches all lower case letters + ~["\n","\r"] matches any character except the new line characters + + +-------------------------------------------------------------------------------- + +The following patterns are ignored when they appear between tokens: + +" " +"\t" +"\n" +"\r" +"//" (~["\n","\r"])* ("\n"|"\r\n") +"/*" (~["*"])* "*" (~["/"] (~["*"])* "*")* "/" +"," +"|" + + +-------------------------------------------------------------------------------- + +The definition of a word is: + +WORD: LETTER (LETTER | DIGIT)* +LETTER: ["a"-"z","A"-"Z","_","-"] +DIGIT: ["0"-"9"] + + +-------------------------------------------------------------------------------- + +The definition of a non-negative integer number is: + +DECIMAL_LITERAL: ["1"-"9"] (["0"-"9"])* + + + +-------------------------------------------------------------------------------- +The definition of a non-negative real number is: +FLOATING_POINT_LITERAL: (["0"-"9"])+ "." (["0"-"9"])* (EXPONENT)? + | "." (["0"-"9"])+ (EXPONENT)? + | (["0"-"9"])+ (EXPONENT)? +#EXPONENT: ["e","E"] (["+","-"])? (["0"-"9"])+ + + +-------------------------------------------------------------------------------- + +The following words are keywords: + +NETWORK: "network" +VARIABLE: "variable" +PROBABILITY: "probability" +PROPERTY: "property" +VARIABLETYPE: "type" +DISCRETE: "discrete" +DEFAULTVALUE: "default" +TABLEVALUES: "table" + + +-------------------------------------------------------------------------------- + +A property is defined as: + +PROPERTYSTRING: PROPERTY (~[";"])* ";" + + +-------------------------------------------------------------------------------- + +The productions of the grammar are: + +CompilationUnit() : + NetworkDeclaration() + ( VariableDeclaration() | ProbabilityDeclaration() )* + EOF + +NetworkDeclaration() : + NETWORK WORD NetworkContent() + +NetworkContent() : + "{" ( Property() )* "}" + +VariableDeclaration() : + VARIABLE ProbabilityVariableName() VariableContent() + +VariableContent(String name) : + "{" ( Property() | VariableDiscrete() )* "}" + +VariableDiscrete() : + VARIABLETYPE DISCRETE + "[" DECIMAL_LITERAL "]" "{" VariableValuesList() "}" ";" + +void VariableValuesList() : + ProbabilityVariableValue() + ( ProbabilityVariableValue() )* + +ProbabilityVariableValue() : WORD + +ProbabilityDeclaration() : + PROBABILITY ProbabilityVariablesList() ProbabilityContent() + +ProbabilityVariablesList() : + "(" ProbabilityVariableName() ( ProbabilityVariableName() )* ")" + +ProbabilityVariableName() : + +ProbabilityContent() + "{" ( Property() | ProbabilityDefaultEntry() | ProbabilityEntry() + | + ProbabilityTable() )* "}" + +ProbabilityEntry() : + ProbabilityValuesList() FloatingPointList() ";" + +ProbabilityValuesList() : + "(" ProbabilityVariableValue() ( ProbabilityVariableValue() )* + ")" + +ProbabilityDefaultEntry() : + FloatingPointList() ";" + +ProbabilityTable() : + FloatingPointList() ";" + +FloatingPointList() : + FloatingPointToken() ( FloatingPointToken() )* + +FloatingPointToken() : + +Property() : \ No newline at end of file diff --git a/code/drafts/.ipynb_checkpoints/EventLattice-checkpoint.ipynb b/code/drafts/.ipynb_checkpoints/EventLattice-checkpoint.ipynb new file mode 100644 index 0000000..829cb1d --- /dev/null +++ b/code/drafts/.ipynb_checkpoints/EventLattice-checkpoint.ipynb @@ -0,0 +1,113 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "id": "57fc5921-9d6b-4b43-a8f6-743a03650d63", + "metadata": {}, + "outputs": [], + "source": [ + "import event_lattice as el" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "00f0eb68", + "metadata": {}, + "outputs": [], + "source": [ + "def zoom_event(event_str, lattice, lower_op=el.sum_op, upper_op=el.prod_op):\n", + " event = el.Event.from_str(event_str)\n", + " event_class = lattice.event_class(event)\n", + " propagated = lattice.propagated_value(\n", + " event, lower_op=lower_op, upper_op=upper_op)\n", + "\n", + " print(\n", + " f\"Event: {event}\\n\\tClass: {event_class} \\n\\tValue: {propagated}\")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "cdd8c6d6", + "metadata": {}, + "outputs": [], + "source": [ + "smodels = el.Lattice.parse({\n", + " \"A\": 2,\n", + " \"ab\": 3,\n", + " \"ac\": 5\n", + "})\n", + "\n", + "lattice = el.Lattice(smodels)\n", + "\n", + "print(lattice)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2b445339", + "metadata": {}, + "outputs": [], + "source": [ + "zoom_event(\"abc\", lattice)\n", + "zoom_event(\"a\", lattice)\n", + "zoom_event(\"b\", lattice)\n", + "zoom_event(\"bc\", lattice)\n", + "zoom_event(\"ac\", lattice)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f1b85255", + "metadata": {}, + "outputs": [], + "source": [ + "from itertools import *\n", + "\n", + "lits = lattice.literals()\n", + "for len_lit in range(len(lits)+1):\n", + " events = list(\"\".join(c) for c in combinations(lits, len_lit))\n", + " for event in events:\n", + " zoom_event(event, lattice)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "07973a47", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3.9.13 ('base')", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.13" + }, + "vscode": { + "interpreter": { + "hash": "a59afa236e16843183c59a167f072b6fa0409044b3c4938e82ac98aad91bf217" + } + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/code/drafts/.ipynb_checkpoints/Untitled-checkpoint.ipynb b/code/drafts/.ipynb_checkpoints/Untitled-checkpoint.ipynb new file mode 100644 index 0000000..363fcab --- /dev/null +++ b/code/drafts/.ipynb_checkpoints/Untitled-checkpoint.ipynb @@ -0,0 +1,6 @@ +{ + "cells": [], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/code/drafts/.ipynb_checkpoints/Untitled1-checkpoint.ipynb b/code/drafts/.ipynb_checkpoints/Untitled1-checkpoint.ipynb new file mode 100644 index 0000000..99b15c0 --- /dev/null +++ b/code/drafts/.ipynb_checkpoints/Untitled1-checkpoint.ipynb @@ -0,0 +1,164 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 2, + "id": "5e6ea3be-67e9-43ac-bed4-abbae7c4352b", + "metadata": { + "tags": [] + }, + "outputs": [ + { + "data": { + "image/png": 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DataFrames\n", + "using Latexify\n", + "using Plots\n", + "using StatsPlots\n", + "\n", + "data = CSV.read(\"sample.csv\", DataFrame)\n", + "groups = groupby(data, :event)\n", + "totals = combine(groups, :event => length => :count)\n", + "\n", + "x = -π:0.1:π\n", + "y = sin.(x)\n", + "plot(totals.count)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e1c270c9-4270-414b-acce-33a3cc3b85c4", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.9.3", + "language": "julia", + "name": "julia-1.9" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/code/drafts/.ipynb_checkpoints/sample_analysis-checkpoint.jl b/code/drafts/.ipynb_checkpoints/sample_analysis-checkpoint.jl new file mode 100644 index 0000000..a3133a3 --- /dev/null +++ b/code/drafts/.ipynb_checkpoints/sample_analysis-checkpoint.jl @@ -0,0 +1,15 @@ +using CSV +using DataFrames +using Latexify +using Plots +using StatsPlots + +data = CSV.read("sample.csv", DataFrame) +groups = groupby(data, :event) +totals = combine(groups, :event => length => :count) + +x = -π:0.1:π +y = sin.(x) +@df totals plot(:event, :count) +println("Press ENTER to exit.") +readline() \ No newline at end of file diff --git a/code/drafts/.ipynb_checkpoints/teste-checkpoint.ipynb b/code/drafts/.ipynb_checkpoints/teste-checkpoint.ipynb new file mode 100644 index 0000000..c62a6aa --- /dev/null +++ b/code/drafts/.ipynb_checkpoints/teste-checkpoint.ipynb @@ -0,0 +1,116 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "4" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "2+2" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "x = np.linspace(-6, 6)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "y = np.sin(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.plot(x,y)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3.9.15 ('base')", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.15" + }, + "orig_nbformat": 4, + "vscode": { + "interpreter": { + "hash": "a59afa236e16843183c59a167f072b6fa0409044b3c4938e82ac98aad91bf217" + } + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/code/drafts/EventLattice.ipynb b/code/drafts/EventLattice.ipynb new file mode 100644 index 0000000..8cd7d9e --- /dev/null +++ b/code/drafts/EventLattice.ipynb @@ -0,0 +1,154 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 4, + "id": "57fc5921-9d6b-4b43-a8f6-743a03650d63", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The autoreload extension is already loaded. To reload it, use:\n", + " %reload_ext autoreload\n" + ] + } + ], + "source": [ + "%load_ext autoreload\n", + "%autoreload 1\n", + "%aimport event_lattice" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "00f0eb68", + "metadata": {}, + "outputs": [], + "source": [ + "def zoom_event(event_str, lattice):\n", + " event = event_lattice.Event.from_str(event_str)\n", + " event_class = lattice.event_class(event)\n", + " propagated = lattice.extended_value(\n", + " event)\n", + "\n", + " print(\n", + " f\"Event: {event}\\n\\tClass: {event_class} \\n\\tValue: {propagated}\")" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "cdd8c6d6", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "{\n", + "\t'stable_models': {\n", + "\t\t A: 2,\n", + "\t\tab: 3,\n", + "\t\tac: 5 \n", + "\t}\n", + "\t'literals': { A,B,C,a,b,c } \n", + "}\n" + ] + } + ], + "source": [ + "smodels = event_lattice.Lattice.parse({\n", + " \"A\": 2,\n", + " \"ab\": 3,\n", + " \"ac\": 5\n", + "})\n", + "\n", + "lattice = event_lattice.Lattice(smodels)\n", + "\n", + "print(lattice)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "2b445339", + "metadata": {}, + "outputs": [ + { + "ename": "TypeError", + "evalue": "__init__() missing 1 required positional argument: 'lattice'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m/tmp/ipykernel_361713/2581811254.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mzoom_event\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"abc\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlattice\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 2\u001b[0m \u001b[0mzoom_event\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"a\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlattice\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0mzoom_event\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"b\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlattice\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mzoom_event\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"bc\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlattice\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0mzoom_event\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"ac\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlattice\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m/tmp/ipykernel_361713/1675915232.py\u001b[0m in \u001b[0;36mzoom_event\u001b[0;34m(event_str, lattice)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mzoom_event\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mevent_str\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlattice\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0mevent\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mevent_lattice\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mEvent\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfrom_str\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mevent_str\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mevent_class\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlattice\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mevent_class\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mevent\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 4\u001b[0m propagated = lattice.extended_value(\n\u001b[1;32m 5\u001b[0m event)\n", + "\u001b[0;32m~/sci/projetos/zugzwang/code/python/event_lattice.py\u001b[0m in \u001b[0;36mevent_class\u001b[0;34m(self, event)\u001b[0m\n\u001b[1;32m 146\u001b[0m \u001b[0;34m@\u001b[0m\u001b[0mcache\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 147\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mevent_class\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mevent\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 148\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mEventsClass\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstable_core\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mevent\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 149\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 150\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mrelated\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mTypeError\u001b[0m: __init__() missing 1 required positional argument: 'lattice'" + ] + } + ], + "source": [ + "zoom_event(\"abc\", lattice)\n", + "zoom_event(\"a\", lattice)\n", + "zoom_event(\"b\", lattice)\n", + "zoom_event(\"bc\", lattice)\n", + "zoom_event(\"ac\", lattice)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f1b85255", + "metadata": {}, + "outputs": [], + "source": [ + "from itertools import *\n", + "\n", + "lits = lattice.literals()\n", + "events = []\n", + "for len_lit in range(len(lits)+1):\n", + " events = events + list(\"\".join(c) for c in combinations(lits, len_lit))\n", + "for event in events:\n", + " zoom_event(event, lattice)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "07973a47", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.15" + }, + "vscode": { + "interpreter": { + "hash": "a59afa236e16843183c59a167f072b6fa0409044b3c4938e82ac98aad91bf217" + } + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/code/drafts/Untitled.ipynb b/code/drafts/Untitled.ipynb new file mode 100644 index 0000000..67adba8 --- /dev/null +++ b/code/drafts/Untitled.ipynb @@ -0,0 +1,50 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.002516442" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "a = 0.001; ca = 1 - a;\n", + "b = 0.002; cb = 1 - b;\n", + "\n", + "0.95 * a * b + 0.94 * a* cb + 0.29 * ca * b + 0.001 * ca * cb" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "665bdd88", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.9.3", + "language": "julia", + "name": "julia-1.9" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/code/drafts/Untitled1.ipynb b/code/drafts/Untitled1.ipynb new file mode 100644 index 0000000..a28c4e4 --- /dev/null +++ b/code/drafts/Untitled1.ipynb @@ -0,0 +1,160 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 5, + "id": "5e6ea3be-67e9-43ac-bed4-abbae7c4352b", + "metadata": { + "tags": [] + }, + "outputs": [ + { + "data": { + "image/png": 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"using Plots\n", + "using StatsPlots\n", + "\n", + "data = CSV.read(\"sample.csv\", DataFrame)\n", + "groups = groupby(data, :event)\n", + "totals = combine(groups, :event => length => :count)\n", + "\n", + "x = -π:0.1:π\n", + "y = sin.(x)\n", + "plot(totals.count, labels=totals.event)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e1c270c9-4270-414b-acce-33a3cc3b85c4", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.9.3", + "language": "julia", + "name": "julia-1.9" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/code/drafts/__init__.py b/code/drafts/__init__.py new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/code/drafts/__init__.py diff --git a/code/drafts/__pycache__/event_lattice.cpython-39.pyc b/code/drafts/__pycache__/event_lattice.cpython-39.pyc new file mode 100644 index 0000000..a72fa05 Binary files /dev/null and b/code/drafts/__pycache__/event_lattice.cpython-39.pyc differ diff --git a/code/drafts/algebra.py b/code/drafts/algebra.py new file mode 100644 index 0000000..2cae6bb --- /dev/null +++ b/code/drafts/algebra.py @@ -0,0 +1,30 @@ +from itertools import combinations, product + +def fmt(expr): + """Doc string""" + return ",".join(f"{x:>2}" for x in expr) + +def c(expr): + """Doc string""" + def litcomp(x): + if x == "⊤": + return "⊥" + elif x == "⊥": + return "⊤" + elif x[0] == "¬": + return x[1:] + else: + return f"¬{x}" + return [litcomp(x) for x in expr] + +def domain(symbols, unary="¬"): + """Doc string""" + atoms = list(symbols) + literals = [ + [f"{u}{a}" for u in unary] + + [a, "⊤", "⊥"] for a in atoms ] + return product(*literals) + +d = sorted(domain("abc")) +for x in d: + print(f"{fmt(x)} | {fmt(c(x))}") diff --git a/code/drafts/api_01.py b/code/drafts/api_01.py new file mode 100644 index 0000000..f119ed2 --- /dev/null +++ b/code/drafts/api_01.py @@ -0,0 +1,26 @@ +from clingo.symbol import Number +from clingo.control import Control + +class Context: + def inc(self, x): + return Number(x.number + 1) + + def seq(self, x, y): + return [x, y] + +def on_model(m): + print(m) + +ctl = Control() +ctl.add("base", [], """\ +p(@inc(10)). +q(@seq(1,2)). +""") + +ctl.ground( + [("base", [])], + context=Context()) + +s = ctl.solve(on_model=on_model) + +print(s) \ No newline at end of file diff --git a/code/drafts/arithfunclang.jl b/code/drafts/arithfunclang.jl new file mode 100644 index 0000000..c088dae --- /dev/null +++ b/code/drafts/arithfunclang.jl @@ -0,0 +1,113 @@ +using ParserCombinator + +# the AST nodes we will construct, with evaluation via calc() + +abstract type Node end +Base.:(==)(n1::Node, n2::Node) = n1.val == n2.val +calc(n::Float64) = n +# repr(n::Float64) = string(n) + +struct Const <: Node val end +calc(c::Const) = c.val +# repr(c::Const) = repr(c.val) + +struct Inv <: Node val end +calc(i::Inv) = 1.0/calc(i.val) +# repr(i::Inv) = "1.0/$(repr(i.val))" + +struct Prd <: Node val end +calc(p::Prd) = Base.prod(map(calc, p.val)) +# repr(p::Prd) = join(map(repr, p.val), "×") + +struct Neg <: Node val end +calc(n::Neg) = -calc(n.val) +# repr(n::Neg) = "-$(repr(n.val))" + +struct Sum <: Node val end +calc(s::Sum) = Base.sum(map(calc, s.val)) +# repr(s::Sum) = join(map(repr, s.val), "+") + +struct TermSequence <: Node val end +calc(ts::TermSequence) = map(calc, ts.val) +# repr(ts::TermSequence) = join(map(repr, ts.val), ",") + +struct Functor <: Node val end +calc(f::Functor) = nothing +# repr(f::Functor) = "$(ts.val)" + +struct Function <: Node val end +calc(f::Function) = Const(0.0) +# repr(f::Function) = "$(repr(f.val[1]))($(repr(f.val[2])))" + +# the grammar (the combinators!) + +#= + sum ::= prd (add | sub)* + add ::= "+" prd + sub ::= "-" prd + + prd ::= neg (mul | div)* +=# +@with_names begin + spc = Drop(Star(Space())) + @with_pre spc begin + sum = Delayed() + + term_sequence = StarList(sum, E",") |> TermSequence + functor = p"[a-z][a-zA-Z0-9_]*" > Functor + func = functor + E"(" + term_sequence + E")" |> Function + + val = E"(" + sum + E")" | (PFloat64() > Const) + + + neg = Delayed() # allow multiple (or no) negations (eg ---3) + neg.matcher = func | val | (E"-" + neg > Neg) + + mul = E"*" + neg + div = E"/" + neg > Inv + prd = neg + Star(mul | div) |> Prd + + add = E"+" + prd + sub = E"-" + prd > Neg + sum.matcher = prd + Star(add | sub) |> Sum + + arith_lang = (func | val | neg | prd | sum) + Eos() + func_lang = func + Eos() + tl_lang = term_sequence + Eos() + end +end + + +# and test + +source = "f(1,2,3+4)*1+g(5,10,h(100,3))" +grammar = arith_lang + +debug, task = make(Debug, source, grammar; delegate=NoCache) +try + expr = once(task) + println("---\n\"$source\"\tparses to\t\"$(expr)\"\n---") +catch ParserException + println("Error parsing after \"$(source[1:debug.max_iter - 1])\" and before \"$(source[debug.max_iter:end])\".\n") + + #println(grammar) + parse_dbg(source, Trace(grammar)) +end + +x = Any[ + Sum( + Any[ + Prd( + Any[ + Const(2.0) + ] + ), + Prd( + Any[ + Const(3.0), + Const(4.0) + ] + ) + ] + ) + ] \ No newline at end of file diff --git a/code/drafts/arithlang.jl b/code/drafts/arithlang.jl new file mode 100644 index 0000000..dafb29d --- /dev/null +++ b/code/drafts/arithlang.jl @@ -0,0 +1,56 @@ +using ParserCombinator + + +# the AST nodes we will construct, with evaluation via calc() + +abstract type Node end +Base.:(==)(n1::Node, n2::Node) = n1.val == n2.val +calc(n::Float64) = n + +struct Const <: Node val end +calc(c::Const) = c.val + +struct Inv<:Node val end +calc(i::Inv) = 1.0/calc(i.val) + +struct Prd<:Node val end +calc(p::Prd) = Base.prod(map(calc, p.val)) + +struct Neg<:Node val end +calc(n::Neg) = -calc(n.val) + +struct Sum<:Node val end +calc(s::Sum) = Base.sum(map(calc, s.val)) + + +# the grammar (the combinators!) + +spc = Drop(Star(Space())) +@with_pre spc begin + sum = Delayed() + val = E"(" + sum + E")" | (PFloat64() > Const) + + neg = Delayed() # allow multiple (or no) negations (eg ---3) + neg.matcher = val | (E"-" + neg > Neg) + + mul = E"*" + neg + div = E"/" + neg > Inv + prd = neg + (mul | div)[0:end] |> Prd + + add = E"+" + prd + sub = E"-" + prd > Neg + sum.matcher = prd + (add | sub)[0:end] |> Sum + + all = sum + Eos() +end + + +# and test + +prog = "1+2*3/4" +expr = parse_one(prog, all)[1] + +# this prints 2.5 +value = calc(expr) + +println("$prog → $expr = $value") \ No newline at end of file diff --git a/code/drafts/drafts.ipynb b/code/drafts/drafts.ipynb new file mode 100644 index 0000000..002dba9 --- /dev/null +++ b/code/drafts/drafts.ipynb @@ -0,0 +1,590 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "44" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# A\n", + "sum([6, 6, 1, 5, 3, 6, 3, 9, 5])" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "31" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# indep\n", + "sum([5, 2, 2, 3, 4, 3, 3, 5, 4])" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "sum([2,4,2])" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "10-element Vector{Int64}:\n", + " 4\n", + " 3\n", + " 3\n", + " 2\n", + " 5\n", + " 1\n", + " 4\n", + " 2\n", + " 5\n", + " 4" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "rand(1:5, 10)" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "154//75" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "(88*7) // (100*3)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.5366666666666666" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "float((7 *23)//(300))" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "ename": "ErrorException", + "evalue": "syntax: invalid identifier name \"?\"", + "output_type": "error", + "traceback": [ + "syntax: invalid identifier name \"?\"\n", + "\n", + "Stacktrace:\n", + " [1] top-level scope\n", + " @ ~/sci/projetos/zugzwang/code/drafts/drafts.ipynb:1" + ] + } + ], + "source": [ + "?Rational" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-61//100" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "1-(21*23)//300" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "1000" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "35+644+68+211+7+35" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "391//750" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "θ = (68 * 23)//(3000)\n" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.5213333333333333" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "float(θ)" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "4853//3000" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "θ_1 = (211 * 23) // 3000" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "using StatsBase" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [], + "source": [ + "ds = 0.001 .* [0, 35, 644, 68, 211, 0, 0, 7, 35];\n", + "de = (1/23) .* [0, 0, 7, 3*θ, 3*(1-θ), 0, 0, 3, 10];" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.18670312228405667" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "rmsd(ds, de)" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.5213333333333333" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "float(θ)" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "3.1029411764705883" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "211/68" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.3222748815165877" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "68/211" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "7.666666666666667" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "23/3" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ab: 253//500, 0.506\n", + "ac: -2543//3000, -0.8476666666666667\n" + ] + } + ], + "source": [ + "n_ab = 66\n", + "n_ac = 241\n", + "\n", + "t_ab = (23*n_ab) // 3000\n", + "println(\"ab: $t_ab, $(float(t_ab))\")\n", + "\n", + "t_ac = 1 - (23 * n_ac // 3000) \n", + "println(\"ac: $t_ac, $(float(t_ac))\")" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a: 0.03402, b: -0.03298, c: 0.30903\n" + ] + }, + { + "data": { + "image/png": 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+ "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/html": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "using Plots\n", + "\n", + "t = 0:0.005:1\n", + "a = 0.03402\n", + "b = -0.03298\n", + "c = 0.30903\n", + "e = (xi -> a*(xi^2) + b*xi + c).(t)\n", + "\n", + "println(\"a: $a, b: $b, c: $c\")\n", + "plot(t,e)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.48471487360376253" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "0.03298/(0.03402 * 2)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "1.2167205406993828" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "0.08281/(0.03403 * 2)" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "b: -0.08280907372400756; 2a: 0.06805293005671077; argmin: 1.2168333333333334 => 0.30699680718336486\n" + ] + } + ], + "source": [ + "a = 18 /529\n", + "b = -21903/264500\n", + "c = 188207311/529000000\n", + "\n", + "err(t) = c + b*t + a*(t^2)\n", + "m = -b / (2 * a)\n", + "\n", + "println(\"b: $b; 2a: $(2*a); argmin: $m => $(err(1))\")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.9.3", + "language": "julia", + "name": "julia-1.9" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + }, + "orig_nbformat": 4 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/code/drafts/drafts.jl b/code/drafts/drafts.jl new file mode 100644 index 0000000..ec82458 --- /dev/null +++ b/code/drafts/drafts.jl @@ -0,0 +1,67 @@ +using Symbolics +using Latexify + +# # γ = 0.5 +# no_s = [ +# 0 , +# 23 , +# 614 , +# 165 , +# 169 , +# 0 , +# 0 , +# 4 , +# 25 , +# ] + +# γ = 0.8 +no_s = [ + 0 , + 28 , + 632 , + 246 , + 59 , + 0 , + 0 , + 5 , + 27 , +] +pr_s = (x -> x // 1000).(no_s) + +@variables θ +num_e = [ + 0 , + 0 , + 7 , + 3 * θ , + 3 * (1 - θ) , + 0 , + 0 , + 3 , + 10 , +] +pr_e = (x -> x // 23).(num_e) + +target = expand(sum( (x -> x^2).(pr_s - pr_e) )) +println(latexify(target)) + +# using Plots +# g(t) = (20869963/66125000) + (477/52900)*t + (18/529)*(t^2) +# t = 0:0.1:1 +# plot(t, g.(t)) + + + +function solve2(a, b, c) + delta = sqrt(Complex(b^2 - 4 * a * c)) + return ( (-b - delta)/(2*a), (-b + delta)/(2*a) ) +end + +# # g = target ~ 0 + +a = float(18//529) +b = -float(21903//264500) +c = float(188207311//529000000) + +println("a: $a b: $b c: $c") +println("̂θ = $(-b/a)") diff --git a/code/drafts/event_lattice.py b/code/drafts/event_lattice.py new file mode 100644 index 0000000..899d260 --- /dev/null +++ b/code/drafts/event_lattice.py @@ -0,0 +1,301 @@ +import math +from functools import cache +from itertools import accumulate, combinations, chain, groupby +import operator + + + +def uniform_op(x): + n = len(list(x)) + return 1.0 if n == 0 else 1.0/n + + +def max_op(x): + return max(x) + + +def min_op(x): + return min(x) + + +def sum_op(x): + return sum(x) + + +def stableprod_op(x): + log_x = map(math.log, x) + return math.exp(sum(log_x)) + + +def prod_op(x): + return list(accumulate(x, func=lambda a,b: a*b))[-1] + + +class Event: + """Events. + + An event is a set of literals - atoms and negated atoms. + + The convention is that atoms are represented by lower case single letters + and a negated atom by upper case single letters. + """ + + @staticmethod + def _parse(text): + return frozenset(text) + + @staticmethod + def parse(text): + """Convert a string to an event. + + Each letter in the string represents a literal. + """ + return Event(Event._parse(text)) + + + def __init__(self, literals): + """Instantiate from a (frozen) set of literals. + For example: e = Event(frozenset("abc")).""" + self._literals = frozenset(literals) + + + def literals(self): + return self._literals + + + def __iter__(self): + return self._literals.__iter__() + + @cache + def is_consistent(self): + """True if this event is consistent.""" + return all(x.swapcase() not in self._literals for x in self._literals) + + + def co(self): + """Negation of this event. + + Negation is case based: A = not a; a = not A.""" + return Event(x.swapcase() for x in self._literals) + + def invert(self): + """Negation of this event. + + See the method "co" + """ + return self.co() + + def __repr__(self) -> str: + return ''.join(str(x) for x in sorted(self._literals)) if len(self._literals) > 0 else '0' + + def latex(self): + """LaTeX representation of this even. + + Negation is represented by overline and the empty event by + + """ + return ''.join( + (str(x) if x.islower() else f"\co{{{x.lower()}}}") \ + for x in sorted(self._literals) + ) if len(self._literals) > 0 else "\set{}" + + def __hash__(self) -> int: + return self._literals.__hash__() + + + def __eq__(self, other): + """Event equality test.""" + return self._literals.__eq__(other._literals) + + def __or__(self, other): + """Event union operation.""" + return Event(self._literals | other._literals) + + def __le__(self, other): + """Event subset test.""" + return self._literals.__le__(other._literals) + + + def __lt__(self, other): + """Event strict subset test.""" + return self._literals.__lt__(other._literals) + + + def __ne__(self, other): + """Event not-equal test.""" + return self._literals.__ne__(other._literals) + + + def __ge__(self, other): + """Event superset test.""" + return self._literals.__ge__(other._literals) + + + def __gt__(self, other): + """Event strict superset test.""" + return self._literals.__gt__(other._literals) + + +class Lattice: + + @staticmethod + def parse(d): + """Input stable models. + + The input format is a dictionary associating a stable model in string form to an weight. + + For example: + + input_dict = { + "A": 0.3, + "ab": 0.2, + "ac": 0.5 + } + smodels = Lattice.parse(input_dict) + """ + result = dict() + for k, v in d.items(): + key = Event.parse(k) + result[key] = v + return result + + + @staticmethod + def close_literals(events): + """Closed set of literals entailed by a set of events. + + Includes the literals in the set of events and any missing negation.""" + base_lits = list(accumulate(events, func=operator.or_))[-1] + lits = set() + for x in base_lits.literals(): + lits.add(x) + lits.add(x.swapcase()) + return sorted(lits) + + def __init__(self, smodels_dict): + """Create an Events lattice.""" + self._smodels = smodels_dict + self._literals = Lattice.close_literals(self._smodels.keys()) + + def literals(self): + """The literals in this lattice.""" + return self._literals + + @cache + def stable_models(self): + """The stable models that generate this lattice.""" + return self._smodels.keys() + + #@cache + def events(self): + """All the events of this lattice.""" + return chain.from_iterable(map(Event, combinations(self._literals, r)) for r in range(len(self._literals)+1)) + + @cache + def stable_core(self, event): + """The stable core of an event in this lattice.""" + return set(filter(lambda sm: sm <= event or event <= sm, self.stable_models())) + + # @cache + # def event_class(self, event): + # """The equivalence class of an event.""" + # return EventsClass(self.stable_core(event), self) + + @cache + def classes(self): + """The classes of this lattice. + + Each class is presented as a key:value pair where the "key" is the stable core of the elements in "value".""" + map_ev_classes = [(e, tuple(self.stable_core(e))) for e in self.events() if e.is_consistent()] + groups = dict() + for e,c in map_ev_classes: + if c in groups.keys(): + groups[c].add(e) + else: + groups[c] = set([e]) + inconsistent = list(e for e in self.events() if not e.is_consistent()) + inconsistent_repr = inconsistent[0] + groups[(inconsistent_repr,)] = set(inconsistent) + return groups + + + def related(self, u, v): + """Tests if two events are related.""" + u_consistent = u.is_consistent() + v_consistent = v.is_consistent() + if u_consistent and (u_consistent == v_consistent): + return self.stable_core(u) == self.stable_core(v) + else: + return u_consistent == v_consistent + + def extended_value(self, event:Event, + op=prod_op): + """TODO: well...""" + value = 0 + # + # INCONSISTENT EVENTS + # + if not event.is_consistent(): + return value + # + # CONSISTENT EVENTS + # + score = self.stable_core(event) + len_score = len(score) + # CONSISTENT, INDEPENDENT + if len_score == 0: + value = 0 + elif len_score == 1: + value = self._smodels[score[0]] + else: + value = op(map(lambda sm: self._smodels[sm], score)) + + return value + + def __repr__(self): + smodels_repr = ',\n\t\t'.join(f"{k}: {v:<}" for k,v in self._smodels.items()) + lits_repr = ','.join(sorted(self._literals)) + + return "{\n" +\ + f"\t'stable_models': {{\n\t\t {smodels_repr} \n\t}}\n" +\ + f"\t'literals': {{ {lits_repr} }} \n" +\ + "}" + +# class EventsClass: +# def __init__(self, core, lattice:Lattice): +# self._core = core +# self._lattice = lattice + +# def __repr__(self): +# core_repr = "" if len(self._core) == 0 else ",".join(str(x) for x in self._core) +# return f"<{core_repr}>" + +# def __contains__(self, event:Event): +# return self.lattice.stable_core(event) == self._core + +if __name__ == "__main__": + def zoom_event(event_str, lattice): + event = Event.parse(event_str) + event_class = lattice.event_class(event) + propagated = lattice.extended_value( + event) + + print( + f"Event: {event}\n\tClass: {event_class} \n\tValue: {propagated}") + + smodels = Lattice.parse({ + "A": 2, + "ab": 3, + "ac": 5 + }) + + lattice = Lattice(smodels) + + ev_classes = lattice.classes() + for k,g in ev_classes.items(): + print(f"{tuple(s.latex() for s in k)} {set(e.latex() for e in g)}") + # zoom_event("abc", lattice) + # zoom_event("a", lattice) + # zoom_event("b", lattice) + # zoom_event("bc", lattice) + # zoom_event("ac", lattice) \ No newline at end of file diff --git a/code/drafts/explore_01.py b/code/drafts/explore_01.py new file mode 100644 index 0000000..f894bdd --- /dev/null +++ b/code/drafts/explore_01.py @@ -0,0 +1,13 @@ +from clingo.control import Control +from clingox.program import Program, ProgramObserver, Remapping + +prg = Program() +ctl_a = Control() +ctl_a.register_observer(ProgramObserver(prg)) +print(f"<1>\n{prg}\n") + +prog = "code/asp/alarm.lp" +ctl_a.load(prog) +ctl_a.ground([('base', [])]) +print(f"<2>\n{prg}\n") + diff --git a/code/drafts/model_sample-[0.0;0.3;0.2].csv b/code/drafts/model_sample-[0.0;0.3;0.2].csv new file mode 100644 index 0000000..e8b9869 --- /dev/null +++ b/code/drafts/model_sample-[0.0;0.3;0.2].csv @@ -0,0 +1,1001 @@ +event +A +A +A +A +A +A +A +ab +A +A +A +A +A +ac +A +A +ac +A +A +A +ab +A +A +A +A +A +A +ac +A +A +A +ac +A +A +ac +ac +A +ac +A +A +ac +A +A +A +A +A +A +A +A +A +ac +ab +A +A +A +A +A +ac +A +A +A +ac +A +A +A +ac +A +A +A +ac +A +A +A +A +ac +A +A +A +ab +A +A +A +ab +A +ab +A +A +A +ab +A +A +A +ac +ac +A +ab +A +A +ac +A +A +A +A +ac +A +A +A +A +A +A +ac +A +ac +A +A +ac +ac +A +ac 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+AB +bc +ABC +ac +AC +aBC +aB +AB +A +cAB +AB +B +B +bC +bcA +A +a +cAB +aC +AC +bAC +bAC +cA +bC +AB +bc +abC +bAC +B +AB +bc +B +b +bcA +AC +bA +a +A +ac +bC +bA +bcA +cB +A +a +bcA + +bc +bAC +bcA +aBC +abc +b +a +cA +aB + +cAB +b +aBC +A +aC +ab +bcA +bC +cA + +aB +bcA +AB +ac +aBC +acB +C +A diff --git a/code/drafts/sample_analysis.jl b/code/drafts/sample_analysis.jl new file mode 100644 index 0000000..79da0f0 --- /dev/null +++ b/code/drafts/sample_analysis.jl @@ -0,0 +1,77 @@ +using CSV +using DataFrames +using Latexify +using Plots +using StatsPlots + +function format_event(event) + symbols = [] + if event == "z" + symbols = "\\bot" + elseif event == "i" + symbols = "\\indepclass" + else + for c in event + if isuppercase(c) + push!(symbols, "\\co{$(lowercase(c))}") + else + push!(symbols, c) + end + end + end + return join(symbols, "") +end + +cores = Dict( + "indepclass" => [ "B", "C", "bc", "Ba", "Bc", "BC", "Ca", "Cb", "BCa" ], + "A" => [ "A", "Ab", "Ac", "AB", "AC", "Abc", "ACb", "ABc", "ABC" ], + "ab" => [ "b", "ab", "Cab" ], + "ac" => [ "c", "ac", "Bac" ], + "A,ab" => [ ], + "A,ac" => [ ], + "ab,ac" => [ "a", "abc" ], + "A,ab,ac" => [ "Δ" ], +) + +core_of = Dict{String,String}( + "B" => "i", # independent class + "C" => "i", + "bc" => "i", + "Ba" => "i", + "Bc" => "i", + "BC" => "i", + "Ca" => "i", + "Cb" => "i", + "BCa" => "i", + "A" => "A", + "Ab" => "A", + "Ac" => "A", + "AB" => "A", + "AC" => "A", + "Abc" => "A", + "ACb" => "A", + "ABc" => "A", + "ABC" => "A", + "b" => "ab", + "ab" => "ab", + "Cab" => "ab", + "c" => "ac", + "ac" => "ac", + "Bac" => "ac", + "a" => "ab,ac", + "abc" => "ab,ac", + "Δ" => "A,ab,ac", +) + +α = 0.1 +β = 0.3 +γ = 0.8 + +data = CSV.read("model_sample-[$α;$β;$γ].csv", DataFrame) +transform!(data, :event => ByRow(e -> get(core_of, e, "z")) => :core) +groups = groupby(data, :core) +totals = combine(groups, :core => length => :count) +for row in copy.(eachrow(totals)) + row_fmt = format_event(row.core) + println("$(row_fmt) & $(row.count)") +end \ No newline at end of file diff --git a/code/drafts/sampling.jl b/code/drafts/sampling.jl new file mode 100644 index 0000000..7bf838c --- /dev/null +++ b/code/drafts/sampling.jl @@ -0,0 +1,91 @@ +using Random +using Combinatorics + +neg(a) = islowercase(a) ? uppercase(a) : lowercase(a) + +function random_sample(atoms) + result = Char[] + for a in atoms + if rand(Bool) + push!(result, rand(Bool) ? a : neg(a)) + end + end + return length(result) > 0 ? join(result) : "Δ" +end + +function random_sample(n::Int, events) + result = String[] + for _ in 1:n + x = rand(events) + push!(result, join(x, "")) + end + return result +end + +function model_sample(n::Int; α=0.01, β=0.3, γ=0.2) + literals = [ 'a', 'b', 'c', 'A', 'B', 'C' ] + samples = [] + for _ in 1:n + sample = [] + if rand() <= α # RCE + sample = Random.shuffle(literals)[1:rand(0:3)] + while !is_consistent(sample) + sample = Random.shuffle(literals)[1:rand(0:3)] + end + if sample == [] + push!(sample , 'Δ') + else + sort!(sample) + end + elseif rand() <= β + # a + push!(sample, "a") + if rand() <= γ + push!(sample, "b") + else + push!(sample, "c") + end + else + # A + push!(sample, "A") + end + push!(samples, join(sort(sample), "")) + end + return samples +end + + +function is_consistent(event) + for (i,x) in enumerate(event) + for y in event[i:end] + if y == neg(x) + return false + end + end + end + return true +end + +using DelimitedFiles + +literals = [ 'a', 'b', 'c', 'A', 'B', 'C' ] +omega = collect(powerset(literals)) +consistent = filter(is_consistent, omega) +consistent = join.(consistent) + + +α = 0.1 +β = 0.3 +γ = 0.5 + +events = model_sample(1000; + α = α, + β = β, + γ = γ) + +println( events ) + +open("model_sample-[$α;$β;$γ].csv", "w") do io + writedlm(io, [ "event" ]) + writedlm(io, events) +end \ No newline at end of file diff --git a/code/drafts/symbops.py b/code/drafts/symbops.py new file mode 100644 index 0000000..25fd6f8 --- /dev/null +++ b/code/drafts/symbops.py @@ -0,0 +1,41 @@ +from unicodedata import numeric +from sympy import * +from sympy.plotting import plot + +def variants(expr, with_plot=False): + print(f"Expr: {latex(expr)}") + print(f"Simplify: {latex(simplify(expr))}") + print(f"Expand: {latex(expand(expr))}") + print(f"Factor: {latex(factor(expr))}") + if with_plot: + plot(expr, (d, 0, 1, 10),ylabel="$\\mathrm{P(expr \\mid \\alpha = 0.3)}$") + +init_printing(use_unicode=True) + +a, d = symbols('a d') +A = 1 - a +D = 1 - d + +wab = a * d +wac = a * D +wA = A + +wabc = wab * wac +wAb = wA +wa = wab + wac +wb = wab +wc = wac +wE = wab + wab + wA + +z = wabc + 9 * wA + wab + wac + wa + wb +wc + wE +pabc = wabc/z + +z_03 = z.subs(a, 0.3) +wabc_03 = wabc.subs(a, 0.3) +pabc_03 = pabc.subs(a, 0.3) + +variants(z_03) +variants(wabc_03) +variants(pabc_03) +print(solve(wabc_03 - 0.0015 * z_03, d)) + diff --git a/code/drafts/teste.ipynb b/code/drafts/teste.ipynb new file mode 100644 index 0000000..c62a6aa --- /dev/null +++ b/code/drafts/teste.ipynb @@ -0,0 +1,116 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "4" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "2+2" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "x = np.linspace(-6, 6)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "y = np.sin(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.plot(x,y)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3.9.15 ('base')", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.15" + }, + "orig_nbformat": 4, + "vscode": { + "interpreter": { + "hash": "a59afa236e16843183c59a167f072b6fa0409044b3c4938e82ac98aad91bf217" + } + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/code/julia/BIFParser.jl b/code/julia/BIFParser.jl new file mode 100644 index 0000000..553980a --- /dev/null +++ b/code/julia/BIFParser.jl @@ -0,0 +1,50 @@ +using ParserCombinator + +# AST Nodes with evaluation via calc +abstract type Node end + +Base.:(==)(n1::Node, n2::Node) = n1.val == n2.val + +calc(n::Float64) = n + +struct Inv <: Node val end +calc(i::Inv) = 1.0/calc(i.val) + +struct Prd <: Node val end +calc(p::Prd) = Base.prod(map(calc, p.val)) + +struct Neg <: Node val end +calc(n::Neg) = -calc(n.val) + +struct Sum <: Node val end +calc(s::Sum) = Base.sum(map(calc, s.val)) + + +# The Grammar +spc = Drop(Star(Space())) + +@with_pre spc begin + sum = Delayed() + val = E"(" + sum + E")" | PFloat64() + + + neg = Delayed() # allow eg ---3 + neg.matcher = val | (E"-" + neg > Neg) + + mul = E"*" + neg + div = E"/" + neg > Inv + prd = neg + (mul | div)[0:end] |> Prd + + add = E"+" + prd + sub = E"-" + prd > Neg + sum.matcher = prd + (add | sum)[0:end] |> Sum + + all = sum + Eos() +end + +# Test + +expr = "1 + 2 * 3 / 4" +parsed = parse_one(expr, all) +value = calc(parsed[1]) +println("Expression: $(expr)\nAST : $parsed\nValue : $value") diff --git a/code/julia/asia2.bif b/code/julia/asia2.bif new file mode 100644 index 0000000..13deaca --- /dev/null +++ b/code/julia/asia2.bif @@ -0,0 +1,60 @@ +network asia2 { +} +variable asia { + type discrete [ 2 ] { yes, no }; +} +variable tub { + type discrete [ 2 ] { yes, no }; +} +variable smoke { + type discrete [ 2 ] { yes, no }; +} +variable lung { + type discrete [ 2 ] { yes, no }; +} +variable bronc { + type discrete [ 2 ] { yes, no }; +} +variable either { + type discrete [ 2 ] { yes, no }; +} +variable xray { + type discrete [ 2 ] { yes, no }; +} +variable dysp { + type discrete [ 2 ] { yes, no }; +} +probability ( asia ) { + table 0.01, 0.99; +} +probability ( tub | asia ) { + (yes) 0.05, 0.95; + (no) 0.01, 0.99; +} +probability ( smoke ) { + table 0.5, 0.5; +} +probability ( lung | smoke ) { + (yes) 0.1, 0.9; + (no) 0.01, 0.99; +} +probability ( bronc | smoke ) { + (yes) 0.6, 0.4; + (no) 0.3, 0.7; +} +probability ( either | lung, tub ) { + (yes, yes) 1.0, 0.0; + (no, yes) 1.0, 0.0; + (yes, no) 1.0, 0.0; + (no, no) 0.0, 1.0; +} +probability ( xray | either ) { + (yes) 0.98, 0.02; + (no) 0.05, 0.95; +} +probability ( dysp | bronc, either ) { + (yes, yes) 0.9, 0.1; + (no, yes) 0.7, 0.3; + (yes, no) 0.8, 0.2; + (no, no) 0.1, 0.9; +} diff --git a/code/julia/asia2.lp b/code/julia/asia2.lp new file mode 100644 index 0000000..e34a711 --- /dev/null +++ b/code/julia/asia2.lp @@ -0,0 +1,36 @@ +% prob(asia, "0.01"). +asia ; -asia. +% prob(tub, [asia], "0.05"). +tub ; -tub :- asia. +% prob(tub, [-asia], "0.01"). +tub ; -tub :- -asia. +% prob(smoke, "0.5"). +smoke ; -smoke. +% prob(lung, [smoke], "0.1"). +lung ; -lung :- smoke. +% prob(lung, [-smoke], "0.01"). +lung ; -lung :- -smoke. +% prob(bronc, [smoke], "0.6"). +bronc ; -bronc :- smoke. +% prob(bronc, [-smoke], "0.3"). +bronc ; -bronc :- -smoke. +% prob(either, [lung, -tub], "1.0"). +either ; -either :- lung, -tub. +% prob(either, [-lung, tub], "1.0"). +either ; -either :- -lung, tub. +% prob(either, [lung, tub], "1.0"). +either ; -either :- lung, tub. +% prob(either, [-lung, -tub], "0.0"). +either ; -either :- -lung, -tub. +% prob(xray, [either], "0.98"). +xray ; -xray :- either. +% prob(xray, [-either], "0.05"). +xray ; -xray :- -either. +% prob(dysp, [bronc, -either], "0.8"). +dysp ; -dysp :- bronc, -either. +% prob(dysp, [-bronc, either], "0.7"). +dysp ; -dysp :- -bronc, either. +% prob(dysp, [bronc, either], "0.9"). +dysp ; -dysp :- bronc, either. +% prob(dysp, [-bronc, -either], "0.1"). +dysp ; -dysp :- -bronc, -either. \ No newline at end of file diff --git a/code/julia/asplang.jl b/code/julia/asplang.jl new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/code/julia/asplang.jl diff --git a/code/julia/bif2asp.jl b/code/julia/bif2asp.jl new file mode 100644 index 0000000..4375122 --- /dev/null +++ b/code/julia/bif2asp.jl @@ -0,0 +1,163 @@ +using PythonCall + +function is_truthy(x) + falsy_re = r"false|0|-0|nul|undef|no|nan"i + return Bool(match(falsy_re, "$x") === nothing) +end + +BIFReader = pyimport("pgmpy.readwrite").BIFReader + +function cpd_indexes(x) + if length(x) > 0 + h = x[1] + subs = cpd_indexes(x[2:end]) + result = Dict() + if length(subs) > 0 + for subk in keys(subs) + pos_key = [:pos, subk...] + neg_key = [:neg, subk...] + result[pos_key] = [h[2], subs[subk]...] + result[neg_key] = [h[3], subs[subk]...] + end + else + result[ [:pos] ] = [ h[2] ] + result[ [:neg] ] = [ h[3] ] + end + return result + else + return Dict() + end +end + +function compile_nodecpd(states, parents, cpd) + pos_idx = findfirst(states .|> is_truthy) + + par_idxs = cpd_indexes(parents) + idxs = length(par_idxs) > 0 ? + Dict( [:pos, cx...] => [pos_idx, ix...] for (cx, ix) in par_idxs ) : + Dict([:pos] => [ pos_idx ]) + + return Dict( + conf => getindex(cpd, idx...) for (conf, idx) in idxs) +end + +function get_bif(filename) + dag = Dict{String, Any}("source" => basename(filename)) + model = BIFReader(filename).get_model() + nodes = model.nodes() + bn_info = [] + for node in nodes + # states + states = pyconvert(PyList, model.states[node]) + # parents + parents_names = pyconvert(PyList, model.get_parents(node)) + parents = [] + for parent in parents_names + parent_states = pyconvert(PyList, model.states[parent]) + pos_idx = findfirst(parent_states .|> is_truthy) + neg_idx = findfirst(parent_states .|> (x -> !x) ∘ is_truthy) + push!(parents, (parent, pos_idx, neg_idx)) + end + # CPDS + prob_tab = model.get_cpds(node) + prob_cpd = pyconvert(PyArray, prob_tab.values) + + push!(bn_info, Dict( + "node" => pyconvert(String, node), + "parents" => parents_names, + "cpd" => compile_nodecpd(states, parents, prob_cpd) + )) + end + dag["nodes"] = bn_info + return dag +end + +function pretty_print(d::Dict, pre=1) + for (k,v) in d + if typeof(v) <: Dict + s = "$(repr(k)) => " + println(join(fill(" ", pre)) * s) + pretty_print(v, pre+1+length(s)) + else + println(join(fill(" ", pre)) * "$(repr(k)) => $(repr(v))") + end + end + nothing +end + +function make_rule(head, parents, case, prob; with_annotations=false) + body = [] + for (ci,si) in enumerate(case[2:end]) + p = getindex(parents, ci) + push!(body, si == :pos ? p : "-$p") + end + body_str = join(body, ", ") + if with_annotations + rule = "$(rpad(prob, 5)) :: $head :- $body_str." + else + rule = "% prob($head, [$body_str], \"$prob\").\n$head ; -$head :- $body_str." + end + return rule +end + +make_fact(head, prob; with_annotations=false) = with_annotations ? + "$(rpad(prob, 5)) :: $head." : + "% prob($head, \"$prob\").\n$head ; -$head." + +function bif2ast(dag; with_annotations=false, logging=false) + rules = [] + for node in dag["nodes"] + parents = node["parents"] + arity = length(parents) + node_head = node["node"] + cpd = node["cpd"] + if logging + println("Node $node_head | $(join(parents, " "))\n\t$(cpd)") + end + if arity == 0 + push!(rules, make_fact(node_head, cpd[[:pos]]; with_annotations)) + else + for (case, prob) in cpd + push!(rules, make_rule(node_head, parents, case, prob; with_annotations)) + end + end + if logging + println("\t Rules: $(length(rules))") + end + end + return join(rules, "\n") +end + +using ArgParse + +arg_parser = ArgParseSettings() +@add_arg_table arg_parser begin + + "filename" + help = "A BIF file to convert." + required = true + + "--outfile", "-o" + help = "Target ASP file." + + "--annotated", "-a" + help = "Generate ASP extended with probabilistic annotations." + action = :store_true +end + +args = parse_args(arg_parser) +filename = args["filename"] +annotations = args["annotated"] +has_astfile = args["outfile"] !== nothing +if !isfile(filename) error("Missing file [$filename].") end + +dag = get_bif(filename) +ast = bif2ast(dag; with_annotations=annotations) + +if has_astfile + open(args["outfile"], "w") do ast_file + write(ast_file, ast) + end +else + println(ast) +end diff --git a/code/julia/dados.dat b/code/julia/dados.dat new file mode 100644 index 0000000..70e50d1 --- /dev/null +++ b/code/julia/dados.dat @@ -0,0 +1,3 @@ +bom dia +1 2 3 4 +4 5 6 7 \ No newline at end of file diff --git a/code/julia/drafts.jl b/code/julia/drafts.jl new file mode 100644 index 0000000..45951d8 --- /dev/null +++ b/code/julia/drafts.jl @@ -0,0 +1,20 @@ +using ParserCombinator + +struct TS val; end + +num_seq = (StarList!(PInt(), E",") |> TS) + Eos() + +source = "1" +grammar = num_seq + +debug, task = make(Debug, source, grammar; delegate=NoCache) +try + once(task) + expr = parse_one(source, grammar)[1] + println("---\n\"$source\"\n\tparses to\n$(expr)\n---") +catch ParserException + println("Error parsing after \"$(source[1:debug.max_iter - 1])\" and before \"$(source[debug.max_iter:end])\".\n") + + #println(grammar) + parse_dbg(source, Trace(grammar)) +end \ No newline at end of file diff --git a/code/julia/forward.jl b/code/julia/forward.jl new file mode 100644 index 0000000..6f749b1 --- /dev/null +++ b/code/julia/forward.jl @@ -0,0 +1,187 @@ +using Combinatorics +using Symbolics +using Latexify +Latexify.set_default(;env=:raw) +#= + +\mu(e) :- + \mu(e,t), + \mu_TC(t). + +\mu_TC(t). + +\mu(e,t) :- + \mu([e], t), + #[e]. + +#[e]. + +\mu([e], t) :- \mu(s, t), s \in [[e]] + +\mu(s, t) :- \theta_s,t. + +\theta_s,t. +=# + + +neg(a) = islowercase(a[1]) ? uppercase(a) : lowercase(a) +literals(atoms) = join(sort(collect(atoms * neg.(atoms))), "") + +function is_consistent(event) + for (i,x) in enumerate(event) + for y in event[i:end] + if y == neg(x) + return false + end + end + end + return true +end + + +omega(atoms) = join.(powerset(literals(atoms)), "") +consistent(atoms) = join.(filter(is_consistent, omega(atoms))) + +sbf_atoms = "abc" + +@variables θ +sbf_theta = Dict( + ("A", "A") => 1, + ("ab", "a") => θ, + ("ac", "a") => 1 - θ +) + +sbf_totalchoices = unique(values(sbf_theta)) +sbf_probabilities = Dict( + "a" => 3 // 10, + "A" => 7 // 10 +) + +mu_st(s::String, t::String; theta=sbf_theta) = get(theta, (s,t), 0) + +sbf_core = Dict{String,String}( + "B" => "i", # independent class + "C" => "i", + "bc" => "i", + "Ba" => "i", + "Bc" => "i", + "BC" => "i", + "Ca" => "i", + "Cb" => "i", + "BCa" => "i", + "A" => "A", + "Ab" => "A", + "Ac" => "A", + "AB" => "A", + "AC" => "A", + "Abc" => "A", + "ACb" => "A", + "ABc" => "A", + "ABC" => "A", + "b" => "ab", + "ab" => "ab", + "Cab" => "ab", + "c" => "ac", + "ac" => "ac", + "Bac" => "ac", + "a" => "ab,ac", + "abc" => "ab,ac", + "" => "A,ab,ac", +) + +stablecore(e::String; core::Dict{String,String}=sbf_core) = Set( + string.(split(get(core, e, "⊥"), ",")) + ) + +classes(core=sbf_core) = Dict( + e => Set(k for k in keys(core) if core[k] == core[e]) for e in keys(core) + ) + +sbf_classes = classes(sbf_core) + +mu_class(e::String, t::String; theta=sbf_theta, core=sbf_core) = + sum(mu_st(s,t, theta=theta) for s in stablecore(e, core=core)) + +mu_class(e::String; theta=sbf_theta, core=sbf_core, tc_probabilities=sbf_probabilities) = + sum(mu_class(e, t, theta=theta,core=core) * p for (t, p) in tc_probabilities) + +function mu(e::String, t::String; + theta=sbf_theta, + core=sbf_core, + classes=sbf_classes) + class_size = length(classes[e]) + if class_size > 0 + return mu_class(e, t, theta=sbf_theta, core=sbf_core) // class_size + else + return 0 + end +end + +mu(e::String; + theta=sbf_theta, + core=sbf_core, + classes=sbf_classes, + tc_probabilities=sbf_probabilities) = + sum(mu(e, t, theta=theta, core=core, classes=classes) * p for (t, p) in tc_probabilities) + +sbf_quoset = unique(values(sbf_classes)) + + +Z = 0 // 1 +for qs in sbf_quoset + class_size = length(qs) + if class_size > 0 + e = [x for x in qs][1] + mce = mu_class(e) + global Z = Z + mce + end +end +Z = simplify(Z) +println("Z = $Z") + +# for qs in sbf_quoset +# class_size = length(qs) +# if class_size > 0 +# e = [x for x in qs][1] +# e_fmt = rpad(e,3) +# cs_fmt = rpad(class_size, 3) +# sc_fmt = rpad(join(stablecore(e), ", "), 20) +# mu_ceA = rpad(latexify(mu_class(e, "A")), 20) +# mu_cea = rpad(latexify(mu_class(e, "a")), 20) +# mce = mu_class(e) +# mu_ce = rpad(latexify(mce), 20) +# mu_eA = rpad(latexify(mu(e, "A")), 20) +# mu_ea = rpad(latexify(mu(e, "a")), 20) +# mu_e = rpad(latexify(mu(e)), 20) +# h_line="#[$e_fmt] = $cs_fmt; SC: $sc_fmt" +# c_line = "[$e_fmt]\tA: $mu_ceA\ta: $mu_cea\tμ~: $mu_ce" +# e_line = "'$e_fmt'\tA: $mu_eA\ta: $mu_ea\tμ~: $mu_e" +# println(join([h_line, c_line, e_line], "\n\t")) +# end +# end + +for qs in sbf_quoset + class_size = length(qs) + if class_size > 0 + e = [x for x in qs][1] + e_fmt = rpad(e,3) + sc_fmt = rpad(join(stablecore(e), ", "), 20) + pr_e = simplify(mu(e) // Z) + pr_c = simplify(pr_e * class_size) + pe_fmt = rpad(latexify(pr_e), 20) + pce_fmt = rpad(latexify(pr_c), 20) + h_line="P(E = $e_fmt) = $pe_fmt; P(E ∈ [$e_fmt]) = $pce_fmt | SC: $sc_fmt." + println(h_line) + end +end + +#= +for e in consistent(sbf_atoms) + e_fmt = rpad(e,3) + sc_fmt = rpad(join(stablecore(e), ", "), 20) + class_size = length(sbf_classes[e] + cs_fmt = rpad(class_size), 3) + println("[[$e_fmt]] #$cs_fmt: $sc_fmt\tA: $(rpad(latexify(mu(e, "A")),20))\ta:$(rpad(latexify(mu(e, "a")), 20))") +end +=# + diff --git a/code/julia/miguel.jl b/code/julia/miguel.jl new file mode 100644 index 0000000..bff25b9 --- /dev/null +++ b/code/julia/miguel.jl @@ -0,0 +1,7 @@ +open("dados.dat") do file + t = read(file, String) + m = read(file, Array{Float32, 2}(undef, 2, 4)) +end + +println(m) +println(t) \ No newline at end of file diff --git a/code/problog/alarm.pl b/code/problog/alarm.pl new file mode 100644 index 0000000..6e2a0f0 --- /dev/null +++ b/code/problog/alarm.pl @@ -0,0 +1,7 @@ +0.1::burglary. +0.01::earthquake. + +alarm :- burglary. +0.8::alarm :- earthquake. + +query(alarm). \ No newline at end of file diff --git a/code/problog/nats.pl b/code/problog/nats.pl new file mode 100644 index 0000000..3e75b58 --- /dev/null +++ b/code/problog/nats.pl @@ -0,0 +1,2 @@ +n(0). +n(s(X)) :- n(X). \ No newline at end of file diff --git a/code/sasp b/code/sasp new file mode 160000 index 0000000..5bcfee4 --- /dev/null +++ b/code/sasp @@ -0,0 +1 @@ +Subproject commit 5bcfee4d5d465618db31bea4d2e408aa27e94e89 diff --git a/code/zugzwang_code.code-workspace b/code/zugzwang_code.code-workspace new file mode 100644 index 0000000..27e7c33 --- /dev/null +++ b/code/zugzwang_code.code-workspace @@ -0,0 +1,11 @@ +{ + "folders": [ + { + "path": "." + }, + { + "path": "../students/amartins/tarefas" + } + ], + "settings": {} +} \ No newline at end of file diff --git a/cspell.json b/cspell.json new file mode 100644 index 0000000..3b8bb03 --- /dev/null +++ b/cspell.json @@ -0,0 +1,16 @@ +{ + "version": "0.2", + "ignorePaths": [], + "dictionaryDefinitions": [], + "dictionaries": [], + "words": [ + "struct", + "testset" + ], + "ignoreWords": [], + "import": [], + "enabled": true, + "enableFiletypes": [ + "latex" + ] +} diff --git a/notes.md b/notes.md new file mode 100644 index 0000000..7336e76 --- /dev/null +++ b/notes.md @@ -0,0 +1,180 @@ +# Zugzwang Meetings + +## 2024-01-30 - Exploratory Research Project + +> Apply for FCT funding. + + + +## 2024-01-05 - Next Research Lines + +> After the base-setting work of "_An Algebraic Approach to Stochastic ASP_" these are the next tasks to consider. Is summary: +> 1. **Logic Programming** - Stratified & Non-stratified programs +> 2. **Computer Science** - Inductive Logic Programming +> 3. **Software** - Integration with Potassco and other frameworks +> 4. **Applications** + +#### Line 1: Logic Programming - Stratified & Non-stratified programs + +##### Line 1a + +> _Stratified_ & _non-stratified_ programs are quoted in the "CREDAL" papers as important classes of logic programs. + +Minimal example of a **non-stratified program**. + +The following annotated LP, with clauses $c_1, c_2, c_3$ respectively, is non-stratified (because has a cycle with negated arcs) but no head is disjunctive: +```prolog +0.3::a. % c1 +b :- not c, not a. % c2 +c :- not b. % c3 +``` + +This program has three stable models: +$$ +\begin{aligned} +m_1 &= \set{ a, c } \cr +m_2 &= \set{ \neg a, b } \cr +m_3 &= \set{ \neg a, c } +\end{aligned} +$$ + +> We should **investigate** _What are stratified programs and why are they important?_ and how does our approach deals with such programs? + +##### Line 1b - Investigate the expressiveness of PASP + +Consider: + +- Recursion +- Variables, +- functional symbols, + +##### Line 1c - The equivalence relation + +Consider the cases where only $s \subseteq e$ and $e \subseteq s$. Or other refinements. Also consider the inconsistent and independent events. + +##### Line 1d - Stability of the error function + +Consider alternative error functions. See statistics, Kullback-Leibler divergence + +#### Line 2: Computer Science - Inductive Logic Programming + +> Proceed from scoring programs to support genetic algorithms or other program space exploration methods. + +Scoring programs, as described in our paper, is just a step into **Inductive Logic Programming**. To go further, we need to explore algorithms that: + +1. Use **background knowledge**, expressed as a PLP. +2. Consult **positive examples** that should be _soft_ induced. +3. Consult **negative examples** that should be _soft_ excluded. +4. Generate **PLPs** that are scored. +5. Recombine the **best scored** into a new _population_, using recombination rules. + +> In order to do that, **PLPs must be expressed as data structures** to be manipulated. Also **recombination rules** must investigated before become formally expressed and supported with adequate methods. + +#### Line 3: Software - Integration with Potassco and other frameworks + +> Support annotated programs with zugzwang semantics. + +- Bayesian Networks (BII Alice) + - Generate an annotated asp program from a bayesian network and run it trough `clingo`. + - Recover the stable models from the previous ste and compute the respective probabilities. +- Program Manipulation + - Annotated ASP program _representation_ and a _parser_. + +#### Line 4: Applications + +> Apply zugzwang to a few showcases, besides the theoretic corner stones (non-stratified, disjunctive, bayes networks), preferably based in real world scenarios, with complex structure and large datasets. + +- (Stochastic) Plan Generation +- Yale-Shooting Problem +- (Stochastic) Situation Calculus +- Frame Problem +- Latent Facts - and core assumptions. +- Given a **Bayesian Network** (or a **Markov Networks**): + - Represent it. (**done** for BNs; MNs?) + - Solve the common probability tasks: join (**done**), marginals, conditionals, parameter learning, inferring unobserved variables, sample generation, _etc._ +- Given a _solved_ ASP specification: + - What is the marginal probability of the atom `a`? (**done**) + - What other probability queries are important to consider? +- Given an _unsolved_ ASP specification: + - What is the probability (distribution?) of the probabilistic fact `a`? + - What other questions are relevant? _E.g._ the distribution family of a fact? +- Given a _solved_ ASP specification and a set of _samples_: + - How do the probabilities inferred from the specification match the ones from the empiric distribution? (**done** might see alternative approaches) +- Given two _solved_ ASP specification and a set of _samples_: + - Which specification best describes the empiric distribution? (**done**) + +## 2024-01-05 - Publish Paper "AASASP" + +> Target conferences to publish paper "AASASP" + +| Conference | Abstract Deadline | Conference Date | Location | OBS | +|-----------:|:------------------|:----------------|------------------------------:|-----------| +| IJCAR 2024 | 2024-01-29 | 2024-07-3:6 | Nancy, France | Picked | +| ECAI'24 | 2024-04-19 | 2024-10-19:24 | Santiago de Compostela, Spain | | +| KR 2024 | 2024-04-24 | 2024-11-2:8 | Hanoi, Vietnam | | +| GECCO 24 | 2024-02-05 | 2024-07-14:18 | Melboune, Australia | | +| ICLP 24 | 2024-04-15 | | | preferred | +| JELIA 25 | | | | | +| ICFP 24 | 2024-03-01 | 2024-09-2:7 | Milan, Italy | | + +## 2023-02-28 - Looking for Application Examples + +What **applications** are we looking for? + +- (Stochastic) Plan Generation +- Yale-Shooting Problem +- (Stochastic) Situation Calculus +- Frame Problem +- Given a **Bayesian Network** (or a **Markov Networks**): + - Represent it. + - Solve the common probability tasks: marginals, conditionals, parameter learning, inferring unobserved variables, sample generation, _etc._ +- Given a _solved_ ASP specification: + - What is the marginal probability of the atom `a`? + - What other probability queries are important to consider? +- Given an _unsolved_ ASP specification: + - What is the probability (distribution?) of the probabilistic fact `a`? + - What other questions are relevant? E.g. the distribution family of a fact? +- Given a _solved_ ASP specification and a set of _samples_: + - How do the probabilities inferred from the specification match the ones from the empiric distribution? +- Given two _solved_ ASP specification and a set of _samples_: + - Which specification best describes the empiric distribution? + +What should be the **task for the scholarship student**? _Use the `Python` API of `clingo`._ + +1. Read a string and extract probability annotations; Associate those annotations with the respective atoms. +2. Call `clingo` to get stable models. +3. Support **computation of the equivalence classes**: _Which functions and relations?_ +4. Compute event probability using _weighted model counting_ on the equivalence classes. +5. Read a Bayesian Network from a file (`BIF`, `DSC`, `NET`, `RDA`, `RDS`, ...) and generate an annotated "ASP" specification. + +2022 - AAAI - Inference and Learning with Model Uncertainty in Probabilistic Logic Programs +- Is **"Epistemic Uncertainty (EU)"** the right framework for Zugzwang? How relevant are the epistemic questions in this paper to our work? +- EU can be represented by _Credal Sets, Subjective Logic_ and _Beta Distributions_? +- **Experiments** made with BNs from [(Kaplan and Ivanovska 2018)](https://www.sciencedirect.com/science/article/pii/S0888613X17302384) and larger networks from the [BNLearn repository](https://www.bnlearn.com/bnrepository). +- **Are networks, Bayesian Networks in particular, a "good enough" pool of "example applications" to us, for now?** + +## 2023-01-10 - 15:00 + +- Paper +- Project +- Latent Facts + +## 2022-12-12 + +- Is the project proposal ok? How long/detailed should it be? +- Initial exploratory code `event_lattice.py` and `EventLattice.ipynb` done. +- Start writing paper: Introduction, state of the art, motivation + - Identify key problems + - Target Conferences + - KR; + - [ICLP](https://waset.org/language-planning-conference-in-april-2023-in-lisbon); + - [ECAI](https://ecai2023.eu/) +- Next task for prototype: + - Get stable models from potassco/s(casp) + - other? + + +## 2022-12-05 + +- Created shared folder (gdrive:zugzwang) +- Refine project tasks, for Bachelor, M.Sc., Ph.D. students and for researchers. \ No newline at end of file diff --git a/notes.pdf b/notes.pdf new file mode 100644 index 0000000..8da7bde Binary files /dev/null and b/notes.pdf differ diff --git a/pex2024/FCT-Sessão_esclarecimento-2024-01-23.md b/pex2024/FCT-Sessão_esclarecimento-2024-01-23.md new file mode 100644 index 0000000..4c86558 --- /dev/null +++ b/pex2024/FCT-Sessão_esclarecimento-2024-01-23.md @@ -0,0 +1,36 @@ +# Sessão de Esclarecimento - Projetos IC&DT + +- Prorrogação do prazo (?) ~1mês ~21 março +- Balcão dos Fundos +- Custos Simplificados +- Formulário de Candidatura +- Condições de Elegibilidade + +## Balcão dos Fundos + +- ponto de acesso apresentação candidatura +- entidade / super-utilizador / utilizador ; perfis +- entidade convida investigador responsável +- **linha dos fundos** - esclarecimentos + +## Modelo de Custos Simplificados + +- 4320€ ETI/mês +- (o que é um ETI?) +- custo unitário = custos diretos + custos indiretos +- Consultar o Manual Frascati (2015) +- ETI = FTE +- (custo unitário) = (média despesas I&D 2019, 2020, 2021) / (média ETIs investigadores, técnicos, equivalente 2019, 2020, 2021) +- evidências: (i'm lost) + +## Formulário de Candidatura + +- 14 secções sequenciais! +- Enquadramento ENEI ou EREI (wtf?) (regional?) +- Sistema de Apoio (?) +- CV narrativo (< 3000 chars) + + +## Condições de Elegibilidade + +- enquadramento RIS3 ou RIS3 regional diff --git a/pex2024/candidatura.md b/pex2024/candidatura.md new file mode 100644 index 0000000..d3951f2 --- /dev/null +++ b/pex2024/candidatura.md @@ -0,0 +1,10 @@ +# Candidatura PeX - Zugzwang + +## Tarefas + +- [ ] Submeter +- [ ] Redigir +- [ ] Preparar + - [ ] Consultar documentação + - [ ] Reunir requisitos + - [ ] Delinear proposta \ No newline at end of file diff --git a/pex2024/documents/Application_Guide_PeX_2023.pdf b/pex2024/documents/Application_Guide_PeX_2023.pdf new file mode 100644 index 0000000..cfca14f Binary files /dev/null and b/pex2024/documents/Application_Guide_PeX_2023.pdf differ diff --git a/pex2024/documents/Aviso_de_Abertura_PEX_2023.pdf b/pex2024/documents/Aviso_de_Abertura_PEX_2023.pdf new file mode 100644 index 0000000..a460546 Binary files /dev/null and b/pex2024/documents/Aviso_de_Abertura_PEX_2023.pdf differ diff --git a/pex2024/documents/Data_Protection_PEX_2023.pdf b/pex2024/documents/Data_Protection_PEX_2023.pdf new file mode 100644 index 0000000..8e0d76e Binary files /dev/null and b/pex2024/documents/Data_Protection_PEX_2023.pdf differ diff --git a/pex2024/documents/Ethics_Self_Assessment_Guide.pdf b/pex2024/documents/Ethics_Self_Assessment_Guide.pdf new file mode 100644 index 0000000..ce1f2a5 Binary files /dev/null and b/pex2024/documents/Ethics_Self_Assessment_Guide.pdf differ diff --git a/pex2024/documents/Guiao_CIENCIAVITAE.pdf b/pex2024/documents/Guiao_CIENCIAVITAE.pdf new file mode 100644 index 0000000..14126e5 Binary files /dev/null and b/pex2024/documents/Guiao_CIENCIAVITAE.pdf differ diff --git a/pex2024/documents/Regulation_on_projects_FCT_EN.pdf b/pex2024/documents/Regulation_on_projects_FCT_EN.pdf new file mode 100644 index 0000000..faac185 Binary files /dev/null and b/pex2024/documents/Regulation_on_projects_FCT_EN.pdf differ diff --git a/sample.csv b/sample.csv new file mode 100644 index 0000000..240b732 --- /dev/null +++ b/sample.csv @@ -0,0 +1,100 @@ +c +a +B +c +ab +Ac +aBC +BC +Ab +b +c +AB +Ac +B +bC +Ab + +aB +B +a +aC +B +AB +B + +C +bc + +B +bc +AC +ABC +a +b +ac +AC +B +bC +b +C + +aC +AbC + +C +AC +bc + + +C +b + +b +abc +a +abc + +B +a +C +aB + +A +Bc +C +AB +aBc +A +b + +a +a +c +b +a +A +Ab +aC + +B +C +Ab +Bc +A +Ac +c +B +AC +ABC +Ab +aC +bC + + +C +a + +ABC +b +aB diff --git a/students/amartins/2023-03-13 | Tarefa 01.eml b/students/amartins/2023-03-13 | Tarefa 01.eml new file mode 100644 index 0000000..5568cef --- /dev/null +++ b/students/amartins/2023-03-13 | Tarefa 01.eml @@ -0,0 +1,172 @@ +MIME-Version: 1.0 +Date: Mon, 13 Mar 2023 10:09:44 +0000 +References: + +In-Reply-To: +Message-ID: +Subject: Re: Tarefa 01 +From: Francisco Coelho +To: ALICE MARTINS +Content-Type: multipart/alternative; boundary="00000000000071009e05f6c54f6f" + +--00000000000071009e05f6c54f6f +Content-Type: text/plain; charset="UTF-8" +Content-Transfer-Encoding: quoted-printable + +Tem aqui um reposit=C3=B3rio de red= +es de +bayes, nesses formatos. +Pode testar a biblioteca fazendo experi=C3=AAncias com os exemplos do +reposit=C3=B3rio. + +Quest=C3=B5es simples que pode tentar responder com a biblioteca: +- quantos n=C3=B3s tem a rede? +- quantos n=C3=B3s s=C3=A3o descendentes? quantos s=C3=A3o ascendentes? +- qual =C3=A9 o n=C3=BAmero m=C3=A9dio de arestas "in"? e "out"? + +Um exerc=C3=ADcio com mais "f=C3=B4lego". +O dot +=C3=A9 um formato para descrever grafos, suportado em Python pelo pydot + e usado pelo graphviz + para visualiza=C3=A7=C3=A3o. +Por outro lado, networkx =C3=A9 uma biblioteca de +an=C3=A1lise de grafos e redes. + +Passe uma rede lida com o bnlearn para o networkx e visualize-a com o +graphviz passando pelo pydot. + +Entretanto, estamos interessados no Potassco . Pode +instalar no seu sistema e consultar o guia respetivo +. + +-- +Francisco Coelho +Departamento de Inform=C3=A1tica, Universidade de =C3=89vora +NOVA LINCS +High Performance Computing Chair + + +ALICE MARTINS escreveu no dia domingo, 12/03/2023 +=C3=A0(s) 20:18: + +> Ol=C3=A1 professor, +> espero que se encontre bem. +> Da minha pesquisa sobre este assunto, pude concluir que para realizarmos = +a +> leitura de redes Bayesianas de um ficheiro, precisamos de importar a +> biblioteca bnlearn no python atrav=C3=A9s do comando: +> *import bnlearn* +> +> Atrav=C3=A9s dessa biblioteca, podemos ler ficheiros atrav=C3=A9s dos com= +andos: +> *read.bif(), read.dsc(), read.net ()* +> +> Para al=C3=A9m de toda esta pesquisa, tamb=C3=A9m aprendi como programar = +em Python. +> Por favor, diga-me o que mais posso fazer para terminar esta tarefa. +> +> Atenciosamente, +> Alice Martins +> +> Francisco Coelho escreveu no dia ter=C3=A7a, 7/03/2023 =C3= +=A0(s) +> 11:07: +> +>> Ol=C3=A1 Alice, +>> +>> Aqui est=C3=A1: "Read a Bayesian Network from a file (`BIF`, `DSC`, `NET= +`, +>> `RDA`, `RDS`, ...)" em Python. +>> +>> Cumprimentos, +>> -- +>> Francisco Coelho +>> Departamento de Inform=C3=A1tica, Universidade de =C3=89vora +>> NOVA LINCS +>> High Performance Computing Chair +>> +> + +--00000000000071009e05f6c54f6f +Content-Type: text/html; charset="UTF-8" +Content-Transfer-Encoding: quoted-printable + +
Tem aqui um reposi= +t=C3=B3rio de redes de bayes, nesses formatos.
Pode testar a biblioteca fazendo expe= +ri=C3=AAncias com os exemplos do reposit=C3=B3rio.

Quest=C3=B5es simples que pode te= +ntar responder com a biblioteca:
- quantos n=C3=B3s tem a rede?
- quantos n=C3=B3s s=C3=A3o d= +escendentes? quantos s=C3=A3o ascendentes?
- qual =C3=A9 o n=C3=BAmero m=C3=A9dio de= + arestas "in"? e "out"?

Um exerc=C3=ADcio com mais "f=C3=B4lego= +".
O dot =C3=A9 um formato para descrever grafos, suportado em Pyth= +on pelo pydot e usado pelo = +graphviz para visualiza=C3=A7=C3= +=A3o.
Por= + outro lado, networkx =C3=A9 uma bibl= +ioteca de an=C3=A1lise de grafos e redes.

Passe uma rede lida com o bnlearn para o netwo= +rkx e visualize-a com o graphviz passando pelo pydot.

Entretanto, estamos interessados n= +o Potassco. Pode instalar no seu sist= +ema e consultar o g= +uia respetivo.

--
Francisco Coelho
Departamento de Inform=C3=A1tica,=C2=A0<= +span style=3D"font-family:monospace">Universidade de =C3=89vora
NOVA LINCS
High Performance Computing Chair

ALICE MARTINS <l52768@alunos= +.uevora.pt> escreveu no dia domingo, 12/03/2023 =C3=A0(s) 20:18:
= +
Ol=C3=A1 professor,
espero que se encontre bem.
Da m= +inha pesquisa sobre este assunto, pude concluir que para realizarmos a leit= +ura de redes Bayesianas de um ficheiro, precisamos de importar a biblioteca= + bnlearn no python atrav=C3=A9s do comando:
import bnlearn= +

Atrav=C3=A9s dessa biblioteca, podemos ler fichei= +ros atrav=C3=A9s dos comandos:
read.bif(), read.dsc(), read.net()

Para al=C3=A9m de toda esta pesquisa, tamb=C3=A9m aprendi como prog= +ramar em Python.
Por favor, diga-me o que mais posso fazer p= +ara terminar esta tarefa.

Atenciosamente,
Alice Martins

Francisco Coelho <fc@uevora.pt> escreveu no dia ter=C3=A7a= +, 7/03/2023 =C3=A0(s) 11:07:
Ol=C3=A1 Alice,

Aqui est=C3=A1: "Read a Bayesian Network from a fil= +e (`BIF`, `DSC`, `NET`, `RDA`, `RDS`, ...)" em Python.

Cumprimentos,
= +
--
Francisco Coelho
D= +epartamento de Inform=C3=A1tica,=C2=A0Universidade de =C3=89vora
NOVA LINCS= +
High Performance Com= +puting Chair
+
+
+ +--00000000000071009e05f6c54f6f-- \ No newline at end of file diff --git a/students/amartins/tarefas/Untitled.ipynb b/students/amartins/tarefas/Untitled.ipynb new file mode 100644 index 0000000..363fcab --- /dev/null +++ b/students/amartins/tarefas/Untitled.ipynb @@ -0,0 +1,6 @@ +{ + "cells": [], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/students/amartins/tarefas/Untitled1.ipynb b/students/amartins/tarefas/Untitled1.ipynb new file mode 100644 index 0000000..3b2222c --- /dev/null +++ b/students/amartins/tarefas/Untitled1.ipynb @@ -0,0 +1,424 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "id": "a1e66bad-47aa-49b5-ac59-a2183646e74e", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "0dbacf2a-1b98-4a20-89f7-1b46ac725a13", + "metadata": {}, + "outputs": [], + "source": [ + "x = np.linspace(0,1)\n", + "y = x*(1-x)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "178862b2-2fc8-42ec-b8e9-ceb8b78ea7a9", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plt.plot(x,y)" + ] + }, + { + "cell_type": "code", + "execution_count": 94, + "id": "26e02fe3-a079-4b64-b986-56f37f967e85", + "metadata": {}, + "outputs": [], + "source": [ + "from sympy import pprint, symbols, expand, factor, simplify, collect, cancel, apart, Matrix, solve, Poly, roots, degree\n", + "from sympy.printing.latex import latex\n", + "from sympy.plotting import plot" + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "id": "baf8a3e6-0a39-410a-85c8-8c536b7603ae", + "metadata": {}, + "outputs": [], + "source": [ + "t,x = sympy.symbols(\"t x\")" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "5d570fe9-37cf-4680-a608-99e015d9b977", + "metadata": {}, + "outputs": [], + "source": [ + "Z = 8.8 + t*(1-t)" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "de141bca-0e2a-42c7-86a3-355a1eda8287", + "metadata": {}, + "outputs": [], + "source": [ + "pa = 0.7/Z" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "0194f6b5-c206-47d7-bc0d-8b1617186d46", + "metadata": {}, + "outputs": [], + "source": [ + "T = 1 - t" + ] + }, + { + "cell_type": "code", + "execution_count": 58, + "id": "d1b858fc-2432-42e7-ba79-e012e76a6a99", + "metadata": {}, + "outputs": [], + "source": [ + "s = Matrix([\n", + " 37,\n", + " 9,\n", + " 9,\n", + " 3,\n", + " 3,\n", + " 0,\n", + " 0,\n", + " 2,\n", + " 1])\n", + "m = Matrix([\n", + " 0.0,\n", + " 0.0,\n", + " 0.7,\n", + " 0.3 * t,\n", + " 0.3 * T,\n", + " 0.7 + 0.3 * t,\n", + " 0.7 + 0.3 * T,\n", + " 0.3 * t * T,\n", + " 0.7 + 0.3 * t * T\n", + "])\n", + "z = (s.T * m)[0,0]" + ] + }, + { + "cell_type": "code", + "execution_count": 50, + "id": "2b461810-5bbe-4e4c-a938-b3c916dd95ee", + "metadata": {}, + "outputs": [], + "source": [ + "z = (s.T * m)[0,0]" + ] + }, + { + "cell_type": "code", + "execution_count": 51, + "id": "3946e0dc-fc81-4030-9cac-ccdccf05b9e1", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle - 0.9 t \\left(t - 1\\right) + 7.9$" + ], + "text/plain": [ + "-0.9*t*(t - 1) + 7.9" + ] + }, + "execution_count": 51, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "simplify(z)" + ] + }, + { + "cell_type": "code", + "execution_count": 68, + "id": "a34c1fc2-3810-4054-bcee-b3d8db7b852f", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle - 7.9 \\cdot \\left(0.113924050632911 t^{2} - 0.113924050632911 t - 1.0\\right)$" + ], + "text/plain": [ + "-7.9*(0.113924050632911*t**2 - 0.113924050632911*t - 1.0)" + ] + }, + "execution_count": 68, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "factor(z)" + ] + }, + { + "cell_type": "code", + "execution_count": 53, + "id": "89ac28fa-0c47-4a27-a0cc-f7fe0ddf63bd", + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle - 0.9 t^{2} + 0.9 t + 7.9$" + ], + "text/plain": [ + "-0.9*t**2 + 0.9*t + 7.9" + ] + }, + "execution_count": 53, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "expand(z)" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "id": "f575ba56-647a-4e01-8c0f-39499abf6300", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'- 1.3 t^{2} + 1.3 t + 7.5'" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "latex(expand(z))" + ] + }, + { + "cell_type": "code", + "execution_count": 61, + "id": "a4c62a99-9ad7-4b79-9dda-a28092383a07", + "metadata": {}, + "outputs": [], + "source": [ + "zx = z.replace(t,x)" + ] + }, + { + "cell_type": "code", + "execution_count": 70, + "id": "9d67f9d2-48d9-454e-bbfa-4d185eb86200", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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wAwBYweFwqEePHtq3b5++/fbbZJcDqEuXLurevXu7PifBDgBgDa/XqwEDBnA5Fknn8XjadaQuimAHALCK0+m08l+KAY3BDQgAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWcCe7AAD2M8bIGClojIKh6uWQMQqFwm0ho/B6pE8oFF4PGiNjjIKRdRPpJyWum7jnCZnwcshIRuG2+H4hYyQTqS26oHCf+Hl4e+LXUbMtyiHJ4XDELUcmOWIdon0c0X0i2x2RbXJITocjMoX7OuPaHNFlp2r0qe7nqOMxXM64KW49uh2APQh2QAdjjFFV0KgqGFJlIKTKuHm0Lbo9EDSqCoXngWC4TyBoFAiF+wSCIQVCpro9GFJVKDKv6zEi+4VCRoFI4AqEQpF5zfbq7aGQEvoFg5F5JKQFQ3VFIXQETofkdjrldEouh0NOp0PuWPALLztrhEK3yxnr43FF587wNmdkm8shj9Mhl9NZRx+H3K5wX48r3O52OeV1hR/b46puD29zyBt5To/bKY/TKY87vL/XFV72upzyuiOTy0lgxWkrJYJdIBDQ/PnztXr1ah05ckQ9evTQnXfeqUceeUROJ1eT0ToCwZD8gegUlL8qbjkQiqwH695eR9/KYHUoi4W0yLI/2hYX1OJD3OkqOorkdFQHC6dDsWVHbBQqMjqlyAhYjdGq6hGzSP/oqFjc/tG2qIQYELfBUXdzQns0tsYGAuNG9oypHhWMjhxW969ulxJHFKMjjOFRzeoRyFB0JDJudNPEtcfvG4wsNyRkpMpgSAo23C/VeFw1wp47HBK9Lqd87sQQ6Inr56uzzaU0T3juczvl8zSyzR0OrEB7Solg9+STT+qFF17Qv/3bv2nw4MHauXOn7rrrLmVnZ2vmzJnJLg9tJBQyKq8KqqIqGJmHVBFZr6gKJWzz19FWURWSP7Zc92NUB7VQhx1Vio6KRP9IeWqMaMSPfLidzsiIRmRExBVejo6IeGqNkkTmTkdsu8vpkMcZGVlxVY/aRNddTmfcyE3NkZxo30ifyHp1UFN49Cd6qTA6ShQZKULriwbC6MhpdBQ1OvoaihtVDZnqkdn40dZgrF/cyGx0VDYyyhsMhUeAgyGjqpBRMDJaHB3ZjW0LhvcJhOJGjKOjx6GQKgPVy1WB8GhytF9lrH8oMiUuxwtvC+pkZXITq8vpkM/tVJqnOuyleVyRKbycHrfuc7uU7nUpLRISo8u+Ovqme8J90z0uZXjd8rmd/B4hNYLd+++/rwkTJmjcuHGSpL59+2rt2rXauXNnkis7vRkTfqEt8wdVVhVUeWVAZZVBlVWGQ1R5dLm+9qq49sqgyioDqqgKqSzS35/EkavoWb0v7uzb63bKF/fi7Iu82MaW4/p7XY7YaIDX7QoHs8h+nriAVt3HmRDc4tddvFCjBRwOh1yRUU+bhSK3HFTWGB2vjJy41TV6XhkZPa+rveZ2fyB8UhgekQ/WMbpfPXJfGax+7QqGTOx1rj2keZzK8LoTQl9iAHQpLW45w+uOzCPLPpcyPC518rlrtTH6mBpSIthdfvnleuGFF/TFF1/o3HPP1X//939r+/bt+v3vf5/s0lJKZSCkUn9AJ/0BnawM6KQ/HKZi88qgyvzx84b7nPQHFGinUS6v2xk5U3XGzlh9HpfS6ziLjZ4Np9dYr9UWd9kkdvnEEw5UnPUCqcXpdCjNGf49T7ZQyNR5G0dFVUgVgeorBvVdcYj1qwxG+odUHrccvRIRPVGOPwkOP25lm3xdXrdTGV6XOnndSve61MnriszdyvC5leFxKcMXXu/kc6uzLxwMw8tudfK51NkX7tvZG14nLLa+lAh2Dz30kIqKijRw4EC5XC4Fg0EtXLhQ//AP/1DvPn6/X36/P7ZeXFzcHqW2uvgwVlIRDlulFQGV+gOJ7ZEgFl0Obw+q1F+lk/6gSisCCWeRrc3rCl8yyIj8omdEzxS9kV/2+PbI2WR8W7i/O9YWvz3N7SJoAUgZTqcj/DrmbZ+QGQyZWEgsj1wZiV4JqaiKv1oSiMxDKqsKxK6eRK+ShK++BFTmD+pk3JWW6G0q0RHNE2VVrVa7z+2MhL7w6390ORoEO/ncykzzKCvNrcw0tzr7PMqMLIen8Hq6x8UbZiJSItitW7dOq1at0po1azR48GDt2bNHs2bNUn5+vqZMmVLnPoWFhVqwYEE7V5qoMhBSSUWVSioCKo7MSyqqVFwRDmDF5dVtNfuURPq0RRiL/iJFz6wyvOFfnk7euLaa23xudfJGz74S5xlelzycdQFAUricjtjrdGurectNmT8c+E5WhoNh9CpONCBG10v9wbirQ+GrPqVx69F7IsMjm5U6drJlo4wup0OdfdHw51ZWJPB1jguA4fbq5fhgGN3PhhFEhzGnes9U8vXu3Vtz5szRjBkzYm2PP/64Vq1apc8++6zOfeoasevdu7eKioqUlZV1yucMBMMjZcXltQNXfevF8W3lVa16j1i6x6XOaYnD2Z1jZzXhH97O3rjluG2ZaZF2hr4BAB1AZSAUu7oUDXul/mgojF6FCqqkIqBSf/VgR2nc39xSf/hvbmveEZTucdUaDcxMcyvT50kIiZlxwbBzWjgwdsnwKist+eEwJUbsysrKan2sicvlUihUf3Dy+Xzy+Xy12l/bfUhVrmM6UV6l4vIqFZVX6URZpYrKE0Naa97oWn1mED+k7KleT08cag6faXiqg5yXMAYAsEf4o2S8OqOTt0WPY0z4zSnRkFccCX/RwZdSf/WgS6zdX71cHAmOFVXhPBG9d/Foif8Uz1y/zj63stM9yk73qEuGJ7acHVnuku6NbRt1Tk6Lvv66pESwGz9+vBYuXKizzjpLgwcP1u7du7Vo0SLdfffdTX6shzd8Iqcvo9H949N7VronFsiy6gtpcetZkXBm+7vRAABIBoej+jJ0XlZasx8nej97XSOC0eVYe0Vie+zWKn9AkmL3wB86UX7K593/T+OaXXN9UiLYPffcc3r00Uc1ffp0HT16VPn5+brvvvv02GOPNfmxRpx9prp17ZKQpLuke5WVHh45y4oLaZlpbu4dAwDAcl63U13dXnVtwQhiIBhScUUg4UpgbCqr0oka60XlrfcmlHgpcY9dayguLlZ2dnaj77EDAABINQxHAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYImWC3aFDh/SP//iPOvPMM5WRkaELL7xQu3btSnZZAAAAHYY72QU0xvHjxzVq1ChdddVVevPNN5Wbm6uvv/5aXbp0SXZpAAAAHUZKBLsnn3xSvXv31sqVK2Ntffv2TV5BAAAAHVBKXIp9/fXXVVBQoFtuuUW5ubm66KKLtHz58mSXBQAA0KGkRLD75ptv9Pzzz2vAgAHauHGjpk2bpgceeED//u//Xu8+fr9fxcXFCRMAAIDNHMYYk+wiTsXr9aqgoEDvvfderO2BBx7Qjh079P7779e5z/z587VgwYJa7UVFRcrKymqzWgEAAJIlJUbsevTooUGDBiW0nX/++Tpw4EC9+8ydO1dFRUWx6eDBg21dJgAAQFKlxJsnRo0apc8//zyh7YsvvlCfPn3q3cfn88nn87V1aQAAAB1GSozY/epXv9IHH3ygJ554Ql999ZXWrFmjF198UTNmzEh2aQAAAB1GStxjJ0l/+ctfNHfuXH355Zfq16+fZs+erXvuuafR+xcXFys7O5t77AAAgLVSJti1FMEOAADYLiUuxQIAAODUCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFjC3ZKdq6qqdOTIEZWVlalbt27q2rVra9UFAACAJmryiF1paamWLVumK6+8UtnZ2erbt68GDRqkbt26qU+fPrrnnnu0Y8eOtqgVAAAADWhSsPuXf/kX9e3bV8uXL9fPfvYzvfrqq9qzZ48+//xzvf/++5o3b54CgYCuueYaXXvttfryyy/bqm4AAADU4DDGmMZ2vuWWW/TYY4/pJz/5SYP9/H6/VqxYIa/Xq6lTp7a4yNZQXFys7OxsFRUVKSsrK9nlAAAAtLomBbt4JSUlyszMbO162gzBDgAA2K7Z74odPXq0jhw50pq1AAAAoAWaHewKCgp06aWX6rPPPkto3717t66//voWFwYAAICmaXaw+8Mf/qC7775bl19+ubZv364vvvhCt956qwoKCuTz+VqzRgAAADRCiz7Hbt68efJ6vbrmmmsUDAY1duxY7dixQxdffHFr1QcAAIBGavaI3eHDh/XAAw/od7/7nQYNGiSPx6NJkyYR6gAAAJKk2cGuf//+2rZtm1555RXt2rVLr776qqZPn64nn3yyNesDAABAIzX7UuzKlSs1adKk2PrYsWP19ttv64YbbtC3336rpUuXtkqBAAAAaJxmf45dffbv36/rr79ee/fubc2HbTE+xw4AANiu2Zdi69O3b1+9++67rf2wAAAAOIUmBbsDBw40qt8ZZ5whSTp06FDTKwIAAECzNCnYXXLJJbrnnnv04Ycf1tunqKhIy5cv1wUXXKBXX321xQUCAACgcZr05okJEyYoMzNT1157rTwejwoKCpSfn6+0tDQdP35ce/fu1aeffqqCggI9/fTTuu6669qqbgAAANTQpDdPeL1eHTx4UFlZWcrLy9Ott96qY8eOqby8XDk5Obrooos0duxYXXDBBW1Zc7Pw5gkAAGC7Jo3Y9ezZU7t379a1116r0tJSPfHEE8rNzW2r2gAAANAETbrH7sEHH9SNN96okSNHyuFwaPXq1dqxY4fKy8vbqj4AAAA0UpM/x+7TTz/Va6+9pkceeUT9+/fX/v375XA4dM4552jo0KG68MILNXTo0A53fx2XYgEAgO2a/QHF55xzjj744AN16tRJ//M//6M9e/bEpk8++UQlJSWtXWuLEOwAAIDtWv0/T0iSMUYOh6O1H7ZFCHYAAMB2rf6fJyR1uFAHAABwOmiTYAcAAID2R7ADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEsQ7AAAACxBsAMAALAEwQ4AAMASBDsAAABLEOwAAAAsQbADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEsQ7AAAACxBsAMAALAEwQ4AAMASKRnsCgsL5XA4NGvWrGSXAgAA0GGkXLDbsWOHXnzxRQ0ZMiTZpQAAAHQoKRXsSktLdccdd2j58uU644wzkl0OAABAh5JSwW7GjBkaN26crr766lP29fv9Ki4uTpgAAABs5k52AY318ssv66OPPtKOHTsa1b+wsFALFixo46oAAAA6jpQYsTt48KBmzpypVatWKS0trVH7zJ07V0VFRbHp4MGDbVwlAABAcjmMMSbZRZzKhg0bdPPNN8vlcsXagsGgHA6HnE6n/H5/wra6FBcXKzs7W0VFRcrKymrrkgEAANpdSlyK/bu/+zt9/PHHCW133XWXBg4cqIceeuiUoQ4AAOB0kBLBLjMzUxdccEFCW6dOnXTmmWfWagcAADhdpcQ9dgAAADi1lLjHrjVwjx0AALAdI3YAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJYg2AEAAFiCYAcAAGAJgh0AAIAlCHYAAACWINgBAABYgmAHAABgCYIdAACAJQh2AAAAliDYAQAAWIJgBwAAYAmCHQAAgCUIdgAAAJZIiWBXWFioSy65RJmZmcrNzdVNN92kzz//PNllAQAAdCgpEey2bNmiGTNm6IMPPtDmzZsVCAQ0ZswYnTx5MtmlAQAAdBgOY4xJdhFN9f333ys3N1dbtmzRT3/600btU1xcrOzsbBUVFSkrK6uNKwQAAGh/7mQX0BxFRUWSpK5du9bbx+/3y+/3x9aLi4vbvC4AAIBkSolLsfGMMZo9e7Yuv/xyXXDBBfX2KywsVHZ2dmzq3bt3O1YJAADQ/lLuUuyMGTP0H//xH9q+fbt69epVb7+6Rux69+7NpVgAAGCtlLoU+8tf/lKvv/66tm7d2mCokySfzyefz9dOlQEAACRfSgQ7Y4x++ctfav369XrnnXfUr1+/ZJcEAADQ4aREsJsxY4bWrFmj1157TZmZmTpy5IgkKTs7W+np6UmuDgAAoGNIiXvsHA5Hne0rV67UnXfe2ajH4ONOAACA7VJixC4FsicAAEDSpdzHnQAAAKBuBDsAAABLEOwAAAAsQbADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEsQ7AAAACxBsAMAALAEwQ4AAMASBDsAAABLEOwAAAAsQbADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEsQ7AAAACxBsAMAALAEwQ4AAMASBDsAAABLEOwAAAAsQbADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEsQ7AAAACxBsAMAALAEwQ4AAMASBDsAAABLEOwAAAAsQbADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEsQ7AAAACxBsAMAALAEwQ4AAMASBDsAAABLEOwAAAAsQbADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEsQ7AAAACxBsAMAALAEwQ4AAMASBDsAAABLuJNdAIDTRCgkhQKSCUqhYPU8YTkgmVBkMnHLoXCfhPUa20MNbJcJr8fmSmyTGtiuuO1xHI4ay44a7Y76l+P3cTgkhzOy7Kxej83jt8VPkTanS3K44padNZZd1X2c0ckdWXdLTs7vAZsQ7ICOzhgpWCkF/HFzvxSojJtHplAgslwVnkKReWxbVe1+9S0n7B/ZFpuCcctVNdZrbo8EN3RcTnftsJew7pJcHsnpCa+73PUse6ofq9ayJ9zX5ZVcvnCbyxs3b8Gy2xeuEQDBDmiQMeFwEqiQqirC84BfCpRH5vHt8ZNfqorrU197sKpGSPPXDnGhqmR/F9pe/MiS0117xKrOyVG9X4PboyNiUvXIWR2jZvVul2qNttUczZPqGekzjdhWz+hiQ6OTsbZg3EhlZB6KX47b1pBoCE9lTnc4MLojk8srudMkd2Te4LZIOHSnRebp4bknvXrdkxbZHpni1z3pkZ9bx6nrBNoYwQ6pKxiQqk5KlWVSVVk4MFWVxy3XmAcq6thWc7lGW6A88se2A4n9AfPGzeNGMJzREQ134rLLmzhq4vRERjw89fSL3xZtd1eP4MSWo+ueGuvxl/xqXAqsK8Sh7USDYexyd3RENVQd6mq2mRqjrrFR26q49ara24JxI7nB6GhuVY0R4crq0eNGLce1RUeoa57wRGutOpmc77HDGQmA6ZInIxz8osvutEhbJCBGl2v1z5C8narn8cvRffhdwSkQ7ND2glWSv0SqLI3MT4anqrIay2XhPnUtV0X6xZbLwiNc7c3liztb9yWewdc8w3f7apzZ12iPjRzUmNdqiw9vPu6JQtM5HNVhWt5kV9M6jKke8Q5WxY2mx92qEKioHgk/1bZgZR2j6uW1R+RrrsfqCYVfm9o0WDrqD36nXM6QvJ3rX3Z52rButCeCHWozJvyC5S+VKkvC8/hgFlsujcyL45ajfUuq2+Jf/NqCwxV58Yo/+42bJ5wtZ8T1q69/tD2yXzSIEaqAjsPhiFxKTWJQNSbx1oyqsnDwi472x0b/KxKvAlTVs62qLO7E9mT1SWygPPqE4dfUylKptfOj0xN+HfVlhoOeL1PyReeZkjezRltW7X7RPm4fI4tJRLCzSSgUDlkVReEpuhwLaHFhq2ZASxhRK22b+21cvvALgLdzM842G1h2eXkRAdD+HI7ICWBa2z5PKNjIKxk1rmrEh8P45arI41SWVV/SDlVJFSfCU0s5PbXDXl0B8FTB0ZfJSGIzpFSwW7p0qZ5++mkdPnxYgwcP1u9//3uNHj062WW1nkBlXCA7EQlodQS1+tr9xa1fk6dT3C9j59pnc974X8zONX5xa2znFxQAms7pqn4dVV7rPnagMjH81TrxL6m+KlNfW7S9sjT8mKEqqfx4eGopT4aU1kVKy27i1EVKyzot/+6kTLBbt26dZs2apaVLl2rUqFFatmyZrrvuOu3du1dnnXVWsssLC1ZJ5Seqf6Bjwauo4UAWbY8Nt7eQOy18tpOWHf7BTghj8QEsq3rZG1mPLUfmfIQAANgrejk7/YyWP1YoGA6HCcGwuPpqUew2nZIabXX0i/49jF6eLvlb82rydAr/LUzv0vRg6MsKv3EsxTiMqfme/Y7p0ksv1cUXX6znn38+1nb++efrpptuUmFh4Sn3Ly4uVnZ2toqKipSVldVw56qK6nDWqOlEeF5Z0sKvMsKXFRfMIuEsLbvutrRsyZed2O72tU4dAAAkQ/RNdxXxAyMnaqw3MEVHD1vK2zkx7KWfEZ4yIvP0rlJG1+rl9DPC65701nn+ZkiJKFpZWaldu3Zpzpw5Ce1jxozRe++917QH+2iV5KoZ3E4krrdo5MwRd3bQpZ4AFt9WI6z5shglAwCc3lyecEDK6Nq8/YOBGrc1NWIqj+sbfXdz9BJz8aGmPb87LTHopXepsR4Jguff0Lyvr6GnbvVHbAM//PCDgsGg8vIS7y3Iy8vTkSNH6tzH7/fL76/+OIyioiJJUvHrD0m+xtxo76oOZ/HzutrSukjp0Wv62c0PZlVK3mcwAQBgFbfkzpE650idm7hrsEqqKJH8JyK3UBWHb6uKDQSdCIfGhPXI3AQkf7l08pCkUwTCud8pMzNTjlZ8A2BKBLuoml+4Mabeb0ZhYaEWLFhQq733vzRlePZEE/oCAAA0wT9l6+jRo+rWrVurPWRKBLucnBy5XK5ao3NHjx6tNYoXNXfuXM2ePTu2fuLECfXp00cHDhxQdnZ2m9aL1lNcXKzevXvr4MGDp743Eh0Gxy01cdxSE8ctNUWPm9fbup/FmBLBzuv1atiwYdq8ebNuvvnmWPvmzZs1YcKEOvfx+Xzy+Wq/iSA7O5sf/BSUlZXFcUtBHLfUxHFLTRy31NSal2GlFAl2kjR79mxNnjxZBQUFGjFihF588UUdOHBA06ZNS3ZpAAAAHULKBLvbbrtNx44d029/+1sdPnxYF1xwgd544w316dMn2aUBAAB0CCkT7CRp+vTpmj59erP29fl8mjdvXp2XZ9FxcdxSE8ctNXHcUhPHLTW11XFLmQ8oBgAAQMOcyS4AAAAArYNgBwAAYAmCHQAAgCWsCnZLly5Vv379lJaWpmHDhmnbtm0N9t+yZYuGDRumtLQ09e/fXy+88EI7VYp4TTlur776qq655hp169ZNWVlZGjFihDZu3NiO1SKqqb9vUe+++67cbrcuvPDCti0QdWrqcfP7/Xr44YfVp08f+Xw+nX322frXf/3XdqoWUU09bqtXr9bQoUOVkZGhHj166K677tKxY8faqVps3bpV48ePV35+vhwOhzZs2HDKfVotkxhLvPzyy8bj8Zjly5ebvXv3mpkzZ5pOnTqZb7/9ts7+33zzjcnIyDAzZ840e/fuNcuXLzcej8f86U9/aufKT29NPW4zZ840Tz75pPnwww/NF198YebOnWs8Ho/56KOP2rny01tTj1vUiRMnTP/+/c2YMWPM0KFD26dYxDTnuN14443m0ksvNZs3bzb79u0zf/3rX827777bjlWjqcdt27Ztxul0mmeeecZ88803Ztu2bWbw4MHmpptuaufKT19vvPGGefjhh82f//xnI8msX7++wf6tmUmsCXbDhw8306ZNS2gbOHCgmTNnTp39f/Ob35iBAwcmtN13333msssua7MaUVtTj1tdBg0aZBYsWNDapaEBzT1ut912m3nkkUfMvHnzCHZJ0NTj9uabb5rs7Gxz7Nix9igP9WjqcXv66adN//79E9qeffZZ06tXrzarEfVrTLBrzUxixaXYyspK7dq1S2PGjEloHzNmjN57770693n//fdr9R87dqx27typqqqqNqsV1Zpz3GoKhUIqKSlR165d26JE1KG5x23lypX6+uuvNW/evLYuEXVoznF7/fXXVVBQoKeeeko9e/bUueeeqwcffFDl5eXtUTLUvOM2cuRIfffdd3rjjTdkjNH//d//6U9/+pPGjRvXHiWjGVozk6TUBxTX54cfflAwGFReXl5Ce15eno4cOVLnPkeOHKmzfyAQ0A8//KAePXq0Wb0Ia85xq+mf//mfdfLkSd16661tUSLq0Jzj9uWXX2rOnDnatm2b3G4rXnZSTnOO2zfffKPt27crLS1N69ev1w8//KDp06frxx9/5D67dtKc4zZy5EitXr1at912myoqKhQIBHTjjTfqueeea4+S0QytmUmsGLGLqvmPdI0xDf5z3br619WOttXU4xa1du1azZ8/X+vWrVNubm5blYd6NPa4BYNB3X777VqwYIHOPffc9ioP9WjK71soFJLD4dDq1as1fPhwXX/99Vq0aJFeeuklRu3aWVOO2969e/XAAw/oscce065du/TWW29p3759/G/1Dq61MokVp845OTlyuVy1zl6OHj1aKwFHde/evc7+brdbZ555ZpvVimrNOW5R69at0y9+8Qu98soruvrqq9uyTNTQ1ONWUlKinTt3avfu3br//vslhQODMUZut1ubNm3Sz372s3ap/XTWnN+3Hj16qGfPnsrOzo61nX/++TLG6LvvvtOAAQPatGY077gVFhZq1KhR+vWvfy1JGjJkiDp16qTRo0fr8ccf54pUB9SamcSKETuv16thw4Zp8+bNCe2bN2/WyJEj69xnxIgRtfpv2rRJBQUF8ng8bVYrqjXnuEnhkbo777xTa9as4Z6RJGjqccvKytLHH3+sPXv2xKZp06bpvPPO0549e3TppZe2V+mnteb8vo0aNUp/+9vfVFpaGmv74osv5HQ61atXrzatF2HNOW5lZWVyOhP/vLtcLknVo0DoWFo1kzT57RYdVPTt4CtWrDB79+41s2bNMp06dTL79+83xhgzZ84cM3ny5Fj/6FuLf/WrX5m9e/eaFStW8HEnSdDU47ZmzRrjdrvNkiVLzOHDh2PTiRMnkvUlnJaaetxq4l2xydHU41ZSUmJ69eplJk6caD799FOzZcsWM2DAADN16tRkfQmnpaYet5UrVxq3222WLl1qvv76a7N9+3ZTUFBghg8fnqwv4bRTUlJidu/ebXbv3m0kmUWLFpndu3fHPqKmLTOJNcHOGGOWLFli+vTpY7xer7n44ovNli1bYtumTJlirrjiioT+77zzjrnooouM1+s1ffv2Nc8//3w7VwxjmnbcrrjiCiOp1jRlypT2L/w019Tft3gEu+Rp6nH73//9X3P11Veb9PR006tXLzN79mxTVlbWzlWjqcft2WefNYMGDTLp6emmR48e5o477jDfffddO1d9+nr77bcb/FvVlpnEYQzjsgAAADaw4h47AAAAEOwAAACsQbADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEsQ7AAAACxBsAMAALAEwQ4AGmHt2rVKS0vToUOHYm1Tp07VkCFDVFRUlMTKAKCawxhjkl0EAHR0xhhdeOGFGj16tBYvXqwFCxboD3/4gz744AP17Nkz2eUBgCTJnewCACAVOBwOLVy4UBMnTlR+fr6eeeYZbdu2jVAHoENhxA4AmuDiiy/Wp59+qk2bNumKK65IdjkAkIB77ACgkTZu3KjPPvtMwWBQeXl5yS4HAGphxA4AGuGjjz7SlVdeqSVLlujll19WRkaGXnnllWSXBQAJuMcOAE5h//79GjdunObMmaPJkydr0KBBuuSSS7Rr1y4NGzYs2eUBQAwjdgDQgB9//FGjRo3ST3/6Uy1btizWPmHCBPn9fr311ltJrA4AEhHsAAAALMGbJwAAACxBsAMAALAEwQ4AAMASBDsAAABLEOwAAAAsQbADAACwBMEOAADAEgQ7AAAASxDsAAAALEGwAwAAsATBDgAAwBIEOwAAAEv8f2NOLTnGmcXPAAAAAElFTkSuQmCC\n", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p1 = plot(zx, xlim=(0.0, 1.0), ylim=(0,10), show=False, legend=True)\n", + "p2 = plot( (t*T).replace(t, x), xlim=(0.0,1.0), show=False)\n", + "p1.append(p2[0])\n", + "p1.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 78, + "id": "27da5481-766b-4062-a947-aa1d565a6735", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "{-2.50462606288666: 1, 3.50462606288666: 1}" + ] + }, + "execution_count": 78, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "roots(z, t)" + ] + }, + { + "cell_type": "code", + "execution_count": 101, + "id": "6927383c-e62a-436e-bbc9-79bbe7f4273f", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.3⋅t⋅(1 - t) + 0.7\n", + "───────────────────\n", + "0.9⋅t⋅(1 - t) + 7.9\n", + "----------------------------------------------------------------------------------------------------\n", + "0.3⋅t⋅(t - 1) - 0.7\n", + "───────────────────\n", + "0.9⋅t⋅(t - 1) - 7.9\n", + "----------------------------------------------------------------------------------------------------\n", + " 2 \n", + " 0.3⋅t 0.3⋅t 0.7 \n", + "- ────────────────────── + ────────────────────── + ──────────────────────\n", + " 2 2 2 \n", + " - 0.9⋅t + 0.9⋅t + 7.9 - 0.9⋅t + 0.9⋅t + 7.9 - 0.9⋅t + 0.9⋅t + 7.9\n", + "----------------------------------------------------------------------------------------------------\n", + " ⎛ 2 ⎞\n", + "0.0886075949367088⋅⎝0.428571428571429⋅t - 0.428571428571429⋅t - 1.0⎠\n", + "─────────────────────────────────────────────────────────────────────\n", + " 2 \n", + " 0.113924050632911⋅t - 0.113924050632911⋅t - 1.0 \n", + "----------------------------------------------------------------------------------------------------\n", + "t⋅(0.3 - 0.3⋅t) + 0.7\n", + "─────────────────────\n", + "t⋅(0.9 - 0.9⋅t) + 7.9\n", + "----------------------------------------------------------------------------------------------------\n", + " 2 \n", + "0.3⋅t - 0.3⋅t - 0.7\n", + "────────────────────\n", + " 2 \n", + "0.9⋅t - 0.9⋅t - 7.9\n", + "----------------------------------------------------------------------------------------------------\n", + " 0.244725738396624 \n", + "0.333333333333333 + ────────────────────────────────────────────────\n", + " 2 \n", + " 0.113924050632911⋅t - 0.113924050632911⋅t - 1.0\n", + "----------------------------------------------------------------------------------------------------\n" + ] + } + ], + "source": [ + "e = (0.7 + 0.3 * t * T)/z;\n", + "for ei in (e, simplify(e), expand(e), factor(e), collect(e, t), cancel(e), apart(e)):\n", + " pprint(ei, use_unicode=True)\n", + " print(100*\"-\")" + ] + }, + { + "cell_type": "code", + "execution_count": 102, + "id": "ba9fc88f-028c-476f-aa94-22c2e15d6e2c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "sympy.core.add.Add" + ] + }, + "execution_count": 102, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "z.func" + ] + }, + { + "cell_type": "code", + "execution_count": 103, + "id": "b4bc5ad7-3bb9-44d0-9b70-d27d6c9876f2", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(7.90000000000000, 0.9*t*(1 - t))" + ] + }, + "execution_count": 103, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "z.args" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "74c9de1c-01fe-42b6-9b92-a787cfc246b2", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.15" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/students/amartins/tarefas/__pycache__/bninput.cpython-311.pyc b/students/amartins/tarefas/__pycache__/bninput.cpython-311.pyc new file mode 100644 index 0000000..8c968ab Binary files /dev/null and b/students/amartins/tarefas/__pycache__/bninput.cpython-311.pyc differ diff --git a/students/amartins/tarefas/__pycache__/bninput.cpython-39.pyc b/students/amartins/tarefas/__pycache__/bninput.cpython-39.pyc new file mode 100644 index 0000000..3e10148 Binary files /dev/null and b/students/amartins/tarefas/__pycache__/bninput.cpython-39.pyc differ diff --git a/students/amartins/tarefas/__pycache__/tarefa1.cpython-311.pyc b/students/amartins/tarefas/__pycache__/tarefa1.cpython-311.pyc new file mode 100644 index 0000000..c237b1c Binary files /dev/null and b/students/amartins/tarefas/__pycache__/tarefa1.cpython-311.pyc differ diff --git a/students/amartins/tarefas/__pycache__/tarefa2.cpython-311.pyc b/students/amartins/tarefas/__pycache__/tarefa2.cpython-311.pyc new file mode 100644 index 0000000..c427a1a Binary files /dev/null and b/students/amartins/tarefas/__pycache__/tarefa2.cpython-311.pyc differ diff --git a/students/amartins/tarefas/asia.lp b/students/amartins/tarefas/asia.lp new file mode 100644 index 0000000..509ec29 --- /dev/null +++ b/students/amartins/tarefas/asia.lp @@ -0,0 +1,43 @@ +%* 0.01 :: asia. *% +asia ; -asia. + +%* 0.05 :: tub :- asia. *% +tub ; -tub :- asia. +%* 0.01 :: tub :- -asia. *% +tub ; -tub :- -asia. + +%* 0.5 :: smoke. *% +smoke ; -smoke. + +%* 0.1 :: lung :- smoke. *% +lung ; -lung :- smoke. +%* 0.01 :: lung :- -smoke. *% +lung ; -lung :- -smoke. + +%* 0.6 :: bronc :- smoke. *% +bronc ; -bronc :- smoke. +%* 0.3 :: bronc :- -smoke. *% +bronc ; -bronc :- -smoke. + +%* 1.0 :: either :- lung, tub. *% +either ; -either :- lung, tub. +%* 1.0 :: either :- -lung, tub. *% +either ; -either :- -lung, tub. +%* 1.0 :: either :- lung, -tub. *% +either ; -either :- lung, -tub. +%* 0.0 :: either :- -lung, -tub. *% +either ; -either :- -lung, -tub. +%* 0.98 :: xray :- either. *% +xray ; -xray :- either. +%* 0.05 :: xray :- -either. *% +xray ; -xray :- -either. + +%* 0.9 :: dysp :- bronc, either. *% +dysp ; -dysp :- bronc, either. +%* 0.7 :: dysp :- -bronc, either. *% +dysp ; -dysp :- -bronc, either. +%* 0.8 :: dysp :- bronc, -either. *% +dysp ; -dysp :- bronc, -either. +%* 0.1 :: dysp :- -bronc, -either. *% +dysp ; -dysp :- -bronc, -either. + diff --git a/students/amartins/tarefas/asia2.bif b/students/amartins/tarefas/asia2.bif new file mode 100644 index 0000000..1543625 --- /dev/null +++ b/students/amartins/tarefas/asia2.bif @@ -0,0 +1,60 @@ +network unknown { +} +variable asia { + type discrete [ 2 ] { yes, no }; +} +variable tub { + type discrete [ 2 ] { yes, no }; +} +variable smoke { + type discrete [ 2 ] { yes, no }; +} +variable lung { + type discrete [ 2 ] { yes, no }; +} +variable bronc { + type discrete [ 2 ] { yes, no }; +} +variable either { + type discrete [ 2 ] { yes, no }; +} +variable xray { + type discrete [ 2 ] { yes, no }; +} +variable dysp { + type discrete [ 2 ] { yes, no }; +} +probability ( asia ) { + table 0.01, 0.99; +} +probability ( tub | asia ) { + (yes) 0.05, 0.95; + (no) 0.01, 0.99; +} +probability ( smoke ) { + table 0.5, 0.5; +} +probability ( lung | smoke ) { + (yes) 0.1, 0.9; + (no) 0.01, 0.99; +} +probability ( bronc | smoke ) { + (yes) 0.6, 0.4; + (no) 0.3, 0.7; +} +probability ( either | lung, tub ) { + (yes, yes) 1.0, 0.0; + (no, yes) 1.0, 0.0; + (yes, no) 1.0, 0.0; + (no, no) 0.0, 1.0; +} +probability ( xray | either ) { + (yes) 0.98, 0.02; + (no) 0.05, 0.95; +} +probability ( dysp | bronc, either ) { + (yes, yes) 0.9, 0.1; + (no, yes) 0.7, 0.3; + (yes, no) 0.8, 0.2; + (no, no) 0.1, 0.9; +} diff --git a/students/amartins/tarefas/asia2.pkl b/students/amartins/tarefas/asia2.pkl new file mode 100644 index 0000000..478d9c1 Binary files /dev/null and b/students/amartins/tarefas/asia2.pkl differ diff --git a/students/amartins/tarefas/basic.lp b/students/amartins/tarefas/basic.lp new file mode 100644 index 0000000..6af29c0 --- /dev/null +++ b/students/amartins/tarefas/basic.lp @@ -0,0 +1,3 @@ +2 { numero(1..6) } 2. + +par(2 * X) :- numero(X). \ No newline at end of file diff --git a/students/amartins/tarefas/bninput.py b/students/amartins/tarefas/bninput.py new file mode 100644 index 0000000..87aa889 --- /dev/null +++ b/students/amartins/tarefas/bninput.py @@ -0,0 +1,99 @@ +import pickle + +from pgmpy.readwrite import BIFReader +import networkx as nx +import pydot +from networkx.drawing.nx_pydot import graphviz_layout +from pgmpy.models import BayesianModel +from test import * + +import unittest + +def summary_dag(filename): + """A função "summary_dag" tem como argumento uma string com o nome do ficheiro a ler e + retorna um modelo com os dados recolhidos do respetivo ficheiro. + """ + file = BIFReader(filename) + model = file.get_model() # Cria um objeto BayesianModel + nodes = list(model.nodes()) # Converte o NodeView para uma lista + + asc = 0 + des = 0 + nodes_in = 0 + nodes_out = 0 + + for x in nodes: + # Obtem os nós pais e filhos de cada nós + parents = model.get_parents(x) + children = model.get_children(x) + + if len(parents) != 0: + des += 1 + if len(children) != 0: + asc += 1 + nodes_in += len(children) + nodes_out += len(parents) + + return { + 'dag_file': filename, + 'nodes': nodes, + 'count_parents': des, + 'count_children': asc, + 'count_in_edges': nodes_in, + 'count_out_edges': nodes_out, + 'bnmodel': model + } + +def summary_str(d): + s = "DAG file: " + d["dag_file"] + "\n" + s += " Number of nodes: " + str(len(d["nodes"])) + "\n" + s += " Number of parents: " + str(d["count_parents"]) + "\n" + s += " Number of children: " + str(d["count_children"]) + "\n" + s += " Average in-degree: " + str(d["count_in_edges"]/len(d["nodes"])) + "\n" + s += " Average out-degree: " + str(d["count_out_edges"]/len(d["nodes"])) + "\n" + s += " Nodes:" + "\n" + nodes = d["nodes"] + for x in nodes: + s += " " + str(x) + "\n" + return s + +def show_dag(model): + G = nx.DiGraph() + dot = pydot.Dot(graph_type='digraph') + for node in model['nodes']: + G.add_node(node) + dot.add_node(pydot.Node(node)) + for edge in model['bnmodel'].edges: + G.add_edge(edge[0], edge[1]) + dot.add_edge(pydot.Edge(edge[0], edge[1])) + # objeto pydot + + # defina o layout do grafo + pos = graphviz_layout(G, prog='dot') + # desenhe o grafo usando o networkx + nx.draw(G, pos, with_labels=True) + +def save_dag(model, target_filename): + G = nx.DiGraph() + for node in model.nodes: + G.add_node(node) + for edge in model.edges: + G.add_edge(edge[0], edge[1]) + # objeto pydot + dot = pydot.Dot(graph_type='digraph') + for node in model.nodes: + dot.add_node(pydot.Node(node)) + for edge in model.edges: + dot.add_edge(pydot.Edge(edge[0], edge[1])) + # defina o layout do grafo + pos = graphviz_layout(G, prog='dot') + # renderize o grafo com o pydot + graph = pydot.graph_from_dot_data(dot.to_string())[0] + graph.write_png(target_filename) + +if __name__ == "__main__": + summary = summary_dag("asia2.bif") + pickle.dump(summary, open("asia2.pkl", "wb")) + message = summary_str(summary) + print(message) + unittest.main() \ No newline at end of file diff --git a/students/amartins/tarefas/bridge_pyasp.py b/students/amartins/tarefas/bridge_pyasp.py new file mode 100644 index 0000000..0404e6d --- /dev/null +++ b/students/amartins/tarefas/bridge_pyasp.py @@ -0,0 +1,22 @@ +from clingo import Control + +ctl = Control(["0"]) +ctl.add("base", [], "2 { numero(1..6) } 2.") +ctl.add("base", [], "%* 0.01 :: asia. *%") +ctl.add("base", [], "par(2 * X) :- numero(X).") +ctl.ground([("base", [])]) +print(ctl.solve(on_model=print)) + + +class LineInfo: + def __init__(self, statement, probability): + self.statement = statement + self.probability = probability + +[ + LineInfo("asia ; -asia. ", 0.01), + ... + LineInfo("bronc ; -bronc :- smoke.", 0.6), + ... + +] \ No newline at end of file diff --git a/students/amartins/tarefas/graph_asia2.png b/students/amartins/tarefas/graph_asia2.png new file mode 100644 index 0000000..5f3d141 Binary files /dev/null and b/students/amartins/tarefas/graph_asia2.png differ diff --git a/students/amartins/tarefas/notas.md b/students/amartins/tarefas/notas.md new file mode 100644 index 0000000..2235c4a --- /dev/null +++ b/students/amartins/tarefas/notas.md @@ -0,0 +1,67 @@ +# Notas com Alice Martins + +## 2023-05-04 + +### Problemas + +#### Tarefa 1A + +1. Está a contar os nós "in" e os nós "out" ao contrário, certo? `len(children)` diz quantas arestas saem de `x`. +2. Em + ```python + if len(parents) != 0: + asc += 1 + if len(children) != 0: + des += 1 + ``` + deve ser + ```python + if len(parents) != 0: + des += 1 + if len(children) != 0: + asc += 1 + ``` + porque o nó `x` é (mais) um descendente se tem ascendentes (e igualmente para ascendente). + +#### Tarefa 1B + +1. Não deve misturar o código para definir o grafo `G` com o código para o grafo `dot`. **Exceto** se quiser percorrer apenas uma vez `model.nodes` e `model.edges`. + +### Sugestões + +- **Use sempre, e só, o inglês no seu código**, não só porque não se podem usar acentos, como `nós_in`, mas também porque misturar português com inglês fica estranho (isto é, _feio_!). É melhor `nodes_in` do que `nos_in`. +- Use nomes descritivos nas suas variáveis: `node` em vez de `x`. +- Em vez de `asia = bn.import_DAG("asia2.bif")` talvez `dag = bn.import_DAG("asia2.bif")` seja melhor (**porquê?**) +- O código também fica mais simples se usar "f-strings": Em vez de + ```python + print("Número de nós:", len(nos)) + ``` + use + ```python + print(f"Número de nós: {len(nos)}") + ``` +### Desafios + +1. **Torne o seu código geral.** + 1. Coloque a parte essencial da tarefa 1A numa função, `summary_dag(filename)`, que tem argumento uma `string` com o nome do ficheiro e devolve um `dict` com chaves `dag_file`, `nodes`, `count_parents`, `count_children`, `count_in_edges`, `count_out_edges`. + 2. Use uma função `summary_str(d)` que tem argumento um `dict` como acima e devolve uma `string` com as perguntas e as respostas "bem formatadas". Por exemplo: + ```text + DAG file: asia2.bif + Number of nodes: 8 + Number of parents: 6 + Number of children: 6 + Average in-degree: 1.0 + Average out-degree: 1.0 + Nodes: + asia + tub + ... + ``` + 3. Por fim, combine as duas funções: + ```python + summary = summary_dag("asia2.bif") + message = summary_str(summary) + print(message) + ``` + 4. Coloque a parte essencial da tarefa 1B em funções `show_dag(model)` e `save_dag(model, target_filename)`. +2. **A biblioteca `bnlearn` é excessiva para estas tarefas.** Não só precisa de várias dependências irrelevantes para o que pretendemos, por exemplo o `pytorch`, como demora imenso tempo a ler o ficheiro. _Procure uma alternativa mais simples._ (**n.b.** não sei se existe!) \ No newline at end of file diff --git a/students/amartins/tarefas/t2profile.txt b/students/amartins/tarefas/t2profile.txt new file mode 100644 index 0000000..74ebe91 --- /dev/null +++ b/students/amartins/tarefas/t2profile.txt @@ -0,0 +1,9733 @@ +%* 0.01 :: asia. *% +asia ; -asia. + +%* 0.05 :: tub :- asia. *% +%* 0.01 :: tub :- -asia. *% +tub ; -tub :- asia. + +%* 0.5 :: smoke. *% +smoke ; -smoke. + +%* 0.1 :: lung :- smoke. *% +%* 0.01 :: lung :- -smoke. *% +lung ; -lung :- smoke. + +%* 0.6 :: bronc :- smoke. *% +%* 0.3 :: bronc :- -smoke. *% +bronc ; -bronc :- smoke. + +%* 1.0 :: either :- lung, tub. *% +%* 1.0 :: either :- -lung, tub. *% +%* 1.0 :: either :- lung, -tub. *% +%* 0.0 :: either :- -lung, -tub. *% +either ; -either :- lung, tub. + +%* 0.98 :: xray :- either. *% +%* 0.05 :: xray :- -either. *% +xray ; -xray :- either. + +%* 0.9 :: dysp :- bronc, either. *% +%* 0.7 :: dysp :- -bronc, either. *% +%* 0.8 :: dysp :- bronc, -either. *% +%* 0.1 :: dysp :- -bronc, -either. *% +dysp ; -dysp :- bronc, either. + + + 2414684 function calls (2368463 primitive calls) in 6.029 seconds + + Ordered by: cumulative time + + ncalls tottime percall cumtime percall filename:lineno(function) + 358 0.018 0.000 14.166 0.040 __init__.py:1() + 3133/1 0.083 0.000 6.032 6.032 {built-in method builtins.exec} + 1 0.000 0.000 6.032 6.032 tarefa2.py:1() + 1 0.000 0.000 3.050 3.050 bninput.py:10(summary_dag) + 1 0.000 0.000 3.047 3.047 BIF.py:38(__init__) + 1 0.000 0.000 3.031 3.031 BIF.py:304(get_values) + 1 0.000 0.000 3.031 3.031 parallel.py:1000(__call__) + 2908/7 0.019 0.000 2.985 0.426 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+1,81 @@ +from pgmpy.readwrite import BIFReader +import networkx as nx +import pydot +from networkx.drawing.nx_pydot import graphviz_layout + +def summary_dag(filename): + file = BIFReader(filename) + model = file.get_model() # Cria um objeto BayesianModel + nodes = model.nodes() + + asc = 0 + des = 0 + nodes_in = 0 + nodes_out = 0 + + for x in nodes: + # Obtem os nós pais e filhos de cada nós + parents = model.get_parents(x) + children = model.get_children(x) + + if len(parents) != 0: + des += 1 + if len(children) != 0: + asc += 1 + nodes_in += len(children) + nodes_out += len(parents) + + return {'dag_file': filename, 'nodes': nodes, 'count_parents': des, 'count_children': asc, 'count_in_edges': nodes_in, 'count_out_edges': nodes_out} + +def summary_str(d): + s = "DAG file: " + d["dag_file"] + "\n" + s += " Number of nodes: " + str(len(d["nodes"])) + "\n" + s += " Number of parents: " + str(d["count_parents"]) + "\n" + s += " Number of children: " + str(d["count_children"]) + "\n" + s += " Average in-degree: " + str(d["count_in_edges"]/len(d["nodes"])) + "\n" + s += " Average out-degree: " + str(d["count_out_edges"]/len(d["nodes"])) + "\n" + s += " Nodes:" + "\n" + nodes = d["nodes"] + for x in nodes: + s += " " + str(x) + "\n" + return s + +def show_dag(model): + G = nx.DiGraph() + for node in model.nodes: + G.add_node(node) + for edge in model.edges: + G.add_edge(edge[0], edge[1]) + # objeto pydot + dot = pydot.Dot(graph_type='digraph') + for node in model.nodes: + dot.add_node(pydot.Node(node)) + for edge in model.edges: + dot.add_edge(pydot.Edge(edge[0], edge[1])) + # defina o layout do grafo + pos = graphviz_layout(G, prog='dot') + # desenhe o grafo usando o networkx + nx.draw(G, pos, with_labels=True) + +def save_dag(model, target_filename): + G = nx.DiGraph() + for node in model.nodes: + G.add_node(node) + for edge in model.edges: + G.add_edge(edge[0], edge[1]) + # objeto pydot + dot = pydot.Dot(graph_type='digraph') + for node in model.nodes: + dot.add_node(pydot.Node(node)) + for edge in model.edges: + dot.add_edge(pydot.Edge(edge[0], edge[1])) + # defina o layout do grafo + pos = graphviz_layout(G, prog='dot') + # renderize o grafo com o pydot + graph = pydot.graph_from_dot_data(dot.to_string())[0] + graph.write_png(target_filename) + + +summary = summary_dag("asia2.bif") +message = summary_str(summary) +print(message) \ No newline at end of file diff --git a/students/amartins/tarefas/tarefa1a.py b/students/amartins/tarefas/tarefa1a.py new file mode 100644 index 0000000..f53898e --- /dev/null +++ b/students/amartins/tarefas/tarefa1a.py @@ -0,0 +1,38 @@ +import bnlearn as bn + +asia = bn.import_DAG("asia2.bif") + +model = asia['model'] # Cria um objeto BayesianModel + + + +nos = model.nodes() +print("Número de nós:", len(nos)) #resposta à pergunta "quantos nós tem a rede?" + +print("Lista de nós:", nos) + +asc = 0 +des = 0 +nos_in = 0 +nos_out = 0 + +for x in nos: + # Obtem os nós pais e filhos de cada nós + parents = model.get_parents(x) + children = model.get_children(x) + + if len(parents) != 0: + asc += 1 + if len(children) != 0: + des += 1 + nos_in += len(children) + nos_out += len(parents) + +#resposta à pergunta "quantos nós são descendentes? quantos são ascendentes?" +print("Nós ascendentes: ", asc) +print("Nós descendentes: ", des) +# resposta à pergunta "qual é o número médio de arestas "in"? e "out"?" +nos_in = float(nos_in/len(nos)) +nos_out = float(nos_out/len(nos)) +print("Número médio de arestas in: ", nos_in) +print("Número médio de arestas out: ", nos_out) \ No newline at end of file diff --git a/students/amartins/tarefas/tarefa1b.py b/students/amartins/tarefas/tarefa1b.py new file mode 100644 index 0000000..f0ddec1 --- /dev/null +++ b/students/amartins/tarefas/tarefa1b.py @@ -0,0 +1,35 @@ +import bnlearn as bn +import networkx as nx +import pydot +from networkx.drawing.nx_pydot import graphviz_layout + +asia = bn.import_DAG("asia2.bif") +model = asia['model'] + +# DiGraph do networkx +G = nx.DiGraph() + +for node in model.nodes: + G.add_node(node) + +for edge in model.edges: + G.add_edge(edge[0], edge[1]) + +# objeto pydot +dot = pydot.Dot(graph_type='digraph') + +for node in model.nodes: + dot.add_node(pydot.Node(node)) + +for edge in model.edges: + dot.add_edge(pydot.Edge(edge[0], edge[1])) + +# defina o layout do grafo +pos = graphviz_layout(G, prog='dot') + +# desenhe o grafo usando o networkx +nx.draw(G, pos, with_labels=True) + +# renderize o grafo com o pydot +graph = pydot.graph_from_dot_data(dot.to_string())[0] +graph.write_png('graph_asia2.png') diff --git a/students/amartins/tarefas/tarefa2.md b/students/amartins/tarefas/tarefa2.md new file mode 100644 index 0000000..2eb0254 --- /dev/null +++ b/students/amartins/tarefas/tarefa2.md @@ -0,0 +1,87 @@ +# Tarefa 2: Ler Redes Bayesianas, Escrever Programas Lógicos + +> **Estado da Tarefa.** Importação de Redes Bayesianas - OK; Construção de Programa Lógico a Partir de uma RB - Em Curso. + +## Importar uma Rede Bayesiana + +Passos: + +- [x] Implementar +- [ ] Testar e Documentar +- [x] Usar + +Função `summary_dag(filename)` no módulo `bninput`. **Deve ser testada e documentada.** + +## Construir um Programa Lógico dada uma Rede Bayesiana + +Passos: + +- [x] Implementar +- [ ] Testar e Documentar +- [ ] Usar + +### 2023-07-20 + +O ficheiro `tarefa2.py` está **quase** adequado para esta tarefa. Em particular, tem código para converter a descrição de uma bn em _algo que se assemelha a um programa lógico_. No entanto: + +**Criar funções.** À semelhança do que fez no `bninput`, deve **colocar o código "essencial" em funções**. Isto é, o essencial de + +```python +if __name__ == "__main__": + summary = summary_dag("asia2.bif") + model = summary["bnmodel"] + probabilities = get_yes_probabilities(model) + for node, yes_prob in probabilities.items(): + parents = model.get_parents(node) + s = "" + if len(parents) == 0: +... +``` + +deve ir para uma função. A minha sugestão é que o argumento dessa função seja um `model` que poderá resultar de, por exemplo, `summary_dag(...)`. + +**Adaptar a notação dos programas lógicos.** + +A sintaxe para os programas lógicos é a seguinte: +```prolog +f. /* Facto Determinista */ +h :- b1, ..., bN. /* Regra Determinista */ +p::f. /* Facto Probabilístico */ +p::h :- b1, ..., bN./* Regra Probabilística */ +``` + +em que `p` é uma probabilidade (um `float` entre 0 e 1); `f` é um "facto" (por exemplo, `asia`) e `h :- b1, ..., bN` é uma "regra" em que `h` é a "cabeça" (_"head"_) e o "corpo" (_"body"_) tem "literais" (factos ou negações de factos) `b1, ..., bN`. O símbolo "`,`" denota a _conjunção_ ($\wedge$), "`-`" a negação ($\neg$) e "`:-`" (em vez de "`<-`", e lê-se "_if_" ou "se") denota $\leftarrow$. + +Além disso, em relação ao que o seu programa produz, cada regra e cada facto termina em "`.`". Portanto, **falta acertar a sintaxe com a dos programas lógicos.** + +**Sintaxe, parte 2** + +Há, ainda, um aspeto adicional: Os programas que processam os programas lógicos não suportam (mais ou menos, em geral, por enquanto) factos e regras probabilísticas. Isso significa que a sintaxe +```prolog +p::f. /* Facto Probabilístico */ +p::h :- b1, ..., bN./* Regra Probabilística */ +``` +está "errada" para esses programas. O que podemos fazer, por enquanto, é escrever +```prolog +%* p::f. *% +f ; -f. +%* p::h. *% +h ; -h :- b1, ..., bN. +``` + +Por exemplo, +```prolog +%* 0.01::asia. *% +asia ; -asia. +``` +em vez de +```prolog +0.01::asia. +``` + +Nestes exemplos a sintaxe dos programas lógicos está acrescentada com "`;`" para denotar a disjunção ($\vee$) e "`%* ... *%`" para blocos de comentários. Isto é, +```prolog +%* 0.01::asia. *% +asia ; -asia. +``` +diz que temos um **facto disjuntivo**, `asia ; -asia` que indica que ou "acontece" `asia` ou "acontece" não `asia`. O comentário `%* 0.01::asia. *%` serve para "transportar" a informação sobre as probabilidades. Esta informação será tratada posteriormente, talvez na tarefa 4 ou na 5. \ No newline at end of file diff --git a/students/amartins/tarefas/tarefa2.pdf b/students/amartins/tarefas/tarefa2.pdf new file mode 100644 index 0000000..d751b67 Binary files /dev/null and b/students/amartins/tarefas/tarefa2.pdf differ diff --git a/students/amartins/tarefas/tarefa2.py b/students/amartins/tarefas/tarefa2.py new file mode 100755 index 0000000..87ab094 --- /dev/null +++ b/students/amartins/tarefas/tarefa2.py @@ -0,0 +1,79 @@ +#!env python +import pickle +from pgmpy.readwrite import BIFReader +import networkx as nx +import pydot +from networkx.drawing.nx_pydot import graphviz_layout +from bninput import * + +def get_yes_probabilities(model): + nodes = model.nodes() + + yes_probabilities = {} + + for node in nodes: + cpd = model.get_cpds(node) + state_names = cpd.state_names[node] + yes_index = state_names.index("yes") + yes_prob = cpd.values[yes_index] + yes_probabilities[node] = yes_prob + + return yes_probabilities + +def compile_bn2asp(model): + """_summary_ + + Args: + model (_type_): _description_ + + Returns: + _type_: _description_ + """ + probabilities = get_yes_probabilities(model) + sf = "" + for node, yes_prob in probabilities.items(): + parents = model.get_parents(node) + s = "" + if len(parents) == 0: + s += f"%* {yes_prob} :: {node}. *% \n" + s += f"{node} ; -{node}. \n\n" + sf += s + elif len(parents) == 1: + p = parents[0] + s += f"%* {yes_prob[0]} :: {node} :- {p}. *% \n" + s += f"{node} ; -{node} :- {p}. \n" + s += f"%* {yes_prob[1]} :: {node} :- -{p}. *% \n" + s += f"{node} ; -{node} :- -{p}. \n\n" + sf += s + else: + yes1 = yes_prob[0] + yes2 = yes_prob[1] + p0 = parents[0] + p1 = parents[1] + s += f"%* {yes1[0]} :: {node} :- {p0}, {p1}. *% \n" + s += f"{node} ; -{node} :- {p0}, {p1}. \n" + s += f"%* {yes2[0]} :: {node} :- -{p0}, {p1}. *% \n" + s += f"{node} ; -{node} :- -{p0}, {p1}. \n" + s += f"%* {yes1[1]} :: {node} :- {p0}, -{p1}. *% \n" + s += f"{node} ; -{node} :- {p0}, -{p1}. \n" + s += f"%* {yes2[1]} :: {node} :- -{p0}, -{p1}. *% \n" + s += f"{node} ; -{node} :- -{p0}, -{p1}. \n" + + sf += s + return sf + +if __name__ == "__main__": + # summary = summary_dag("/home/fc/sci/projetos/zugzwang/students/amartins/tarefas/asia2.bif") + summary = pickle.load(open("asia2.pkl", "rb")) + # 3.2776401042938232 + + model = summary["bnmodel"] + # 1.1920928955078125e-06 + + sum = compile_bn2asp(model) + # 0.00010275840759277344 + + print(sum) + # 5.269050598144531e-05 + + # TOTAL DE TEMPO: 5.778 SEGUNDOS diff --git a/students/amartins/tarefas/test/test_bninput.py b/students/amartins/tarefas/test/test_bninput.py new file mode 100644 index 0000000..e04034f --- /dev/null +++ b/students/amartins/tarefas/test/test_bninput.py @@ -0,0 +1,13 @@ +import unittest +from bninput import * + +class Test(unittest.TestCase): + def test_valid_file(self): + result = summary_dag('asia2.bif') + + self.assertEqual(result['dag_file'], 'asia2.bif') + self.assertIsInstance(result['nodes'], list) + self.assertIsInstance(result['count_parents'], int) + self.assertIsInstance(result['count_children'], int) + self.assertIsInstance(result['count_in_edges'], int) + self.assertIsInstance(result['count_out_edges'], int) \ No newline at end of file diff --git a/students/amartins/tarefas/v01-tarefa1a.py b/students/amartins/tarefas/v01-tarefa1a.py new file mode 100644 index 0000000..bac9ca2 --- /dev/null +++ b/students/amartins/tarefas/v01-tarefa1a.py @@ -0,0 +1,31 @@ +import bnlearn as bn + +asia = bn.import_DAG("asia2.bif") + +model = asia['model'] # Cria um objeto BayesianModel + + + +nos = model.nodes() +print("Número de nós:", len(nos)) #resposta à pergunta "quantos nós tem a rede?" + +print("Lista de nós:", nos) + +nos_in = 0 +nos_out = 0 + +for x in nos: + # Obtem os nós pais e filhos de cada nós + parents = model.get_parents(x) + children = model.get_children(x) + #resposta à pergunta "quantos nós são descendentes? quantos são ascendentes?" + print("Pais de ", x, ": ", parents) + print("Filhos de ", x, " :", children) + nos_in += len(children) + nos_out += len(parents) + +# resposta à pergunta "qual é o número médio de arestas "in"? e "out"?" +nos_in = nos_in/len(nos) +nos_out = nos_out/len(nos) +print("Número médio de arestas in: ", nos_in) +print("Número médio de arestas out: ", nos_out) \ No newline at end of file diff --git a/students/biblio/final_paper.pdf b/students/biblio/final_paper.pdf new file mode 100644 index 0000000..e99d73f Binary files /dev/null and b/students/biblio/final_paper.pdf differ diff --git a/students/zugzwuang-amartins.code-workspace b/students/zugzwuang-amartins.code-workspace new file mode 100644 index 0000000..883c31e --- /dev/null +++ b/students/zugzwuang-amartins.code-workspace @@ -0,0 +1,11 @@ +{ + "folders": [ + { + "path": "amartins/tarefas" + }, + { + "path": "amartins" + } + ], + "settings": {} +} \ No newline at end of file diff --git a/text/backup/probast_draft/probast_draf.pdf b/text/backup/probast_draft/probast_draf.pdf new file mode 100644 index 0000000..8b9c2dd Binary files /dev/null and b/text/backup/probast_draft/probast_draf.pdf differ diff --git a/text/backup/probast_draft/probast_draf.tex b/text/backup/probast_draft/probast_draf.tex new file mode 100644 index 0000000..a24ed2b --- /dev/null +++ b/text/backup/probast_draft/probast_draf.tex @@ -0,0 +1,323 @@ +\documentclass[bigger]{beamer} +\useinnertheme{circles} +\usefonttheme[onlymath]{serif} +\usefonttheme{structurebold} +\setbeamertemplate{navigation symbols}{} +\usepackage{xcolor} +\setbeamercolor{highlight block}{bg=gray} + +\usepackage{tikz} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[normalem]{ulem} % To strikeout +\usepackage{commath} + +\newcommand{\naf}{\ensuremath{\sim\!\!}} + +\title{Probabilistic Answer Set Programming} +\subtitle{A Research Draft} +\author{Francisco Coelho} +\institute[\texttt{fc@uevora.pt}]{ + NOVA LINCS \&\\ + High Performance Computing Chair \&\\ + Departamento de Informática, Universidade de Évora +} + +\begin{document} + \begin{frame}[plain] + \titlepage + \end{frame} + + \section*{Motivation} + + \begin{frame} + \frametitle{In short\ldots\hfill\small \ldots a word wall. I'm sorry.} + + \begin{itemize} + \item \textbf{Machine Learning} has important limitations: + \begin{itemize} + \item The \emph{one table, conditionally independent rows} assumption. + \item \emph{Background knowledge} is hard to include. + \item \emph{Training} requires ``large'' amounts of data. + \item \emph{Models} are hard do interpret. + \end{itemize} + \item \textbf{Inductive Logic Programming} is based on first order logic --- solves all the problems above but is sensible to \emph{noise}. + \item \textbf{Distribution Semantics} defines the probability of a proposition from probabilities of the (marginally independent) facts. + \item \textbf{Answer Set Programs} resets the common syntax and semantic of logic programs; A ``program'' defines \emph{stable models}, not a computation neither a variable substitution. + \end{itemize} + \end{frame} + + \begin{frame} + \frametitle{\xout{Goals} Wish list} + + \begin{block}{Extend distribution semantics to answer sets} + \begin{itemize} + \item Within a theoretical framework. + \item Computationally applicable to ``real world'' scenarios. + \item Easy to include background knowledge. + \item Perform common tasks such as \emph{marg, mle, map, etc.} + \item Learn program ``parameters'' and ``structure'' from \emph{noisy samples} --- possibly using \emph{templates}. + \item Related to Bayesian Networks, HMMs, \emph{etc.} + \end{itemize} + + \end{block} + \end{frame} + + \section{Development} + + \begin{frame} + \tableofcontents[currentsection] + \end{frame} + + \begin{frame} + \frametitle{The seed on an idea} + We want to define the \textbf{joint distribution} of the stable models. + \begin{enumerate} + \item A \textbf{boolean random variable} can be described by a disjunction $a; \neg a$. + \item This ASP program has two stable models: $a$ and $\neg a$. + \item A program with $n$ such facts $a_i; \neg a_i$ has $2^n$ stable models, the distinct combinations of those choices. + \item \textbf{If each $a_i$ has probability $p_i$ then the probability of a stable model $W$ would be} $$P(W) = \prod_{a_i \in W}p_i \prod_{\neg a_i \in W} (1 - p_i).$$ + \end{enumerate} + \pause + \begin{alertblock}{But this is wrong.} + Even assuming that those facts are marginally independent --- which we will do. + \end{alertblock} + \end{frame} + + \begin{frame}{Problem 1: Disjuntive Clauses} + The ASP program with probabilistic facts + $$ + \begin{aligned} + &b \vee \neg b\\ + &h_1 \vee h_2 \leftarrow b + \end{aligned} + $$ + has \textbf{three} stable models: $\set{\neg b}, \set{b, h_1}$ and $\set{b, h_2}$. + + \begin{block}{How to assign a probability to each model?} + \pause + Possible approaches: + \begin{enumerate} + \item Pre-assign a \textbf{conditional distribution of the head}: + $$P(h_1, h_2 | b).$$ + \item Bayesian learn from \textbf{observations}: + $$P(h_1, h_2 | b,z) \propto P(b, z | h_1, h_2) P(h_1, h_2).$$ + \item Start with the former as \textbf{prior} and \textbf{update} with the latter. + \end{enumerate} + \end{block} + + \end{frame} + + \begin{frame} + \frametitle{Questions to address} + + \begin{itemize} + \item How to \textbf{match} an observation $z$ with a clause case $h,b$? + \item How do observations \textbf{update} the probabilities? + \item Why match observations with clauses and \textbf{not with stable models}? + \item Is this just \textbf{bayesian networking}? + \item How to frame this in a \textbf{sound theoretic setting}? + \item Is this enough to compute the \textbf{joint distribution of the atoms}? + \end{itemize} + \onslide<2-> + \begin{exampleblock}{Counters} + Instead of setting and updating probabilities, we associate \textbf{counters} to disjunctive clauses and their cases. + \end{exampleblock} + + \end{frame} + + \begin{frame} + \frametitle{Bayesian updates: Matching observations} + + \begin{itemize} + \item An \alert{observation} is a subset of the literals from a program\footnote{The set of atoms, $a$, of the program and their classic negations, $\neg a$.}. + \item A \alert{consistent} observation has no subset $\set{p, \neg p}$. + \item A \textbf{consistent} observation $z$ is \alert{relevant} for the clause $h \leftarrow b$ if $b \subseteq z$. + \item A disjunctive clause $$h_1 \vee \cdots \vee h_n \leftarrow b_1 \wedge \cdots \wedge b_m$$ has $n$ \alert{cases}: $\set{h_i, b_1, \ldots, b_m}, i = 1:n$. + \item The \textbf{consistent} observation $z$ \alert{matches} the case $\set{h, b_\ast}$ if + $\set{h, b_\ast} \subseteq z$. + \end{itemize} + The above definitions also apply to \textbf{facts} \emph{i.e.} clauses with an empty body and \textbf{constraints} \emph{i.e.} clauses with no head. + \end{frame} + + \begin{frame} + \frametitle{Bayesian updates: Clauses Update} + A consistent observation \textbf{relevant} for a clause $h_1 \vee \cdots \vee h_n \leftarrow b$ should: + \begin{itemize} + \item Increase the \emph{probability of any matched case}. + \item Decrease the \emph{probability of any unmatched case}. + \end{itemize} + \pause + \begin{block}{Update algorithm} + \begin{enumerate} + \item Associate three \textbf{counters}, $r, u, n$, to each clause $h \leftarrow b$. + \item Associate a \textbf{counter}, $m_i$, to each case $h_i, b$ of each clause. + \item \textbf{Initial} values result from \emph{prior} knowledge. + \item Each \emph{consistent} observation \textbf{increments}: + \begin{itemize} + \item The $r$ counters of \alert{r}elevant clauses. + \item The $u$ counters of \alert{u}nmatched relevant clauses. + \item The $n$ counters of \alert{n}ot relevant clauses. + \item The $m_i$ counters of \alert{m}atched cases $h_i, b$. + \item Clause counters must verify $r \leq u + \sum_i m_i$. + \end{itemize} + \end{enumerate} + \end{block} + \end{frame} + + \begin{frame} + \frametitle{Updates and counters: An example} + Given the following ASP program with \textbf{annotated counters}, + $$ + \begin{aligned} + %&H \leftarrow B&&\text{counters:}~ m_{1:n}; r, u, n \\ + &b \vee \neg b &&\text{counters:}~ 7, 2; 12, 3, 0 \\ + &h_1 \vee h_2 \leftarrow b &&\text{counters:}~ 4 , 3 ; 6, 2, 5 + \end{aligned} + $$ + \onslide*<2>{ + \begin{columns}[t] + \begin{column}{0.5\textwidth} + \begin{block}{Counters of $b \vee \neg b$}\small + $0$ observations where not relevant (because the body is $\top$); + + There where $12$ relevant observations; + + Of those, $b$ was matched by $7$, $\neg b$ by $2$ and $3$ observations matched neither ($\models\naf b, \naf \neg b$). + \end{block} + \end{column} + \begin{column}{0.5\textwidth} + \begin{block}{Counters of $h_1 \vee h_2 \leftarrow b$}\small + There where $11 = 6 + 5$ observations, $6$ relevant to this clause; + + From these, $4$ matched $h_1$, $3$ matched $h_2$ and $2$ matched no case. + \end{block} + \end{column} + \end{columns} + } + \onslide<3-> + \begin{block}{What can be computed?} + \begin{itemize} + \item $P(\neg b) = \frac{2}{12}$ because $\neg b$ matched $2$ of $12$ relevant observations. + \item $P(h_1 | b) = \frac{4}{6}$ because $h_1$ matched $4$ of $6$ relevant observations. + \item $P(b)$ \alert{can't be computed} without further information. \emph{E.g.} supposing that \textbf{observations are independent} then + $$P(b) = \frac{7 + 6}{12 + 0 + 6 + 5}.$$ + \end{itemize} + + \end{block} + \end{frame} + + \begin{frame} + \frametitle{Updates and counters: An example} + Given the following ASP program with \textbf{annotated counters}, + $$ + \begin{aligned} + %&H \leftarrow B&&\text{counters:}~ m_{1:n}; r, u, n \\ + &b \vee \neg b &&\text{counters:}~ 7, 2; 12, 3, 0 \\ + &h_1 \vee h_2 \leftarrow b &&\text{counters:}~ 4 , 3 ; 6, 2, 5 + \end{aligned} + $$ + \begin{block}{Note\ldots} + \onslide*<1>{Counters are local to clauses and, for distinct clauses, may result from distinct sources. \emph{E.g. the relevant counter of $h_1 \vee h_2 \leftarrow b$ and the match counter of $b$ in $b \vee \neg b$.}} + \onslide*<3>{Some observations may have neither $b$ nor $\neg b$: $$P(b) + P(\neg b) < 1.$$} + \onslide*<4>{Since $h_1$ and $h_2$ are not independent, $$\sum_m P(m) \approx 1.02 > 1.$$} + \onslide*<5>{What is missing to compute the \alert{joint distribution} of the program's atoms $$P(H_1, H_2, B)?$$} + \end{block} + \end{frame} + + \begin{frame} + \frametitle{Shortcomming 2: Default Negation} + + \begin{itemize} + \item How to deal with rules with $\naf a$ parts? + \item Should missing elements on observations be replaced with $\naf a$ atoms? + \end{itemize} + \end{frame} + \section{Conclusions} + + \begin{frame} + \tableofcontents[currentsection] + \end{frame} + + \section*{Background Material} + + \begin{frame} + \Huge Background Material + \end{frame} + + \begin{frame}{Machine Learning} + Models are numeric functions: $y \approx f_\theta(x),~\theta_i, x_j, y\in\mathbf{R}$. + \begin{itemize} + \item Amazing achievements. + \item Noise tolerant. + \item (as of today) Huge enterprise funding . + \end{itemize} + but + \begin{itemize} + \item (essentially) Academically solved. + \item Models trained from ``large'' amounts of samples. + \item Hard to add background knowledge. + \item Models are hard to interpret. + \item Single table, independent rows assumption. + \end{itemize} + \end{frame} + + \begin{frame}{Inductive Logic Programming} + Models are logic program: $p_\theta(x, y),~\theta_i, x_j, y\in{\cal A}$. + \begin{itemize} + \item Amazing achievements, at scale. + \item Models trained from ``small'' amounts of samples. + \item Compact, readable models. + \item Background knowledge is easy to incorporate and edit. + \end{itemize} + but + \begin{itemize} + \item as of today, Little enterprise commitment. + \item as of today, Mostly academic interest. + \item Noise sensitive. + \end{itemize} + \end{frame} + + \begin{frame}{Distribution Semantics} + Assigns probability to (marginally independent) facts and derives probability of ground propositions. + + Let $F$ be set of facts, $S\subseteq F$, $R$ a set of definite clauses and $p$ a proposition: + $$\small + \begin{aligned} + P_F(S) &= \prod_{f \in S} P(f) \prod_{f \not\in S} \left(1 - P(f) \right) \cr + P(W) &= \sum_{S \subseteq F :~W=M(S\cup R)} P_F(S) \cr + P(p) &= \sum_{S :~ S\cup R ~\vdash~ p} P_F(S) = \sum_{W :~ p\in W} P(W) + \end{aligned} + $$ + \begin{itemize} + \item Amazing achievements, at scale. + \item Lots of tools and research. + \item The best of both ``worlds''? + \end{itemize} + + \end{frame} + + \begin{frame}{Answer Set Programming} + A ``program'' defines stable models \emph{i.e.} minimal sets of derived ground atoms\footnote{Alternative \xout{fact} definition: $X$ is a stable model of $P$ if $X = \text{Cn}(P^X)$.}. + \begin{itemize} + \item Pure declarative language, unlike Prolog. + \item Uses \emph{generate \& test} methods instead of proofs . + \item Uses both default $\sim\!p$ and classical negation $\neg p$\footnote{Classic negation $\neg a$ in ASP results from replacing the occurrences of $\neg a$ by a new atom $a_\neg$ and adding the restriction $\leftarrow a_\neg, a$.} + \item Clauses can be disjunctive $a ; b \leftarrow c, d.$ + \end{itemize} + \end{frame} + + \subsection*{Stable Sets} + + \begin{frame} + \tableofcontents[currentsection] + \end{frame} + + \subsection*{References} + + \begin{frame} + \tableofcontents[currentsection] + \end{frame} +\end{document} + + diff --git a/text/backup/zz_old/00_DRAFTS.md b/text/backup/zz_old/00_DRAFTS.md new file mode 100644 index 0000000..059618d --- /dev/null +++ b/text/backup/zz_old/00_DRAFTS.md @@ -0,0 +1,21 @@ +# PASP Drafts + +Consider the logic program + +$$ +\begin{aligned} +c_1 &= a \vee \neg a, \cr +c_2 &= b \larr \sim a \land \sim c, \cr +c_3 &= c \larr \sim b +\end{aligned} +$$ + +This program has three stable models: + +$$ +\begin{aligned} +m_1 &= \set{a, c} \cr +m_2 &= \set{b, \neg a} \cr +m_3 &= \set{c, \neg a} +\end{aligned} +$$ \ No newline at end of file diff --git a/text/backup/zz_old/00_PASP.pdf b/text/backup/zz_old/00_PASP.pdf new file mode 100644 index 0000000..6f764a9 Binary files /dev/null and b/text/backup/zz_old/00_PASP.pdf differ diff --git a/text/backup/zz_old/00_PASP.tex b/text/backup/zz_old/00_PASP.tex new file mode 100644 index 0000000..b732c99 --- /dev/null +++ b/text/backup/zz_old/00_PASP.tex @@ -0,0 +1,1047 @@ +\documentclass{beamer} +%------------------------------------------ +\usecolortheme{rose} +%------------------------------------------ +\useinnertheme{circles} +%------------------------------------------ +\setbeamertemplate{navigation symbols}{} +%------------------------------------------ +\AtBeginSection{ + \begin{frame}\small + \tableofcontents[currentsection,subsectionstyle=shaded/shaded/hide] + \end{frame} +} +%------------------------------------------ +\AtBeginSubsection{ + \begin{frame}\small + \tableofcontents[ + currentsection,sectionstyle=shaded/shaded, + currentsubsection,subsectionstyle=show/shaded/hide] + \end{frame} +} +%------------------------------------------ +\usepackage[overridenote]{pdfpc} + +\usepackage{tikz} +\usetikzlibrary{ + % shapes, + % arrows, + % backgrounds, + positioning, +} +\tikzset{ + sm/.style={ + rectangle, + rounded corners, + fill=teal!10, + align=center, + minimum height=1.5em, + }, + sup/.style={ + rectangle, + rounded corners, + fill=blue!10, + align=center, + minimum height=1.5em, + }, + sub/.style={ + rectangle, + rounded corners, + fill=red!10, + align=center, + minimum height=1.5em, + }, + ind/.style={ + align=center, + minimum height=1.5em, + }, +} + +\usepackage[T1]{fontenc} +\usepackage{hyperref} +\hypersetup{% + colorlinks=true, + allcolors=blue, +} +\usepackage{commath} +\usepackage{amssymb} +% +% Local commands +% +\newcommand{\todo}[1]{{\color{orange}TODO #1}} +\newcommand{\naf}{\ensuremath{\sim\!}} +\newcommand{\larr}{\ensuremath{\leftarrow}} +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} +\newcommand{\deft}[1]{\textbf{#1}} +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} +\newcommand{\ent}{\ensuremath{\lhd}} +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} +\newcommand{\uset}[1]{\ensuremath{\left|{#1}\right>}} +\newcommand{\lset}[1]{\ensuremath{\left<{#1}\right|}} +\newcommand{\pr}[1]{\ensuremath{\mathrm{p}\at{#1}}} +\newcommand{\given}{\ensuremath{~\middle|~}} +% +% Identificação deste documento +% +\title{Zugzwang} +\subtitle{Stochastic Adventures in Inductive Logic} +\author{Francisco Coelho} +\institute[\texttt{fc@uevora.pt}]{ + Departamento de Informática, Universidade de Évora\\ + High Performance Computing Chair\\ + NOVA-LINCS +} + +\begin{document} +% +\begin{frame}[plain] +\titlepage +\end{frame} + +\section{Introduction} + + +\begin{frame}{Notation and Assumptions} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item $\co{x} = 1 - x$. + % -------------------------------- + \item \textbf{Probabilistic Atomic Choice (PAC):} $x :: a$ defines $a \lor \neg a$ and probabilities $\pr{a} = x, \pr{\neg a} = \co{x}$. + % -------------------------------- + \item $\delta a$ denotes $a \lor \neg a$ and $\delta\! \set{x :: a, a \in \fml{A}} = \set{\delta a, a \in \fml{A}}$ for a set of atoms $\fml{A}$. + % -------------------------------- + \item \textbf{Closed World Assumption:} $\naf p \models \neg p$. + % -------------------------------- + % \item Probabilistic choices and sub-goals are independent. + % -------------------------------- + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{General Setting} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item \textbf{Atoms} $\fml{A}$, + $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$, and \textbf{literals} $\fml{L} = \fml{A} \cup \co{\fml{A}}$. + % -------------------------------- + \item \textbf{Samples} $z \in \fml{Z} \iff z \subseteq \fml{L}$. + % -------------------------------- + \item \textbf{Events} or \textit{consistent samples} $\fml{E}$ : + $$\fml{E} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$ + % -------------------------------- + \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item $C = C_P = \cset{x_i :: a_i }{ i \in 1:n \land a_i \in \fml{A}}$ \textit{pacs}. + % -------------------------------- + \item $F = F_P$ \textit{facts}. + % -------------------------------- + \item $R = R_P$ \textit{rules}. + % -------------------------------- + \item $\fml{A}_P, \fml{Z}_P$ and $\fml{E}_P$: \textit{atoms}, \textit{samples} and \textit{events} of $P$. + \end{itemize} + % -------------------------------- + \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$. + % -------------------------------- + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{Distribution Semantics} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are $\theta = \cset{t_c}{c \in C}$ where $c=x::a$ and $t_c$ is $a$ or $\neg a$. + % -------------------------------- + %\item For $s\in\fml{S}$ let $\theta_s \subseteq s$ (unique \textit{total choice}) + %\item Define $\fml{S}_\theta = \cset{s \in \fml{S}}{\theta \subset s}$. + % -------------------------------- + + % -------------------------------- + \item \textbf{Total Choice Probability:} + \begin{equation} + \pr{\theta} = \prod_{a \in \theta}x \prod_{\neg a \in \theta}\co{x}.\label{eq:prob.tc} + \end{equation} + % -------------------------------- + \end{itemize} + % -------------------------------- + This is the \emph{distribution semantic} as set by Sato. +\end{frame} +% ================================================================ +\begin{frame} + % -------------------------------- + \begin{block}{Problem Statement} + How to \textit{extend} probability from total choices to stable models, events and samples? + \end{block} + % -------------------------------- + \begin{quotation} + There's a problem right at extending to stable models. + \end{quotation} + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{The Disjunction Case} + % -------------------------------- + \begin{exampleblock}{Disjuntion Example} + The specification + % -------------------------------- + $$ + \begin{aligned} + 0.3 :: a &, \cr + b \lor c &\larr a . + \end{aligned} + $$ + % -------------------------------- + has three stable models, + % -------------------------------- + $$ + \begin{aligned} + s_1 &= \set{\neg a}, & s_2 &= \set{a, b}, & s_3 &= \set{a, c}. + \end{aligned} + $$ + \end{exampleblock} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item\label{prop:unique.ext.tcsm}\textit{Any stable model contains exactly one total choice.~$\blacksquare$} + % -------------------------------- + \item $\pr{\set{\neg a}} = 0.7$ is straightforward. + % -------------------------------- + \item But, no \textit{informed} choice for $x\in\intcc{0,1}$ in + $$ + \begin{aligned} + \pr{\set{a, b}} &= 0.3 x, \cr + \pr{\set{a, c}} &= 0.3 \co{x}. + \end{aligned} + $$ + % -------------------------------- + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{Lack of Information \& Parametrization} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item The specification \textit{lacks information} to set $x\in\intcc{0,1}$ in + $$ + \begin{aligned} + \pr{\set{a, b}} &= 0.3 x, \cr + \pr{\set{a, c}} &= 0.3 \co{x}. + \end{aligned} + $$ + \item A \textit{random variable} captures this uncertainty, \alert{assuming} that the stable models are statistically independent: + $$ + \begin{aligned} + \pr{\set{\neg a} \given X = x } &= 0.7, \cr + \pr{\set{a, b} \given X = x } &= 0.3 x, \cr + \pr{\set{a, c} \given X = x } &= 0.3 \co{x}. + \end{aligned} + $$ + \item Other uncertainties may lead to further conditions: + $$ + \pr{s \given X_1 = x_1, \ldots, X_n = x_n }. + $$ + % -------------------------------- + \end{itemize} + Reducing \textbf{uncertainty}, \textit{e.g.} setting $X = 0.21$, must result from \textbf{external} sources, since the specification lacks information for further assertions. + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{Independence of Stable Models} + % -------------------------------- + + \begin{itemize} + \item[Q:] Why are the stable models assumed statistically independent? + % -------------------------------- + \item[A:] Because dependence can be \textit{explicitly} modelled. + % -------------------------------- + \item So, it is assumed \textit{intention} of the \textit{modeller} to not explicit express such dependences. + % -------------------------------- + \item \textbf{For example:} \todo{Some key examples}. + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}%{Main Research Question} + % -------------------------------- + A \textit{random variable} captures this uncertainty: + $$ + \begin{aligned} + \pr{\set{\neg a} \given X = x } &= 0.7, \cr + \pr{\set{a, b} \given X = x } &= 0.3 x, \cr + \pr{\set{a, c} \given X = x } &= 0.3 \co{x}. + \end{aligned} + $$ + % -------------------------------- + \begin{block}{Main Research Question} + Can \textit{all} specification uncertainties be neatly expressed as that example? + \end{block} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item Follow ASP syntax; for each case, what are the uncertainty scenarios? + % -------------------------------- + \item The disjunction example illustrates one such scenario. + % -------------------------------- + \item \textit{Neat} means a function $d: \fml{S} \to \intcc{0, 1}$ such that + $$ + \sum_{s\in\fml{S}_\theta} d\at{s} = 1 + $$ + for each $\theta \in \Theta$. + % -------------------------------- + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{Leap into Inductive Programming} + % -------------------------------- + Given a method that produces a distribution of samples, $p$, from a specification, $P$ and: + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item $Z$, a dataset (of samples). + % -------------------------------- + \item $e$, the respective empirical distribution. + % -------------------------------- + \item $D$, some probability divergence, \textit{e.g.} Kullback-Leibler. + % -------------------------------- + \end{itemize} + % -------------------------------- + \begin{block}{Specification Performance \& Inductive Programming} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item $D\at{P} = D\at{e, p}$ is a \textbf{performance} measure of $P$. + % -------------------------------- + \item Predictor performance measures, such as accuracy, are common in \textit{Machine Learning} tasks. + % -------------------------------- + \item For \textit{Inductive Programming} this performance can be used, \textit{e.g.} as fitness, by algorithms searching for \textbf{optimal specifications of a dataset}. + % -------------------------------- + \end{itemize} + % -------------------------------- + \end{block} + % -------------------------------- +\end{frame} +% ================================================================ +\section{Extending Probability to Samples} +% ================================================================ +\begin{frame}{Resolution Path} + Prior to \textit{conciliation} with data: + \begin{enumerate} + \item \alert{Hopefully}, \textit{conditional parameters} extend probability from total choices to \textit{standard models}. + \item \textbf{How} to extend it to \textit{events}? + \begin{itemize} + \item $\pr{x} = 0$ for $x$ \textit{excluded} by the specification, including \textit{inconsistent} samples. + \item $\pr{x}$ depends on the $s \in \fml{S}$ that contain/are contained in $x$. + \end{itemize} + \end{enumerate} + \alert{Consider probabilities \textbf{conditional} on the total choice!} +\end{frame} +% ================================================================ +\begin{frame}{Bounds of Events} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item For $x\in\fml{E}$: + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item \textbf{Lower Models:} $\lset{x} = \cset{s\in \fml{S} }{ s \subseteq x}$. + % -------------------------------- + \item \textbf{Upper Models:} $\uset{x} = \cset{s\in \fml{S} }{ x \subseteq s}$. + % -------------------------------- + \end{itemize} + % -------------------------------- + \item\label{prop:lucases} \textbf{Proposition.} Exactly \textit{one} of the following cases takes place: + % -------------------------------- + \begin{enumerate} + % -------------------------------- + \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$ and $x$ is a stable model. Then: + \begin{equation} + \pr{x \given C = \theta_x} = d\at{x}. + \end{equation} + % -------------------------------- + \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$. Then: + \begin{equation} + \pr{x \given C = \theta_s, s \in \lset{x}} = \prod_{s\in\lset{x}} d\at{s}. + \end{equation} + % -------------------------------- + \item\label{prop:lucases.c} $\lset{x} = \emptyset \land \uset{x} \neq \emptyset$. Then: + \begin{equation} + \pr{x \given C = \theta_s, s \in \uset{x}} = \sum_{s\in\uset{x}} d\at{s}. + \end{equation} + % -------------------------------- + \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. Then: + \begin{equation} + \pr{x} = 0. + \end{equation} + % -------------------------------- + \end{enumerate} + because stable models are \textit{minimal}. + % -------------------------------- + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{Conditional on Total Choices} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item A stable model is entailed by an atomic choice plus the facts and rules of the specification. + \item We express that entailment as a \textit{conditional}. For example: + $$\pr{\set{a,b} \given X = x} = \pr{b \given X = x, \Theta = a}\pr{\theta = a}$$ + \item And now $\pr{b \given X = x, \Theta = a} = x$, since $X$ is a proxy for the stable models of the total choice $\theta = a$, we can further. + % -------------------------------- + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{Disjunction Example | The Events Lattice} + \begin{center} + \begin{tikzpicture} + % -------------------------------- + %\draw [help lines, color=gray!20] grid (11,7); + % -------------------------------- + \node at (7, 7) {$\pr{\Theta=a} = 0.3$}; + \node at (7, 6) {$x = \pr{S = ab \given \Theta}$}; + \node at (7, 5) {$\co{x} = \pr{S \not= ab \given \Theta}$}; + \node at (7, 7.5) {$\pr{E = abc \given \Theta} = \pr{S = ab, S = ac \given \Theta }$}; + % -------------------------------- + % \node [rrect] (sub) at (2, 7) {sub}; + % -------------------------------- + % \node [ fill=gray!10] (sup) at (3, 7) {sup}; + % -------------------------------- + % \node (ind) at (4, 7) {ind}; + % -------------------------------- + % + % -------------------------------- + \node [ sub, + pin=45:\textcolor{violet}{$1$} ] + (E) at (5.5,0) {$\emptyset$}; + % -------------------------------- + % + % -------------------------------- + \node [ sub, + pin=0:\textcolor{blue!75}{$1$} ] + (a) at (1.5,1.5) {$a$}; + % -------------------------------- + \node [ sub, + pin=315:\textcolor{blue!50}{$x$}] + (b) at (0,1.5) {$b$}; + % -------------------------------- + \node [ sub, + pin=315:\textcolor{blue!50}{$\co{x}$}] + (c) at (4.5,1.5) {$c$}; + % -------------------------------- + % \node [ sm, + % pin=270:\textcolor{teal}{$1$}] + % (A) at (8.5,1.5) {$\co{a}$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (B) at (9.5, 1.5) {$\co{b}$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (C) at (10.5, 1.5) {$\co{c}$}; + % -------------------------------- + % + % -------------------------------- + \node [ sm, + pin=0:\textcolor{teal}{$x$}] + (ab) at (0,4) {$ab$}; + % -------------------------------- + \node [ sm, + pin=0:\textcolor{teal}{$\co{x}$}] + (ac) at (3,4) {$ac$}; + % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (aB) at (1,4) {$a\co{b}$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (aC) at (2,4) {$a\co{c}$}; + % -------------------------------- + % \node [ sup, + % pin=90:\textcolor{blue!50}{$1$}] + % (Ab) at (4,4) {$\co{a}b$}; + % % -------------------------------- + % \node [ sup, + % pin=90:\textcolor{blue!50}{$1$}] + % (Ac) at (5,4) {$\co{a}c$}; + % % -------------------------------- + % \node [ sup, + % pin=90:\textcolor{blue!50}{$1$}] + % (AB) at (6,4) {$\co{a}\co{b}$}; + % % -------------------------------- + % \node [ sup, + % pin=90:\textcolor{blue!50}{$1$}] + % (AC) at (7,4) {$\co{a}\co{c}$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (bc) at (10,4) {$bc$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (bC) at (11,4) {$b\co{c}$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (Bc) at (9.5,3.5) {$\co{b}c$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (BC) at (10.5,3.5) {$\co{b}\co{c}$}; + % -------------------------------- + % + % -------------------------------- + \node [ sup, + pin=45:\textcolor{blue!75}{$x\co{x}$}] + (abc) at (1.5,6) + {$abc$}; + % -------------------------------- + \node [ sup, + pin=45:\textcolor{blue!50}{$x$}] + (abC) at (0,6) {$ab\co{c}$}; + % -------------------------------- + \node [ sup, + pin=45:\textcolor{blue!50}{$\co{x}$}] + (aBc) at (3,6) {$a\co{b}c$}; + % -------------------------------- + % \node [ ind, + % pin=90:\textcolor{purple}{$0$}] + % (aBC) at (5,6) {$a\co{b}\co{c}$}; + % -------------------------------- + % \node [ sup, + % pin=90:\textcolor{blue!50}{$1$}] + % (Abc) at (7,6) {$\co{a}bc$}; + % % -------------------------------- + % \node [ sup, + % pin=270:\textcolor{blue!50}{$1$}] + % (AbC) at (8,6) {$\co{a}b\co{c}$}; + % % -------------------------------- + % \node [ sup, + % pin=270:\textcolor{blue!50}{$1$}] + % (ABc) at (9,6) {$\co{a}\co{b}c$}; + % % -------------------------------- + % \node [ sup, + % pin=270:\textcolor{blue!50}{$1$}] + % (ABC) at (10,6) {$\co{a}\co{b}\co{c}$}; + % -------------------------------- + % + % -------------------------------- + \draw [->] (ab) to [out=270,in=180] (E); + \draw [->] (ab) to [out=270,in=90] (a); + \draw [->] (ab) to [out=270,in=90] (b); + \draw [->] (ab) to [out=90,in=270] (abc); + \draw [->] (ab) to [out=90,in=270] (abC); + % + \draw [->] (ac) to [out=270,in=180] (E); + \draw [->] (ac) to [out=270,in=90] (a); + \draw [->] (ac) to [out=270,in=90] (c); + \draw [->] (ac) to [out=90,in=270] (abc); + \draw [->] (ac) to [out=90,in=270] (aBc); + % + % \draw [->] (A) to [out=270,in=0] (E); + % % + % \draw [->] (A) to [out=90,in=270] (Abc); + % \draw [->] (A) to [out=90,in=270] (AbC); + % \draw [->] (A) to [out=90,in=270] (ABc); + % \draw [->] (A) to [out=90,in=270] (ABC); + % % + % \draw [->] (A) to [out=90,in=270] (Ab); + % \draw [->] (A) to [out=90,in=270] (Ac); + % \draw [->] (A) to [out=90,in=270] (AB); + % \draw [->] (A) to [out=90,in=270] (AC); + \end{tikzpicture} + \end{center} +\end{frame} +% ================================================================ +\begin{frame}{Disjunction Example | The Events Lattice} + \begin{center} + \begin{tikzpicture} + % -------------------------------- + %\draw [help lines, color=gray!20] grid (11,7); + % -------------------------------- + \node [sm] (sm) at (5, 7) {$\pr{\Theta=\set{\co{a}}} = \co{0.3}$}; + % -------------------------------- + % \node [rrect] (sub) at (2, 7) {sub}; + % -------------------------------- + % \node [ fill=gray!10] (sup) at (3, 7) {sup}; + % -------------------------------- + % \node (ind) at (4, 7) {ind}; + % -------------------------------- + % + % -------------------------------- + \node [ sub, + pin=45:\textcolor{violet}{$1$} ] + (E) at (5.5,0) {$\emptyset$}; + % -------------------------------- + % + % -------------------------------- + % \node [ sub, + % pin=270:\textcolor{blue!75}{$1$} ] + % (a) at (1.5,1.5) {$a$}; + % % -------------------------------- + % \node [ sub, + % pin=270:\textcolor{blue!50}{$x$}] + % (b) at (0,1.5) {$b$}; + % % -------------------------------- + % \node [ sub, + % pin=270:\textcolor{blue!50}{$\co{x}$}] + (c) at (4.5,1.5) {$c$}; + % -------------------------------- + \node [ sm, + pin=45:\textcolor{teal}{$1$}] + (A) at (8.5,1.5) {$\co{a}$}; + % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (B) at (9.5, 1.5) {$\co{b}$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (C) at (10.5, 1.5) {$\co{c}$}; + % -------------------------------- + % + % -------------------------------- + % \node [ sm, + % pin=90:\textcolor{teal}{$x$}] + % (ab) at (0,4) {$ab$}; + % % -------------------------------- + % \node [ sm, + % pin=90:\textcolor{teal}{$\co{x}$}] + % (ac) at (3,4) {$ac$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (aB) at (1,4) {$a\co{b}$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (aC) at (2,4) {$a\co{c}$}; + % -------------------------------- + \node [ sup, + pin=135:\textcolor{blue!50}{$1$}] + (Ab) at (4,4) {$\co{a}b$}; + % -------------------------------- + \node [ sup, + pin=135:\textcolor{blue!50}{$1$}] + (Ac) at (5,4) {$\co{a}c$}; + % -------------------------------- + \node [ sup, + pin=135:\textcolor{blue!50}{$1$}] + (AB) at (6,4) {$\co{a}\co{b}$}; + % -------------------------------- + \node [ sup, + pin=135:\textcolor{blue!50}{$1$}] + (AC) at (7,4) {$\co{a}\co{c}$}; + % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (bc) at (10,4) {$bc$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (bC) at (11,4) {$b\co{c}$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (Bc) at (9.5,3.5) {$\co{b}c$}; + % % -------------------------------- + % \node [ ind, + % pin=270:\textcolor{purple}{$0$}] + % (BC) at (10.5,3.5) {$\co{b}\co{c}$}; + % -------------------------------- + % + % -------------------------------- + % \node [ sup, + % pin=90:\textcolor{blue!75}{$x\co{x}$}] + % (abc) at (1.5,6) + % {$abc$}; + % % -------------------------------- + % \node [ sup, + % pin=90:\textcolor{blue!50}{$x$}] + % (abC) at (0,6) {$ab\co{c}$}; + % % -------------------------------- + % \node [ sup, + % pin=90:\textcolor{blue!50}{$\co{x}$}] + % (aBc) at (3,6) {$a\co{b}c$}; + % % -------------------------------- + % \node [ ind, + % pin=90:\textcolor{purple}{$0$}] + % (aBC) at (5,6) {$a\co{b}\co{c}$}; + % -------------------------------- + \node [ sup, + pin=45:\textcolor{blue!50}{$1$}] + (Abc) at (7,6) {$\co{a}bc$}; + % -------------------------------- + \node [ sup, + pin=45:\textcolor{blue!50}{$1$}] + (AbC) at (8,6) {$\co{a}b\co{c}$}; + % -------------------------------- + \node [ sup, + pin=45:\textcolor{blue!50}{$1$}] + (ABc) at (9,6) {$\co{a}\co{b}c$}; + % -------------------------------- + \node [ sup, + pin=45:\textcolor{blue!50}{$1$}] + (ABC) at (10,6) {$\co{a}\co{b}\co{c}$}; + % -------------------------------- + % + % -------------------------------- + % \draw [->] (ab) to [out=270,in=180] (E); + % \draw [->] (ab) to [out=270,in=90] (a); + % \draw [->] (ab) to [out=270,in=90] (b); + % \draw [->] (ab) to [out=90,in=270] (abc); + % \draw [->] (ab) to [out=90,in=270] (abC); + % % + % \draw [->] (ac) to [out=270,in=180] (E); + % \draw [->] (ac) to [out=270,in=90] (a); + % \draw [->] (ac) to [out=270,in=90] (c); + % \draw [->] (ac) to [out=90,in=270] (abc); + % \draw [->] (ac) to [out=90,in=270] (aBc); + % + \draw [->] (A) to [out=270,in=0] (E); + % + \draw [->] (A) to [out=90,in=270] (Abc); + \draw [->] (A) to [out=90,in=270] (AbC); + \draw [->] (A) to [out=90,in=270] (ABc); + \draw [->] (A) to [out=90,in=270] (ABC); + % + \draw [->] (A) to [out=90,in=270] (Ab); + \draw [->] (A) to [out=90,in=270] (Ac); + \draw [->] (A) to [out=90,in=270] (AB); + \draw [->] (A) to [out=90,in=270] (AC); + \end{tikzpicture} + \end{center} +\end{frame} +% ================================================================ +\begin{frame} + + + \begin{itemize} + \item Consider the ASP program $P = C \land F \land R$ with total choices $\Theta $ and stable models $\fml{S}$. + + \item Let $d : \fml{S} \to \intcc{0,1}$ such that $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$ for each $\theta \in \Theta$. + \end{itemize} +\end{frame} +% ================================================================ +\begin{frame} + For each $z\in\fml{Z}$ only one of the following cases takes place + \begin{enumerate} + \item $z$ is inconsistent. Then \textbf{define} + \begin{equation} + w_d\at{x} = 0.\label{def:w.inconsistent} + \end{equation} + % + \item $z$ is an event and $\lset{z} = \set{z} = \uset{z}$. Then $z$ is a stable model and \textbf{define} + \begin{equation} + w_d\at{z} = w\at{z} = d\at{z} \pr{\theta_z}.\label{eq:prob.sm} + \end{equation} + % + \item $z$ is an event and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define} + \begin{equation} + w_d\at{z} = \sum_{s \in \lset{z}} w_d\at{s}.\label{def:w.disj} + \end{equation} + % + \item $z$ is an event and $\lset{z} = \emptyset \land \uset{z} \neq \emptyset$. Then \textbf{define} + \begin{equation} + w_d\at{z} = \prod_{s \in \uset{z}} w_d\at{s}.\label{def:w.conj} + \end{equation} + % + \item $z$ is an event and $\lset{z} = \emptyset \land \uset{z} = \emptyset$. Then \textbf{define} + \begin{equation} + w_d\at{z} = 0.\label{def:w.empty} + \end{equation} + \end{enumerate} + \end{frame} + % ================================================================ + \begin{frame} + \begin{enumerate} + % + \item The last point defines a ``weight'' function on the samples that depends not only on the total choices and stable models of a PASP but also on a certain function $d$ that must respect some conditions. To simplify the notation we use the subscript in $w_d$ only when necessary. + % + \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an event, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} sample $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$. + % + \item \textbf{We haven't yet defined a probability measure.} To do so we must define a set of samples $\Omega$, a set of events $F\subseteq \pset{\Omega}$ and a function $P:F\to\intcc{0,1}$ such that: + \begin{enumerate} + \item $\pr{E} \in \intcc{0, 1}$ for any $E \in F$. + \item $\pr{\Omega} = 1$. + \item if $E_1 \cap E_2 = \emptyset$ then $\pr{E_1 \cup E_2} = \pr{E_1} + \pr{E_2}$. + \end{enumerate} + % + \item In the following, assume that the stable models are iid. + % + \item Let the sample space $\Omega = \fml{Z}$ and the event space $F = \pset{\Omega}$. Define $Z = \sum_{\zeta\in\fml{Z}} w\at{\zeta}$ and + \begin{equation} + \pr{z} = \frac{w\at{z}}{Z}, z \in \Omega \label{eq:def.prob} + \end{equation} + and + \begin{equation} + \pr{E} = \sum_{x\in E} \pr{x}, E \subseteq \Omega. \label{eq:def.prob.event} + \end{equation} + Now: + \begin{enumerate} + \item $P(E) \in \intcc{0,1}$ results directly from the definitions of $P$ and $w$. + \item $\pr{\Omega} = 1$ also results directly from the definitions. + \item Consider two disjunct events $A, B \subset \Omega \land A \cap B = \emptyset$. Then + $$ + \begin{aligned} + \pr{A \cup B} &= \sum_{x \in A \cup B} \pr{x} \cr + &= \sum_{x \in A} \pr{x} + \sum_{x \in B} \pr{x} - \sum_{x \in A \cap B} \pr{x} \cr + &= \sum_{x \in A} \pr{x} + \sum_{x \in B} \pr{x} &\text{because}~A\cap B = \emptyset \cr + &= \pr{A} + \pr{B}. + \end{aligned} + $$ + \item So $\del{\Omega = \fml{Z}, F = \pset{\Omega}, P}$ is a probability space. {$\blacksquare$} + \end{enumerate} + \end{enumerate} + +\end{frame} +% ================================================================ +\section{Cases \& Examples} +% ================================================================ +\subsection{Programs with disjunctive heads} +% ================================================================ +\begin{frame} + + Consider the program: + $$ + \begin{aligned} + c_1 &= a \lor \neg a, \cr + c_2 &= b \lor c \larr a. + \end{aligned} + $$ + This program has two total choices, + $$ + \begin{aligned} + \theta_1&= \set{ \neg a }, \cr + \theta_2&= \set{ a }. + \end{aligned} + $$ + and three stable models, + $$ + \begin{aligned} + s_1 &= \set{ \neg a }, \cr + s_2 &= \set{ a, b }, \cr + s_3 &= \set{ a, c }. + \end{aligned} + $$ +\end{frame} +% ================================================================ +\begin{frame} + Suppose that we add an annotation $x :: a$, which entails $\co{x} :: \neg a$. This is enough to get $w\at{s_1} = \co{x}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $x$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get: + $$ + \begin{cases} + w\at{s_1 } = &\co{x}\cr + w\at{s_2 } = &dx\cr + w\at{s_3} = &\co{d}x. + \end{cases} + $$ +\end{frame} +% ================================================================ +\begin{frame} + + In this diagram: + \begin{itemize} + \item Negations are represented as \emph{e.g.} $\co{a}$ instead of $\neg a$; Stable models are denoted by shaded nodes as \tikz{\node[fill=gray!50] {$ab$}}. + + \item Events in $\lset{x}$ are \emph{e.g.} \tikz{\node[ circle] {$a$}} and those in $\uset{x}$ are \emph{e.g.} \tikz{\node[draw] {$\co{a}b$}}. The remaining are simply denoted by \emph{e.g.} \tikz{\node {$a\co{b}$}}. + + \item The edges connect stable models with related events. Up arrow indicate links to $\uset{s}$ and down arrows to $\lset{s}$. + + \item The \emph{weight propagation} sets: + $$ + \begin{aligned} + w\at{abc} &= w\at{ab} w\at{ac} = x^2d\co{d}, \cr + w\at{\co{a}\cdot\cdot} &= w\at{\neg a} = \co{x}, \cr + w\at{a} &= w\at{ab} + w\at{ac} = x(d + \co{d}) = x, \cr + w\at{b} &= w\at{ab} = dx, \cr + w\at{c} &= w\at{ac} = \co{d}x, \cr + w\at{\emptyset} &= w\at{ab} + w\at{ac} + w\at{\neg a} = dx + \co{d}x + \co{x} = 1, \cr + w\at{a\co{b}} &= 0. + \end{aligned} + $$ + \item The total weight is + $$ + \begin{aligned} + Z &= w\at{abc} + 8 w\at{\co{a}b}\cr + &+ w\at{ab} + w\at{ac} + w\at{\co{a}}\cr + &+ w\at{a}+ w\at{b}+ w\at{c}\cr + &+ w\at{\emptyset}\cr + % + &= - x^{2} d^{2} + x^{2} d + 2 x d - 7 x + 10 + \end{aligned} + $$ + \item Now, if $x$ has an annotation to \emph{e.g.} $0.3$ we get + $$ + Z = - 0.09 d^{2} + 0.69 d + 7.9 + $$ + \item Now some statistics are possible. For example we get + $$ + \pr{abc \mid x = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9} + $$. + + \item This expression can be plotted for $d\in\intcc{0,1}$ + \begin{center} + \includegraphics[height=15em]{Pabc_alpha03.pdf} + \end{center} + + \item If a data set $E$ entails \emph{e.g.} $\pr{abc \mid E} = 0.0015$ we can numerically solve + $$ + \begin{aligned} + \pr{abc \mid x = 0.3} &= \pr{abc \mid E} \cr + \iff\cr + \frac{0.09 d \del{d - 1}}{0.09 d^{2} - 0.69 d - 7.9} &= 0.0015 + \end{aligned} + $$ + which has two solutions, $d \approx 0.15861$ or $d \approx 0.83138$. + \end{itemize} +\end{frame} +% ================================================================ +\subsection{Non-stratified programs} +% ================================================================ +\begin{frame} + The following LP is non-stratified, because has a cycle with negated arcs: + $$ + \begin{aligned} + c_1 &= a\lor \neg a,\cr + c_2 &= b \larr \naf c \land \naf a, \cr + c_3 &= c \larr \naf b. + \end{aligned} + $$ + This program has three stable models + $$ + \begin{aligned} + s_1 &= \set{ a, c }, \cr + s_2 &= \set{ \neg a, b }, \cr + s_3 &= \set{ \neg a, c }. + \end{aligned} + $$ +\end{frame} + +\begin{frame} + The disjunctive clause $a\lor\neg a$ defines a set of \textbf{total choices} + $$ + \Theta = \set{ + \theta_1 = \set{ a }, + \theta_2 = \set{ \neg a } + }. + $$ +\end{frame} +% ================================================================ +\begin{frame} + + Looking into probabilistic events of the program and/or its models, we define $x = \pr{\Theta = \theta_1}\in\intcc{0, 1}$ and $\pr{\Theta = \theta_2} = \co{x}$. + + Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $\pr{ s_1 } = \pr{\Theta = \theta_1} = x$. However, there is no clear way to assign $\pr{s_2}, \pr{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly, + $$\pr{s_2 \mid \Theta} + \pr{s_3 \mid \Theta} = + \begin{cases} + 0 & \text{if}~\Theta = \theta_1\cr + 1 & \text{if}~\Theta = \theta_2 + \end{cases} + $$ + but further assumptions are not supported \emph{a priori}. So let's \textbf{parameterize} the equation above, + $$ + \begin{cases} + \pr{s_2 \mid \Theta = \theta_2} = &\beta \in \intcc{0, 1} \cr + \pr{s_3 \mid \Theta = \theta_2} = &\co{\beta}, + \end{cases} + $$ + in order to explicit our knowledge, or lack of, with numeric values and relations. +\end{frame} +% ================================================================ +\begin{frame} + Now we are able to define the \textbf{joint distribution} of the boolean random variables $A,B,C$: + + $$ + \begin{array}{cc|l} + A, B, C& P & \text{Obs.}\cr + \hline + a, \neg b, c & x & s_1, \Theta=\theta_1\cr + \neg a, b, \neg c & \co{x}\beta & s_2, \Theta=\theta_2\cr + \neg a, \neg b, c & \co{x}\co{\beta} & s_3, \Theta=\theta_2\cr + \ast & 0&\text{not stable models} + \end{array} + $$ + where $x, \beta\in\intcc{0,1}$. +\end{frame} +% ================================================================ +\section{Conclusions} +% ================================================================ +\begin{frame} + \begin{itemize} + \item We can use the basics of probability theory and logic programming to assign explicit \emph{parameterized} probabilities to the (stable) models of a program. + \item In the covered cases it was possible to define a (parameterized) \emph{family of joint distributions}. + \item How far this approach can cover all the cases on logic programs is (still) an issue \emph{under investigation}. + \item However, it is non-restrictive since \emph{no unusual assumptions are made}. + \end{itemize} +\end{frame} +% ================================================================ +\section*{ASP \& related definitions} +% ================================================================ +\begin{frame} + + \begin{itemize} + \item An \deft{atom} is $r(t_1, \ldots t_n)$ where + \begin{itemize} + \item $r$ is a $n$-ary predicate symbol and each $t_i$ is a constant or a variable. + \item A \deft{ground atom} has no variables; A \deft{literal} is either an atom $a$ or a negated atom $\neg a$. + \end{itemize} + + \item An \deft{ASP Program} is a set of \deft{rules} such as $h_1 \vee \cdots \vee h_m \leftarrow b_1 \wedge \cdots \wedge b_n$. + \begin{itemize} + \item The \deft{head} of this rule is $h_1 \vee \cdots \vee h_m$, the \deft{body} is $b_1 \wedge \cdots \wedge b_n$ and each $b_i$ is a \deft{subgoal}. + \item Each $h_i$ is a literal, each subgoal $b_j$ is a literal or a literal preceded by $\naf\;$ and $m + n > 0$. + \item A \deft{propositional program} has no variables. + \item A \deft{non-disjunctive rule} has $m \leq 1$; A \deft{normal rule} has $m = 1$; A \deft{constraint} has $m = 0$; A \deft{fact} is a normal rule with $n = 0$. + \end{itemize} + + \item The \deft{Herbrand base} of a program is the set of ground literals that result from combining all the predicates and constants of the program. + \begin{itemize} + \item An \deft{event} is a consistent subset (\emph{i.e.} doesn't contain $\set{a, \neg a}$) of the Herbrand base. + \item Given an event $I$, a ground literal $a$ is \deft{true}, $I \models a$, if $a \in I$; otherwise the literal is \deft{false}. + \item A ground subgoal, $\naf b$, where $b$ is a ground literal, is \deft{true}, $I \models \naf b$, if $b \not\in I$; otherwise, if $b \in I$, it is \deft{false}. + \item A ground rule $r = h_1 \vee \cdots \vee h_m \leftarrow b_1 \wedge \cdots \wedge b_n$ is \deft{satisfied} by the event $I$, \emph{i.e.} $I \models r$, iff + $$ + \forall j \exists i~I \models b_j \implies I \models h_i. + $$ + \item A \deft{model} of a program is an event that satisfies all its rules. Denote $\fml{M}_P$ the set of all models of $P$. + \end{itemize} + + \item The \deft{dependency graph} of a program is a digraph where: + \begin{itemize} + \item Each grounded atom is a node. + \item For each grounded rule there are edges from the atoms in the body to the atoms in the head. + \item A \deft{negative edge} results from an atom with $\naf\;$; Otherwise it is a \deft{positive edge}. + \item An \deft{acyclic program} has an acyclic dependency graph; A \deft{normal program} has only normal rules; A \deft{definite program} is a normal program that doesn't contains $\neg$ neither $\naf\;$. + \item In the dependency graph of a \deft{stratified program} no cycle contains a negative edge. + \item \textbf{A stratified program has a single minimal model} that assigns either true or false to each atom. + \end{itemize} + \item Every \emph{definite program} has a unique minimal model: its \deft{semantic}. + \item Programs with negation may have no unique minimal model. + \item Given a program $P$ and an event $I$, their \deft{reduct}, $P^I$, is the propositional program that results from + \begin{enumerate} + \item Removing all the rules with $\naf b$ in the body where $b \in I$. + \item Removing all the $\naf b$ subgoals from the remaining rules. + \end{enumerate} + \item A \deft{stable model} (or \deft{answer set}) of the program $P$ is an event $I$ that is the minimal model of the reduct $P^I$. + \item Denote $\fml{S}_P$ the set of all stable models of program $P$. The \deft{semantics} (or \deft{answer sets}) of a program $P$ is the set $\fml{S}_P$. + \begin{itemize} + \item Some programs, such as $a \leftarrow \naf a$, have no stable models. + \item A stable model is an event closed under the rules of the program. + \end{itemize} + \end{itemize} +\end{frame} +% ================================================================ +\end{document} \ No newline at end of file diff --git a/text/backup/zz_old/00_PROB.pdf b/text/backup/zz_old/00_PROB.pdf new file mode 100644 index 0000000..c4e3126 Binary files /dev/null and b/text/backup/zz_old/00_PROB.pdf differ diff --git a/text/backup/zz_old/Empty File b/text/backup/zz_old/Empty File new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/text/backup/zz_old/Empty File diff --git a/text/backup/zz_old/Pabc_alpha03.pdf b/text/backup/zz_old/Pabc_alpha03.pdf new file mode 100644 index 0000000..e80e2ea Binary files /dev/null and b/text/backup/zz_old/Pabc_alpha03.pdf differ diff --git a/text/backup/zz_old/README.md b/text/backup/zz_old/README.md new file mode 100644 index 0000000..86c88ba --- /dev/null +++ b/text/backup/zz_old/README.md @@ -0,0 +1,67 @@ +# Probabilistic ILP + +**Check** Conformal prediction. + +> Fonte: [Turning 30: New Ideas in Inductive Logic Programming](https://arxiv.org/abs/2002.11002) + +## Introduction + +- How pILP relates to: + - ILP? + - ASP? + - RML? +- What + - tools? + - methods? + - theory? + - Distributed semantics + - applications? + +### Overview of Bibliography and State of the Art + +Recursion; Predicate Invention; Higher order, ASP Hypotheses; Optimality; Prolog, ASP, NNs + +## Context + +### Kanren + +### Inductive Logic Programming + +### Answer Set Programming + +### Relational Machine Learning + +### SAT Solvers + +## Tools + +- [(mini)kanren](http://minikanren.org/) + - in Julia: [MuKanren](https://github.com/latticetower/MuKanren.jl), [YA microkanren in Julia](https://www.philipzucker.com/yet-another-microkanren-in-julia/)!. +- [metagol | archive](https://github.com/metagol/metagol) _superseeded by **popper**._ +- ILP: [popper](https://github.com/logic-and-learning-lab/Popper) +- ASP: [ILASP](https://github.com/ilaspltd/ILASP-releases) +- [Inspire | Kazmi et al. 2017]() +- ASP: [Potassco: clingo, clasp, ...](https://potassco.org/) +- [cplint (on SWISH)](http://cplint.ml.unife.it/) + - exact probabilistic inference (PITA) + - Fabrizio Riguzzi and Terrance Swift. Well-definedness and efficient inference for probabilistic logic programming under the distribution semantics. Theory and Practice of Logic Programming, 13(Special Issue 02 - 25th Annual GULP Conference):279-302, © Cambridge University Press, March 2013. + - Monte Carlo inference (MCINTYRE) + - Fabrizio Riguzzi. MCINTYRE: A Monte Carlo system for probabilistic logic programming. Fundamenta Informaticae, 124(4):521-541, © IOS Press, 2013. + - Metropolis/Hastings sampling + - Arun Nampally and C. R. Ramakrishnan. Adaptive MCMC-Based Inference in Probabilistic Logic Programs. arXiv preprint arXiv:1403.6036, 2014. + - parameter learning (EMBLEM) + - Elena Bellodi and Fabrizio Riguzzi. Expectation Maximization over binary decision diagrams for probabilistic logic programs. Intelligent Data Analysis, 17(2):343-363, © IOS Press, 2013. + - SLIPCOVER algorithm for structure learning + - Elena Bellodi and Fabrizio Riguzzi. Structure learning of probabilistic logic programs by searching the clause space. Theory and Practice of Logic Programming, 15(2):169-212, © Cambridge University Press, 2015. + - LEMUR algorithm for structure learning + - Nicola Di Mauro, Elena Bellodi, and Fabrizio Riguzzi. Bandit-based Monte-Carlo structure learning of probabilistic logic programs. Machine Learning, 100(1):127-156, © Springer International Publishing, July 2015. + +## Methods + +## Theory + +### Distributed Semantics + +## Applications + +### ELearning diff --git a/text/backup/zz_old/README.pdf b/text/backup/zz_old/README.pdf new file mode 100644 index 0000000..24279f1 Binary files /dev/null and b/text/backup/zz_old/README.pdf differ diff --git a/text/backup/zz_old/SCASP-best-practices.pdf b/text/backup/zz_old/SCASP-best-practices.pdf new file mode 100644 index 0000000..ea0feef Binary files /dev/null and b/text/backup/zz_old/SCASP-best-practices.pdf differ diff --git a/text/backup/zz_old/State of the art.txt b/text/backup/zz_old/State of the art.txt new file mode 100644 index 0000000..fe11e38 --- /dev/null +++ b/text/backup/zz_old/State of the art.txt @@ -0,0 +1,42 @@ +https://ceur-ws.org/Vol-3204/paper_9.pdf (Damiano Azzolini, Elena Bellodi and Fabrizio Riguzzi) + + Abduction in ASP an PASP + +https://link.springer.com/article/10.1007/s10489-022-03669-z (Serge Sonfack Sounchio, Laurent Geneste & Bernard Kamsu Foguem) + + evaluate ASP models’ beliefs using experts’ evidence distributions, while reducing the knowledge-intensive load of the expertise process + + +https://hal-emse.ccsd.cnrs.fr/emse-03261182/file/actes_IC_CH_PFIA2021.pdf#page=99 (S. SONFACK SOUNCHIO, L. GENESTE, B. KAMSU FOGUE) + + Hybridation of ASP + + +https://link.springer.com/chapter/10.1007/978-3-031-15707-3_4 (Damiano Azzolini, Elena Bellodi & Fabrizio Riguzzi) + + + “Probabilistic Answer set programming for STAtistical probabilities” + + +https://content.iospress.com/articles/fundamenta-informaticae/fi1975 ( Tarzariol, Alice; Zanazzo, Eugenia; Dovier, Agostino; Policriti, Alberto) + + Use ASP to develop a tool-chain capable of analyzing a data collection of temporally qualified (genetic) mutation profile + + +https://link.springer.com/chapter/10.1007/978-3-030-97454-1_4 (Leopoldo Bertossi & Gabriela Reyes) + + how answer-set programs can be used to declaratively specify counterfactual interventions on entities under classification, and reason about them. + + +https://www.sciencedirect.com/science/article/pii/S0888613X20302012?casa_token=9T4mnIP5OR4AAAAA:_39BvC91LBUO2wPyHMozq1iUdrmqHF7rwk15uXieLGQbd53Ddr-BNZtYGK7JD9wR3VDt7ZKnIw + +(Fabio Gagliardi Cozman, Denis Deratani Mauá) + + + inference algorithm to compute (upper) probabilities given a program. (PASP) + + +https://link.springer.com/chapter/10.1007/978-3-030-99461-7_12 (Germán Vidal) + + + Explanations as Programs in Probabilistic Logic Programming \ No newline at end of file diff --git a/text/backup/zz_old/drafts.pdf b/text/backup/zz_old/drafts.pdf new file mode 100644 index 0000000..325b288 Binary files /dev/null and b/text/backup/zz_old/drafts.pdf differ diff --git a/text/backup/zz_old/drafts.tex b/text/backup/zz_old/drafts.tex new file mode 100644 index 0000000..bcdaf5d --- /dev/null +++ b/text/backup/zz_old/drafts.tex @@ -0,0 +1,83 @@ +\documentclass{standalone} + +\usepackage[x11colors]{xcolor} +% +\usepackage{tikz} +\tikzset{ + event/.style={}, + smodel/.style={fill=gray!25}, + tchoice/.style={draw, circle}, + indep/.style={draw, dashed}, + proptc/.style = {-latex, dashed}, + propsm/.style = {-latex, thick}, + doubt/.style = {gray} +} +\usetikzlibrary{calc, positioning} +% +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=blue, +} + + +\usepackage{commath} + + +% +% Local commands +% +\newcommand{\note}[1]{\marginpar{\scriptsize #1}} +\newcommand{\naf}{\ensuremath{\sim\!}} +\newcommand{\larr}{\ensuremath{\leftarrow}} +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} +\newcommand{\deft}[1]{\textbf{#1}} +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} +\newcommand{\ent}{\ensuremath{\lhd}} +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} +\newcommand{\uset}[1]{\ensuremath{\left|{#1}\right>}} +\newcommand{\lset}[1]{\ensuremath{\left<{#1}\right|}} +\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}} +\newcommand{\given}{\ensuremath{~\middle|~}} + +\begin{document} +\begin{tikzpicture} + \node[event] (E) {$\bot$}; + \node[tchoice, above left = of E] (a) {$a$}; + \node[smodel, above left = of a] (ab) {$ab$}; + \node[smodel, above right = of a] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \draw[proptc] (a) to[bend left] (ab); + \draw[proptc] (a) to[bend right] (ac); + + \draw[propsm] (ab) to[bend left] (abc); + \draw[propsm] (ac) to[bend right] (abc); + + \draw[propsm] (A) to[bend right] (Ac); + \draw[propsm] (A) to[bend right] (Abc); + + \draw[doubt] (ab) to[bend right] (E); + \draw[doubt] (ac) to[bend right] (E); + \draw[doubt] (A) to[bend left] (E); + + \draw[doubt] (ab) to[bend right] (b); + \draw[doubt] (ac) to[bend left] (c); + \draw[doubt] (ab) to[bend left] (a); + \draw[doubt] (ac) to[bend right] (a); + \draw[doubt] (c) to[bend right] (bc); + \draw[doubt] (abc) to[bend left] (bc); + \draw[doubt] (Abc) to (bc); + \draw[doubt] (c) to[bend right] (Ac); +\end{tikzpicture} + +\end{document} \ No newline at end of file diff --git a/text/backup/zz_old/meetings.pdf b/text/backup/zz_old/meetings.pdf new file mode 100644 index 0000000..eae0221 Binary files /dev/null and b/text/backup/zz_old/meetings.pdf differ diff --git a/text/backup/zz_old/probast_draft/probast_draf.pdf b/text/backup/zz_old/probast_draft/probast_draf.pdf new file mode 100644 index 0000000..59fdd2a Binary files /dev/null and b/text/backup/zz_old/probast_draft/probast_draf.pdf differ diff --git a/text/backup/zz_old/probast_draft/probast_draf.tex b/text/backup/zz_old/probast_draft/probast_draf.tex new file mode 100644 index 0000000..a15f2d2 --- /dev/null +++ b/text/backup/zz_old/probast_draft/probast_draf.tex @@ -0,0 +1,434 @@ +\documentclass[bigger]{beamer} +\useinnertheme{circles} +\usefonttheme[onlymath]{serif} +\usefonttheme{structurebold} +\setbeamertemplate{navigation symbols}{} +\usepackage{xcolor} +\setbeamercolor{highlight block}{bg=gray} + +\usepackage{tikz} +\usetikzlibrary{ + automata,% + positioning,% + calc,% + decorations,% + decorations.pathmorphing% +} +\usepackage{tkz-graph} +\newcommand{\qlr}[2]{\ensuremath{\begin{matrix}#1\cr\begin{aligned}\hline #2\end{aligned}\end{matrix}}} +\newcommand{\q}[1]{\mathbf{#1}} +\newcommand{\isep}{~,~} + +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[normalem]{ulem} % To strikeout +\usepackage{commath} + +\newcommand{\naf}{\ensuremath{\sim\!\!}} + +\title{Probabilistic Answer Set Programming} +\subtitle{A Research Draft} +\author{Francisco Coelho} +\institute[\texttt{fc@uevora.pt}]{ + NOVA LINCS \&\\ + High Performance Computing Chair \&\\ + Departamento de Informática, Universidade de Évora +} + +\begin{document} + \begin{frame}[plain] + \titlepage + \end{frame} + + \section*{Motivation} + + \begin{frame} + \frametitle{In short} + + \begin{itemize} + \item Use \textbf{logic programs} to formalize knowledge. + \begin{itemize} + \item logic program = formula = model. + \item \textbf{Observations} not always agree with such models --- errors may result from sensors or from a wrong or incomplete model. + \end{itemize} + \item We can associate \textbf{quantities} to formulas and sub-formulas. + \begin{itemize} + \item And define how observations \textbf{update} those quantities. + \end{itemize} + \item Adequate quantities and updates might be used to \textbf{interpret or evaluate the model} \emph{e.g.} define a joint distribution or measure the accuracy of a clause. + \end{itemize} + \end{frame} + + % \begin{frame} + % \frametitle{Plan} + % \begin{itemize} + % \item Use \textbf{answer set programs} as a logic programming language. + % \item Define the effect of an \textbf{observation} on quantities associated to clauses and their parts. + % \item Later, use those numbers to define \textbf{probabilistic interpretations} or \textbf{performance quantities} of a formula, clauses and atoms. + % \end{itemize} + % \end{frame} + + \section{Development} + + \begin{frame} + \tableofcontents[currentsection] + \end{frame} + + % \begin{frame} + % \frametitle{The seed on an idea} + % We want to define the \textbf{joint distribution} of the stable models. + % \begin{enumerate} + % \item A \textbf{boolean random variable} can be described by a disjunction $a; \neg a$. + % \item This ASP program has two stable models: $a$ and $\neg a$. + % \item A program with $n$ such facts $a_i; \neg a_i$ has $2^n$ stable models, the distinct combinations of those choices. + % \item \textbf{If each $a_i$ has probability $p_i$ then the probability of a stable model $W$ would be} $$P(W) = \prod_{a_i \in W}p_i \prod_{\neg a_i \in W} (1 - p_i).$$ + % \end{enumerate} + % \pause + % \begin{alertblock}{But this is wrong.} + % Even assuming that those facts are marginally independent --- which we will do. + % \end{alertblock} + % \end{frame} + + \begin{frame}{Problem 1: Probabilities} + The stable models of $c_1 \wedge c_2$ where + $$ + \begin{aligned} + c_1 &: b \vee \neg b\\ + c_2 &: h_1 \vee h_2 &\leftarrow b + \end{aligned} + $$ + are $$\set{\neg b}, \set{b, h_1}~\text{and}~\set{b, h_2}.$$ + + \begin{block}{Associate quantities to clauses and update them with observations.} + \onslide*<1>{ + Then compute: + \begin{itemize} + \item The probability of a stable model. + \item The probability of an atom. + \item The joint distribution of all atoms. + \end{itemize} + } + \onslide*<2>{ + \begin{itemize} + \item How to \textbf{match} an observation $z$ with a clause case $h_i,b$? + \item How do observations \textbf{update} the probabilities? + \item Is this enough to compute the \textbf{joint distribution of the atoms}? + \end{itemize} + } + \end{block} + \end{frame} + + \begin{frame} + \frametitle{Matching observations and sub-formulas} + + \begin{itemize} + \item An \alert{observation} is a subset of the literals\footnote{The set of atoms, $a$, of the program and their classic negations, $\neg a$.} from a program. + \item A \alert{consistent} observation has no subset $\set{p, \neg p}$. + \item A \emph{consistent} observation $z$ is \alert{relevant} for the clause $h \leftarrow b$ if $b \subseteq z$. + \item A disjunctive clause $$h_1 \vee \cdots \vee h_n \leftarrow b_1 \wedge \cdots \wedge b_m$$ has $n$ \alert{cases}: $\set{h_i, b_1, \ldots, b_m}, i = 1:n$. + \item The \emph{consistent} observation $z$ and the case $\set{h, b_{1:n}}$ \alert{match} if + $\set{h, b_{1:n}} \subseteq z$. + \end{itemize} + The above definitions apply to \textbf{facts}, $m=0$, and \textbf{constraints}, $n=0$. + \end{frame} + + \begin{frame} + \frametitle{Counters and updates} + A consistent observation \textbf{relevant} for a clause $h_1 \vee \cdots \vee h_n \leftarrow b$ should \textbf{increase the probability of matched cases}. + \begin{block}{Counters and updates} + \onslide*<1>{ + \begin{enumerate} + \item Associate \textbf{counters}, $u, r, n$, to clauses $h \leftarrow b$. + \item Associate a \textbf{counter}, $m_i$, to cases $h_i, b$. + \item \textbf{Initial} values result from \emph{prior} knowledge. + \item Each \emph{consistent} observation \textbf{increments}: + \begin{itemize} + \item The $u$ counters of relevant \alert{u}nmatched clauses (no matched cases). + \item The $r$ counters of \alert{r}elevant clauses. + \item The $n$ counters of \alert{n}ot relevant clauses. + \item The $m_i$ counters of \alert{m}atched cases $h_i, b$. + \item Clause counters must verify $r \leq u + \sum_i m_i$. + \end{itemize} + \end{enumerate} + } + \onslide*<2>{ + \begin{itemize} + \item Literals must be explicitly observed: $\neg b \not= \naf b$. + \item Counters relate a clause structure with observations. + \item So far stable models had no role. + \end{itemize} + } + \end{block} + \end{frame} + + \begin{frame} + \frametitle{Counters and updates: An example} + Given the following clauses with \alert{annotated counters}, + $$ + \begin{aligned} + %&H \leftarrow B&&\text{counters:}~ m_{1:n} ; u, r, n \\ + &b \vee \neg b &&\text{counters:}~ 7, 2 ; 3, 12, 0 \\ + &h_1 \vee h_2 \leftarrow b &&\text{counters:}~ 4, 3 ; 2, 6, 5 + \end{aligned} + $$ + \onslide*<2>{ + \begin{columns}[t] + \begin{column}{0.5\textwidth} + \begin{block}{Counters of $b \vee \neg b$}\small + $0$ observations where not relevant (because the body is $\top$); + + There where $12$ relevant observations; + + Of those, $b$ was matched by $7$, $\neg b$ by $2$ and $3$ observations matched neither ($\models\naf b, \naf \neg b$). + \end{block} + \end{column} + \begin{column}{0.5\textwidth} + \begin{block}{Counters of $h_1 \vee h_2 \leftarrow b$}\small + There where $11 = 6 + 5$ observations, $6$ relevant to this clause; + + From these, $4$ matched $h_1$, $3$ matched $h_2$ and $2$ matched no case. + \end{block} + \end{column} + \end{columns} + } + \onslide*<3>{ + \begin{block}{What can be computed?} + \begin{itemize} + \item $P(\neg b) = \frac{2}{12}$ because $\neg b$ matched $2$ of $12$ relevant observations. + \item $P(h_1 | b) = \frac{4}{6}$ because $h_1, b$ matched $4$ of $6$ relevant observations. + \item \alert{$P(b)$ needs further information}. + \begin{itemize} + \item \emph{E.g.} \textbf{assuming independent observations}, + $$P(b) = \frac{7 + 6}{12 + 0 + 6 + 5}.$$ + \end{itemize} + \end{itemize} + \end{block}} + \onslide*<4>{ + \begin{block}{What can be computed? --- assuming independent observations} + \begin{itemize} + \item $P(b) + P(\neg b) = \frac{13}{23} + \frac{2}{12} \approx 0.73 < 1$ because some observations have neither $b$ nor $\neg b$. + \item $P(h_1, b) = P(h_1 | b) P(b) = \frac{4}{6}\frac{13}{23}$ from above. + \item $P(h_2, b) = P(h_2 | b) P(b)$ is analogous. + \item \alert{But not \emph{e.g.} $P(h_1 | \neg b)$} because no clause relates $h_1$ and $\neg b$. + \end{itemize} + \end{block}} + \onslide*<5->{ + \begin{block}{Also\ldots} + \onslide*<5>{Counters are local to clauses and, for distinct clauses, may result from distinct sources. \emph{E.g. the relevant counter of $h_1 \vee h_2 \leftarrow b$ and the match counter of $b$ in $b \vee \neg b$.}} + \onslide*<6>{Some observations may have neither $b$ nor $\neg b$ so: $$P(b) + P(\neg b) < 1.$$} + \onslide*<7>{Assuming independent observations, since $h_1$ and $h_2$ are not independent, $$\sum_m P(m) > 1.$$} + \onslide*<8>{What's missing to define the \alert{joint distribution} $$P(H_1, H_2, B)?$$} + \end{block} + } + \onslide*<8->{ + \begin{block}{The joint distribution, according to the clauses} + \begin{tikzpicture}[node distance=35mm, ->, >=stealth, state/.style={draw, rounded corners}]\small + \node (B) [state] {\qlr{B}{ + b && \neg b && \naf b \cr + \bullet && \bullet && \cdot + }}; + \node (H1) [state, left of=B] { + \qlr{H_1}{ + ~ && h_1 && \neg h_1 && \naf h_1 \cr + b && \bullet && \circ && \circ \cr + \neg b && \circ && \circ && \circ \cr + \naf b && \circ && \circ && \circ + } + }; + \node (H2) [state, right of=B] { + \qlr{H_2}{ + ~ && h_2 && \neg h_2 && \naf h_2 \cr + b && \bullet && \circ && \circ \cr + \neg b && \circ && \circ && \circ \cr + \naf b && \circ && \circ && \circ + } + }; + \path + (B) edge (H1) + (B) edge (H2) + ; + \end{tikzpicture} + \end{block} + } + \end{frame} + + \begin{frame} + \frametitle{Shortcomming 2: Default Negation} + + \begin{itemize} + \item How to deal with rules with $\naf a$ parts? + \item Should missing elements on observations be replaced with $\naf a$ atoms? + \end{itemize} + \end{frame} + \section{Conclusions} + + \begin{frame} + \tableofcontents[currentsection] + \end{frame} + + \section*{Background Material} + + \begin{frame} + \Huge Background Material + \end{frame} + + \begin{frame}{Machine Learning} + Models are numeric functions: $y \approx f_\theta(x),~\theta_i, x_j, y\in\mathbf{R}$. + \begin{itemize} + \item Amazing achievements. + \item Noise tolerant. + \item (as of today) Huge enterprise funding . + \end{itemize} + but + \begin{itemize} + \item (essentially) Academically solved. + \item Models trained from ``large'' amounts of samples. + \item Hard to add background knowledge. + \item Models are hard to interpret. + \item Single table, independent rows assumption. + \end{itemize} + \end{frame} + + \begin{frame}{Inductive Logic Programming} + Models are logic program: $p_\theta(x, y),~\theta_i, x_j, y\in{\cal A}$. + \begin{itemize} + \item Amazing achievements, at scale. + \item Models trained from ``small'' amounts of samples. + \item Compact, readable models. + \item Background knowledge is easy to incorporate and edit. + \end{itemize} + but + \begin{itemize} + \item as of today, Little enterprise commitment. + \item as of today, Mostly academic interest. + \item Noise sensitive. + \end{itemize} + \end{frame} + + \begin{frame}{Distribution Semantics} + Assigns probability to (marginally independent) facts and derives probability of ground propositions. + + Let $F$ be set of facts, $S\subseteq F$, $R$ a set of definite clauses and $p$ a proposition: + $$\small + \begin{aligned} + P_F(S) &= \prod_{f \in S} P(f) \prod_{f \not\in S} \left(1 - P(f) \right) \cr + P(W) &= \sum_{S \subseteq F :~W=M(S\cup R)} P_F(S) \cr + P(p) &= \sum_{S :~ S\cup R ~\vdash~ p} P_F(S) = \sum_{W :~ p\in W} P(W) + \end{aligned} + $$ + \begin{itemize} + \item Amazing achievements, at scale. + \item Lots of tools and research. + \item The best of both ``worlds''? + \end{itemize} + + \end{frame} + + \begin{frame}{Answer Set Programming} + A program defines stable models. + \begin{itemize} + \item Pure declarative language, unlike Prolog. + \item Uses \emph{generate \& test} methods instead of proofs. + \item Uses both default $\sim\!p$ and classical negation $\neg p$. + \item Clauses can be disjunctive $a \vee b \leftarrow c \wedge d$. + \end{itemize} + \end{frame} + + \begin{frame}{ASP definitions} + \begin{itemize} + \item An \textbf{atom} is $r(t_1, \ldots t_n)$ where + \begin{itemize} + \item $r$ is a $n$-ary predicate symbol. + \item each $t_i$ is a constant or a variable. + \end{itemize} + \item A \textbf{ground atom} has no variables. + \item A \textbf{literal} is either an atom $a$ or a negated atom $\neg a$. + \item An \textbf{ASP Program} is a set of \textbf{rules} such as $h_1 \vee \cdots \vee h_m \leftarrow b_1 \wedge \cdots \wedge b_n$ where + \begin{itemize} + \item Each $h_i$ is a literal, $a$ or $\neg a$. + \item Each $b_j$ is a literal like above or preceded by $\naf~$. + \item $m + n > 0$. + \end{itemize} + \item The \textbf{head} of such rule is $h_1 \vee \cdots \vee h_m$. + \item The \textbf{body} of such rule is $b_1 \wedge \cdots \wedge b_n$. + \item Each $b_i$ is a \textbf{subgoal}. + \end{itemize} + \end{frame} + + \begin{frame}{ASP definitions \hfill(cont.)} + \begin{itemize}\setcounter{enumi}{7} + \item A \textbf{non-disjunctive rule} has $m \leq 1$. + \item A \textbf{normal rule} has $m = 1$. + \item A \textbf{constraint} has $m = 0$. + \item A \textbf{fact} is a normal rule with $n = 0$. + \item The \textbf{dependency graph} of a program is a digraph where: + \begin{itemize} + \item Each grounded atom is a node. + \item For each grounded rule there are edges from the atoms in the body to the atoms in the head. + \end{itemize} + \item A \textbf{negative edge} results from an atom with $\naf~$; Otherwise it is a \textbf{positive edge}. + \item An \textbf{acyclic program} has an acyclic dependency graph. + \end{itemize} + \end{frame} + + \begin{frame}{ASP definitions \hfill(cont.)} + \begin{itemize}\setcounter{enumi}{14} + \item A \textbf{normal program} has only normal rules. + \item A \textbf{definite program} is a normal program that doesn't contains $\neg$ neither $\naf~$. + \item In the dependency graph of a \textbf{stratified program} no cycle contains a negative edge. + \begin{itemize} + \item A stratified program has a single minimal model that assigns either true or false to each atom. + \end{itemize} + \item A \textbf{propositional program} has no variables. + \end{itemize} + \end{frame} + + \begin{frame}{ASP definitions \hfill(cont.)} + \begin{itemize}\setcounter{enumi}{18} + \item The \textbf{Herbrand base} of a program is the set of ground literals that result from combining all the predicates and constants of the program. + \item An \textbf{interpretation} is a consistent subset (\emph{i.e.} doesn't contain $\set{a, \neg a}$) of the Herbrand base. + \item A ground literal is \textbf{true}, $I \models a$, if $a \in I$; otherwise the literal is \textbf{false}. + \item A ground subgoal, $\naf b$, where $b$ is a ground literal, is \textbf{true}, $I \models \naf b$, if $b \not\in I$; otherwise, if $b \in I$, it is \textbf{false}. + \item A ground rule $r = h_1 \vee \cdots \vee h_m \leftarrow b_1 \wedge \cdots \wedge b_n$ is \textbf{satisfied} by the interpretation $I$, \emph{i.e.} $I \models r$, iff + \begin{itemize} + \item $I \not\models b_j$ for some $j$ or $I \models h_i$ for some $i$, + \end{itemize} + \item A \textbf{model} of a program is an interpretation that satisfies all the rules. + \end{itemize} + \end{frame} + + \begin{frame}{Stable Semantics} + \begin{itemize} + \item Every definite program has a unique minimal model; its \emph{semantics}. + \item Programs with negation may have no unique minimal model. + \item Given a program $P$ and an interpretation $I$, their \textbf{reduct}, $P^I$ is the propositional program that results from + \begin{enumerate} + \item Removing all the rules with $\naf b$ in the body where $b \in I$. + \item Removing all the $\naf b$ subgoals from the remaining rules. + \end{enumerate} + \item A \textbf{stable model} of the program $P$ is an interpretation $I$ that is the minimal model of the reduct $P^I$. + \item The \textbf{semantics} (the \textbf{answer sets}) of a program is the set of stable models of that program. + \end{itemize} + \end{frame} + + \begin{frame}{Stable Semantics} + \begin{itemize} + \item A program such as $a \leftarrow \naf a$ may have no stable models. + \item A stable model is a closed interpretation (under the rules of program). + \end{itemize} + \end{frame} + + \subsection*{Stable Sets} + + \begin{frame} + \tableofcontents[currentsection] + \end{frame} + + \subsection*{References} + + \begin{frame} + \tableofcontents[currentsection] + \end{frame} +\end{document} + + diff --git a/text/backup/zz_old/proposal/Objectivos .md b/text/backup/zz_old/proposal/Objectivos .md new file mode 100644 index 0000000..141fa29 --- /dev/null +++ b/text/backup/zz_old/proposal/Objectivos .md @@ -0,0 +1 @@ +O nosso objetivo é o desenvolvimento de um conjunto de ferramentas informáticas para aplicar e avaliar os resultados teóricos a problemas relevantes e bem conhecidos. O alvo inicial é a suite de referência Potassco, que suporta a linguagem ASP-Core-2 e proporciona uma API Python para os programas de grounding (CLASP) e resolução (CLINGO). \ No newline at end of file diff --git a/text/backup/zz_old/proposal/proposal.md b/text/backup/zz_old/proposal/proposal.md new file mode 100644 index 0000000..7f94fb9 --- /dev/null +++ b/text/backup/zz_old/proposal/proposal.md @@ -0,0 +1,49 @@ +# Zugzwang, Stochastic Adventures in Inductive Logic Specifications + +**Abstract.** A major limitation of logical representations is the implicit assumption that the Background Knowledge (BK) is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how ASP specifications with probabilistic facts can lead to characterizations of the joint distribution of the specification's atoms. + +**Sumário** Uma limitação importante das representações lógicas é a suposição implícita de que o _Background Knowledge_ é perfeito. Este é um princípio problemática se os dados tiverem ruído, o que costuma ser o caso. Aqui pretendemos explorar como uma especificação ASP com factos probabilísticos pode levar a uma caracterização da distribuição conjunta dos átomos da especificação. + +## Introduction and Motivation + +Answer Set Programming (ASP) is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. ASP is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints. + +The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts labelled with a probability $p$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 − p$. + +Crucially, a joint distribution of atoms derived from an ASP specification can be used to _quantitatively measure the performance of that specification_ given data observed from the system it is intended to describe. Then, given competing specifications to describe a certain system, these performance measures can be applied in various optimization techniques in order to obtain one that best describes the target system. + +The following example illustrates a critical problem with stochastic semantics of ASP specifications. The specification +```prolog +a::0.3. +b ; c :- a. +``` +has three stable models, `{-a}, {a, b}` and `{a, c}`. While it is quite natural to set `P(-a)=0.7`, there is no further information to assign values to `P(a,b)` and `P(a,c)`. At best, we can use a parameter $\alpha$ such that +$$ +\begin{aligned} +P(a,b) &= 0.3 \alpha,\cr +P(a,c) &= 0.3 (1 - \alpha). +\end{aligned} +$$ + +In summary, if an ASP specification is intended to describe some system and that system can be observed then: + +1. The observations can be used to estimate the value of the parameters (such as $\alpha$ above and others entailed from the laws of probability). +2. Once probabilities are set for each stable model, we can proceed to define a joint distribution of the atoms. +3. If that specification is only one of many possible candidates then the joint distributions together with the observations can be used to score and sort the specifications. + +## Work Plan + +A team of two **(or three?)** researchers and a graduate student, working over six months with adequate resources, should be able to produce an intermediate progress report for a workshop and a final comprehensive paper for a conference, describing: + +- The formalization of the methods outlined above including the parameter estimation from observations and the joint distribution extending the probabilities of the stable models. +- Application and evaluation of this approach to well-known problems, using available software tools, such as + - Problems: **Assim, de momento, não me lembro de nenhum!** + - Software tools: [`s(casp)`](https://ciao-lang.org/playground/scasp.html), [Potassco suit](https://potassco.org/), _etc._ + +## References + +1. Victor Verreet, Vincent Derkinderen, Pedro Zuidberg Dos Martires, Luc De Raedt, Inference and Learning with Model Uncertainty in Probabilistic Logic Programs (2022) +2. Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021) +3. Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020) +4. Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018) +6. Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan & Claypool Publishers (2013) diff --git a/text/backup/zz_old/proposal/proposal.pdf b/text/backup/zz_old/proposal/proposal.pdf new file mode 100644 index 0000000..49b14ed Binary files /dev/null and b/text/backup/zz_old/proposal/proposal.pdf differ diff --git a/text/backup/zz_old/proposal/proposal.tex b/text/backup/zz_old/proposal/proposal.tex new file mode 100644 index 0000000..c296897 --- /dev/null +++ b/text/backup/zz_old/proposal/proposal.tex @@ -0,0 +1,111 @@ +\documentclass[a4paper]{article} + + +\usepackage[ + bibstyle=numeric, + citestyle=numeric +]{biblatex} %Imports biblatex package +\addbibresource{zugzwang.bib} %Import the bibliography file + +\usepackage{commath} +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + citecolor=blue, + linkcolor=blue, +} + +\title{Zugzwang\\\textit{Logic and Artificial Intelligence}} +\author{ + \begin{tabular}{cc} + Francisco Coelho & Bruno Dinis\\ + \texttt{fc@uevora.pt} & \texttt{bruno.dinis@uevora.pt} + \end{tabular} +} + +\begin{document} + +\maketitle + +\nocite{*} + +\begin{abstract} + A major limitation of logical representations is the implicit assumption that the Background Knowledge (BK) is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how ASP specifications with probabilistic facts can lead to characterizations of probability functions on the specification's domain. +\end{abstract} + +\section{Introduction and Motivation } + +Answer Set Programming (ASP) \cite{gebser2012answer} is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. ASP is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints. + +The Distribution Semantics (DS) \cite{riguzzi2022foundations} is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) \cite{riguzzi2022foundations} are the most basic stochastic DS primitive and they take the form of logical facts, $a$, labelled with a probability, such as $p::a$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 - p$. A (consistent) combination of the PFs defines a \textit{total choice} $\theta = \set{p::a, \ldots}$ such that + +\begin{equation} + P(\theta) = \prod_{a\in\theta} p \prod_{a \not\in \theta} (1- p). +\end{equation} + +Our goal is to extend this probability, from total choices, to cover the specification domain. We can foresee two key applications of this extended probability: + +\begin{enumerate} + \item Support any probabilistic reasoning/task on the specification domain. + \item Also, given a dataset and a divergence measure, now the specification can be scored (by the divergence w.r.t. the \emph{empiric} distribution of the dataset), and sorted amongst other specifications. This is a key ingredient in algorithms searching, for example, an \textit{optimal specification} of the dataset. +\end{enumerate} + +This goal faces a critical problem concerning situations where \textit{multiple} standard models result from a given total choice\cite{cozman2020joy}, illustrated by the following example. The specification +$$ +\begin{aligned} + 0.3::a&,\cr + b \vee c& \leftarrow a. +\end{aligned} +$$ +has three stable models, $\set{\neg a}, \set{a, b}$ and $\set{a, c}$. While it is straightforward to set $P(\neg a)=0.7$, there is \textit{no further information} to assign values to $P(a,b)$ and $P(a,c)$. At best, we can use a parameter $\alpha$ such that +$$ +\begin{aligned} +P(a,b) &= 0.3 \alpha,\cr +P(a,c) &= 0.3 (1 - \alpha). +\end{aligned} +$$ + +This uncertainty in inherent to the specification, but can be mitigated with the help of a dataset: the parameter $\alpha$ can be estimated from the empirical distribution. + +In summary, if an ASP specification is intended to describe some system that can be observed then: + +\begin{enumerate} + \item The observations can be used to estimate the value of the parameters (such as $\alpha$ above and others entailed from further clauses). + \item With a probability set for the stable models, we want to extend it to all the samples (\textit{i.e.} consistent sets of literals) of the specification. + \item This extended probability can then be related to the \textit{empirical distribution}, using a probability divergence, such as Kullback-Leibler; and the divergence value used as a \textit{performance} measure of the specification with respect to the observations. + \item If that specification is only but one of many possible candidates then that performance measure can be used, \textit{e.g.} as fitness, by algorithms searching (optimal) specifications of a dataset of observations. +\end{enumerate} + +Currently, we are on the step two above: Extending a probability function (with parameters such as $\alpha$), defined on the stable sets of a specification, to all the samples of the specification. This extension must, of course, respect the axioms of probability so that probabilistic reasoning is consistent with the ASP specification. + +\section{Work Plan} + +A team of two researchers and a undergraduate, master, or Ph.D. student, working over six months with adequate resources, should be able to advance substantial contributions and produce an intermediate progress report for a workshop, a final comprehensive paper for a conference, or start a Ph.D. project with greater reach and depth, describing: + +\begin{itemize} + \item The formalization of the methods outlined above, including the parameter estimation from observations and the probability distribution over the specification samples. + \item Application and evaluation of this approach, using tools such as \hyperlink{https://ciao-lang.org/playground/scasp.html}{s(casp)}, or the \hyperlink{https://potassco.org/}{Potassco suit} to a range of problems from the simple \textit{Burglar, Earthquake, Alarm} to measuring a specification accuracy on a given dataset, or finding an optimal specification for a given dataset given some background knowledge. +\end{itemize} + +While the theoretical work for this project has yet to be completed, there are some relevant tasks that, with different levels of ambition, can be started right now: +\begin{enumerate} + \item \textit{Extract Probability Annotations}. For example, convert the annotated specification \verb!0.3::a. b ; c :- a.! to \verb! a ; -a. b ; c :- a!. This is a simple, syntactical task that can be implemented either with \texttt{prolog} or using \texttt{python} and the API provided by the Potassco suite. + \item \textit{Extend Probability to Stable Models}. Application of the method outlined before, where the probability of total choices is extended to standard models using parameters, which are next estimated with a dataset. + \item \textit{Relate Samples, Stable Models and Total Choices}. Determine which stable models, or total choices, contain and which are contained in a given sample. + \item \textit{Propagate Probability to Samples}. Use of the relation above to assign a probability to an arbitrary event, using an aggregation operation, such as $\max$ or $\prod$, from the relevant stable models. + \item \textit{Process Evaluation on Well-known Cases}. Assessment of the implemented prototype using well-known problems such as the ``A\-larm-Bur\-glar-Earth\-qua\-ke''. +\end{enumerate} + +\section*{References} + +\printbibliography + +% \begin{enumerate} +% \item Victor Verreet, Vincent Derkinderen, Pedro Zuidberg Dos Martires, Luc De Raedt, Inference and Learning with Model Uncertainty in Probabilistic Logic Programs (2022) +% \item Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021) +% \item Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020) +% \item Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018) +% \item Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan \& Claypool Publishers (2013) +% \end{enumerate} + +\end{document} \ No newline at end of file diff --git a/text/backup/zz_old/proposal/zugzwang.bib b/text/backup/zz_old/proposal/zugzwang.bib new file mode 100644 index 0000000..5f1392b --- /dev/null +++ b/text/backup/zz_old/proposal/zugzwang.bib @@ -0,0 +1,48 @@ +@inproceedings{verreet2022inference, + title={Inference and learning with model uncertainty in probabilistic logic programs}, + author={Verreet, Victor and Derkinderen, Vincent and Dos Martires, Pedro Zuidberg and De Raedt, Luc}, + booktitle={Proceedings of the AAAI Conference on Artificial Intelligence}, + volume={36}, + number={9}, + pages={10060--10069}, + year={2022} +} + +@article{cropper2022inductive, + title={Inductive logic programming at 30}, + author={Cropper, Andrew and Duman{\v{c}}i{\'c}, Sebastijan and Evans, Richard and Muggleton, Stephen H}, + journal={Machine Learning}, + volume={111}, + number={1}, + pages={147--172}, + year={2022}, + publisher={Springer} +} + +@article{cozman2020joy, + title={The joy of probabilistic answer set programming: semantics, complexity, expressivity, inference}, + author={Cozman, Fabio Gagliardi and Mau{\'a}, Denis Deratani}, + journal={International Journal of Approximate Reasoning}, + volume={125}, + pages={218--239}, + year={2020}, + publisher={Elsevier} +} + +@book{riguzzi2022foundations, + title={Foundations of probabilistic logic programming: Languages, semantics, inference and learning}, + author={Riguzzi, Fabrizio}, + year={2022}, + publisher={CRC Press} +} + +@article{gebser2012answer, + title={Answer set solving in practice}, + author={Gebser, Martin and Kaminski, Roland and Kaufmann, Benjamin and Schaub, Torsten}, + journal={Synthesis lectures on artificial intelligence and machine learning}, + volume={6}, + number={3}, + pages={1--238}, + year={2012}, + publisher={Morgan \& Claypool Publishers} +} diff --git a/text/paper_01/LLNCS/aasasp-llncs.pdf b/text/paper_01/LLNCS/aasasp-llncs.pdf new file mode 100644 index 0000000..3893398 Binary files /dev/null and b/text/paper_01/LLNCS/aasasp-llncs.pdf differ diff --git a/text/paper_01/LLNCS/aasasp-llncs.tex b/text/paper_01/LLNCS/aasasp-llncs.tex new file mode 100644 index 0000000..cd3e2ec --- /dev/null +++ b/text/paper_01/LLNCS/aasasp-llncs.tex @@ -0,0 +1,1336 @@ +%\documentclass[a4paper, 10pt]{article} +\documentclass{llncs} + +\usepackage[ +bibstyle=numeric, +citestyle=numeric +]{biblatex} %Imports biblatex package +\addbibresource{zugzwang.bib} %Import the bibliography file + +\usepackage[x11colors]{xcolor} + +\usepackage{tikz} +\tikzset{ +event/.style={}, +smodel/.style={fill=gray!25}, +tchoice/.style={draw, circle}, +indep/.style={},%{draw, dashed}, +proptc/.style = {-latex, dashed}, +propsm/.style = {-latex, thick}, +doubt/.style = {gray} +} +\usetikzlibrary{calc, positioning, patterns} + +\usepackage{hyperref} +\hypersetup{ +colorlinks=true, +linkcolor=blue, +citecolor=blue, +urlcolor=blue, +} + +\usepackage{commath} +%\usepackage{amsthm} +\newtheorem{assumption}{Assumption} +%\newtheorem{definition}{Definition} +%\newtheorem{proposition}{Proposition} +%\newtheorem{example}{Example} +%\newtheorem{theorem}{Theorem} +\usepackage{amssymb} +\usepackage[normalem]{ulem} +\usepackage[nice]{nicefrac} +\usepackage{stmaryrd} +\usepackage{acronym} +\usepackage{multicol} +\usepackage{cleveref} +% +% Local commands +% +\newcommand{\naf}{\ensuremath{\sim\!}} +\newcommand{\larr}{\ensuremath{\leftarrow}} +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} +\newcommand{\deft}[1]{\textbf{#1}} +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} +\newcommand{\ent}{\ensuremath{\lhd}} +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} +\newcommand{\uset}[1]{\ensuremath{#1^{\ast}}} +\newcommand{\lset}[1]{\ensuremath{#1_{\ast}}} +\newcommand{\yset}[1]{\ensuremath{\left\langle #1 \right\rangle}} +\newcommand{\stablecore}[1]{\ensuremath{\left\llbracket #1 \right\rrbracket}} +\newcommand{\uclass}[1]{\ensuremath{\intco{#1}}} +\newcommand{\lclass}[1]{\ensuremath{\intoc{#1}}} +\newcommand{\smclass}[1]{\ensuremath{\intcc{#1}}} +\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}} +\newcommand{\err}[1]{\ensuremath{\mathrm{err}\at{#1}}} +\newcommand{\pw}[1]{\ensuremath{\mu\at{#1}}} +\newcommand{\pwcfname}{\ensuremath{\mu_{\textrm{TC}}}} +\newcommand{\pwc}[1]{\ensuremath{\pwcfname\at{#1}}} +\newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}} +\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}} +\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}} +\newcommand{\rep}[2]{\ensuremath{\left\langle #1 \middle| #2 \right\rangle}} +\newcommand{\inconsistent}{\bot} +\newcommand{\given}{\ensuremath{~\middle|~}} +\newcommand{\emptyevent}{\ensuremath{\vartriangle}} +\newcommand{\indepclass}{\ensuremath{\Diamond}} +\newcommand{\probfact}[2]{\ensuremath{#2\mkern-4mu:\mkern-4mu#1}} +\newcommand{\probrule}[3]{\probfact{#1}{#2} \leftarrow #3} +%\newcommand{\tcgen}[1]{\ensuremath{\widehat{#1}}} +\newcommand{\tcgen}[1]{\ensuremath{\left<#1\right>}} +\newcommand{\lfrac}[2]{\ensuremath{{#1}/{#2}}} +\newcommand{\condsymb}[2]{\ensuremath{p_{#1|#2}}} +% +%\newcommand{\oldnote}[1]{\marginpar{\scriptsize #1}} +\newcommand{\oldnote}[1]{\note{#1}} +\newcommand{\todo}[1]{{\color{red!50!black}(\emph{#1})}} +% \newcommand{\oldremark}[2]{\uwave{#1}~{\color{green!40!black}(\emph{#2})}} +\newcommand{\oldremark}[2]{\remark{#1}{#2}} +\newcommand{\oldreplace}[2]{\sout{#1}/{\color{green!20!black}#2}} +\newcommand{\delete}[1]{\xout{#1}} +\newcommand{\franc}[1]{{\color{orange!60!black}#1}} +\newcommand{\bruno}{\color{red!60!blue}} + +% +% Acronyms +% +\acrodef{BK}[BK]{background knowledge} +\acrodef{ASP}[ASP]{answer set programming} +\acrodef{NP}[NP]{normal program} +\acrodef{DS}[DS]{distribution semantics} +\acrodef{PF}[PF]{probabilistic fact} +\acrodef{TC}[TC]{total choice} +\acrodef{SM}[SM]{stable model} +\acrodef{SC}[SC]{stable core} +\acrodef{KL}[KL]{Kullback-Leibler} +\acrodef{SBF}[SBF]{Simple But Fruitful} +\acrodef{RSL}[RSL]{Random Set of Literals} +\acrodef{RCE}[RCE]{Random Consistent Event} +% + +% +% +% +\renewcommand{\remark}[2]{% + \stepcounter{remark}% + \!{\color{red}/\!}% + #1% + {\!\color{red}/}\footnotemark[\arabic{remark}]% + \footnotetext[\arabic{remark}]{{\color{red}/}#2}% + } +\renewcommand{\note}[1]{ + \stepcounter{remark}% + {\!\!\color{red}/}\footnotemark[\arabic{remark}]\!\!% + \footnotetext[\arabic{remark}]{{\color{red}/}#1} +} +% +% +% +\begin{document} +% +% +% +% +% + \title{An Algebraic Approach to Stochastic ASP} + \author{Salvador Abreu\inst{1} \and Francisco Coelho\inst{1} \and Bruno Dinis \inst{1}} + \institute{Universidade de Évora} + \date{} + \maketitle\thispagestyle{empty} + +% +% +% +\begin{abstract} + We address the problem of extending probability from the total choices of an \acs{ASP} program to the \aclp{SM}, and from there to general events. + % + Our approach is algebraic in the sense that it relies on an equivalence relation over the set of events and uncertainty is expressed with variables and polynomial expressions. + % + We illustrate our methods with two examples, one of which shows a connection to bayesian networks. +\end{abstract} +% +% +% +\section{Introduction and Motivation} +% +% +% +A major limitation of logical representations in real world applications is the implicit assumption that the \acl{BK} is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how \acl{ASP} programs with probabilistic facts can lead to characterizations of probability functions on the program's domain, which is not straightforward in the context of \acl{ASP}, as explained below (see also \cite{cozman2020joy,verreet2022inference,baral2009probabilistic,pajunen2021solution}). Unlike current systems such as ProbLog \cite{de2007problog}, P-log \cite{baral2009probabilistic}, LP\textsuperscript{MLN} \cite{lee2016weighted}, or cplint \cite{alberti2017cplint}, that derive a probability distribution from a program, in our system some choices are represented by a parameter that can be later estimated from further information, \emph{e.g.}\ observations. This approach enables later refinement and scoring of a partial program of a model from additional evidence. + +\Ac{ASP} \cite{lifschitz2002answer} is a logic programming paradigm based on the \ac{SM} semantics of \acp{NP} that can be implemented using the latest advances in SAT solving technology. Unlike ProLog, \ac{ASP} is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and both hard and weak constraints. + +The \ac{DS} \cite{sato1995statistical,riguzzi2022foundations} is a key approach to extend logical representations with probabilistic reasoning. +% +Let $\fml{A}$ be a finite set of atoms. A \emph{pre-total choice} is a subset $t^{\ast}$ of \fml{A}. The \emph{\acl{TC}} (TC) associated to $t^{\ast}$ is the set $t := t^{\ast} \cup \set{\co{a} \given a \in \fml{A} \setminus t^{\ast}}$ where $\co{a}$ stands for $\neg a$. \Acp{PF} are the most basic \ac{DS} stochastic primitives and take the form $\probfact{p}{a}$ where each $a\in\fml{A}$ is associated to some $p\in\intcc{0, 1}$. Each \ac{PF} then represents a boolean random variable that is true with probability $p$ and false with probability $\co{p} = 1 - p$. + +%\note{revisit this part. $\co{a}$ não foi definido! Talvez escrever $\neg a$ na definição de $t$?} +Let $F = \set{\probfact{p}{a} \given a \in \fml{A}, p \in \intcc{0, 1}}$. For a \acl{TC} $t$ over $\fml{A}$, define +$$ +P_t := \set{ p \given a \in t^{\ast} \wedge \probfact{p}{a} \in F} \cup + \set{\co{p} \given a \in t \setminus t^{\ast} \wedge \probfact{p}{a} \in F} +$$ + +and + +\begin{equation} + \pr{T = t} = \prod_{p \in P_t} p, + \label{eq:prob.total.choice} +\end{equation} + +where $T$ is a random variable whose values are \aclp{TC}. + +Our goal is to extend this probability (which is, indeed, a product of Bernoulli distributions \cite{Teugels90}), from \aclp{TC}, to cover the program domain. We use the term ``program'' as a set of rules and facts, plain and probabilistic. We can foresee two key applications of this extended probability: + +\begin{enumerate} + \item Support probabilistic reasoning/tasks on the program domain. + \item Given a dataset and a divergence measure, the program can be scored (by the divergence w.r.t.\ the \emph{empiric} distribution of the dataset), and weighted or sorted amongst other programs. These are key ingredients in algorithms searching, for example, optimal models of a dataset. +\end{enumerate} + +To extend probabilities from \aclp{TC} we start with the stance that \emph{a program describes an observable system}, that \emph{the \aclp{SM} are all the possible states} of that system and that \emph{observations (i.e.\ events) are stochastic} --- one observation can be sub-complete (a proper subset of a \ac{SM}) or super-complete (a proper superset of a \ac{SM}), +%\note{We should explain this!} + and might not determine the real state of the system. From here, probabilities must be extended from \acp{TC} to \acp{SM} and then to any event. +% +This extension process starts with a critical problem, illustrated by the example in \cref{sec:example.1}, concerning situations where multiple \acp{SM}, $ab$ and $ac$, result from a single \ac{TC}, $a$, but there is not enough information (in the program) to assign a single probability to each \ac{SM}. We propose to address this issue by using algebraic variables to describe that lack of information and then estimate the value of those variables from empirical data. This lack of uniqueness is also addressed in \cite{cozman2020joy} along a different approach, using credal sets. + +In another related work \cite{verreet2022inference} epistemic uncertainty (or model uncertainty) is considered as a lack of knowledge about the underlying model, that may be mi\-ti\-ga\-ted via further observations. This seems to presuppose a bayesian approach to imperfect knowledge in the sense that having further observations allows to improve/correct the model. Indeed, that approach uses Beta distributions on the total choices in order to be able to learn a distribution on the events +%\remark{events}{Check this: do they learn distributions on the events?} +. This approach seems to be specially fitted to being able to tell when some probability lies beneath some given value. Our approach seems to be similar in spirit, while remaining algebraic in the way that the extension of probabilities is addressed. + +The example in \cref{sec:example.1} uses the code available in the project's repository\footnote{\url{https://git.xdi.uevora.pt/fc/sasp}}, developed with the \textit{Julia} programming language \cite{bezanson2017julia}, and the \textit{Symbolics} \cite{gowda2021high}, and \textit{DataFrames} \cite{bouchetvalat2023dataframes} libraries. +% +% +% +\section{A Simple but Fruitful Example}\label{sec:example.1} +% +% +% +In this section we consider a somewhat simple case that showcases the problem of extending probabilities from \aclp{TC} to \aclp{SM} and then to events. As mentioned before, the main issue arises from the lack of information in the program to assign a single probability to each stable model. This becomes a crucial problem in situations where multiple \aclp{SM} result from a single \acl{TC}. We will come back to this example in \cref{subsec:sbf.example}, after we present our proposal for extending probabilities from \aclp{TC} to \aclp{SM} in \cref{sec:extending.probalilities}. + + +\begin{example}\label{running.example} + Consider $\fml{A} = \set{a, b, c}$ and the following program + %\note{Introduce the notation $\probfact{p}{a}$ and what is the underlying ASP program.} + + \begin{equation} + \begin{aligned} + \probfact{0.3}{a} & ,\cr + b \vee c & \leftarrow a. + \end{aligned} + \label{eq:example.1} + \end{equation} + + %\note{Explain how the SM are defined.} + %\note{Explain our position about negation and be clear about $\co{a} = \neg a$ and not $\co{a} =\,\sim\!\! a$.} + %\note{Introduce the parameterization $\theta_{s,t}$.} + The \emph{standard form} of this program results from replacing annotated facts, such as $\probfact{0.3}{a}$, by the associated disjunctions, $a \vee \neg a$. The \aclp{SM} of the annotated program are the same as the ones from the standard form: $\co{a}, ab$ and $ac$, where $\co{a}$ stands for $\neg a$ (see \cref{fig:running.example}). While it is straightforward to assume $\pr{\co{a}}=0.7$, there is no obvious explicit way to assign values to $\pr{ab}$ and $\pr{ac}$. For instance, we can use a parameter $\theta$ as in + $$ + \begin{aligned} + \pr{ab} & = 0.3 \theta,\cr + \pr{ac} & = 0.3 (1 - \theta) + \end{aligned} + $$ + to express our knowledge that $ab,ac$ are events related in a certain way and, simultaneously, our uncertainty about that relation. The pa\-ra\-me\-ter $\theta=\theta_{s,t}$ depends on both the \acl{SM} $s$ and the \acl{TC} $t$. This uncertainty can then be addressed with the help of adequate distributions, such as empirical distributions from a dataset. +\end{example} + +If an \ac{ASP} program is intended to describe some system then: + +\begin{enumerate} + + \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the program domain. + + \item In the case where some statistical knowledge is available, for example, in the form of a distribution, we consider it as ``external'' knowledge about the parameters, that doesn't affect the extension procedure described below. + + \item Statistical knowledge can be used to estimate parameters and to ``score'' the program. + + \item\label{item:program.selection} If that program is only but one of many possible candidates then that score can be used, \emph{e.g.} as fitness, by algorithms searching (optimal) programs of a dataset of observations. + + \item If observations are not consistent with the program, then we ought to conclude that the program is wrong and must be changed accordingly. +\end{enumerate} + +Currently, we are addressing the problem of extending a probability function (possibly using parameters such as $\theta$ above), defined on the \acp{SM} of a program, to all the events of that program. This extension must satisfy the Kolmogorov axioms of probability so that probabilistic reasoning is consistent with the \ac{ASP} program and follow our interpretation of \aclp{SM} as the states of an observable system. + +As sets, the \acp{SM} can have non-empty intersection. But, as states of a system, we assume that \acp{SM} are disjoint events, in the following sense: + +\begin{assumption}\label{assumption:smodels.disjoint} + \Aclp{SM} are disjoint events: For any set $X$ of \aclp{SM}, + \begin{equation} + \pr{X} = \sum_{s\in X}\pr{s} + \end{equation} +\end{assumption} + +Consider the \aclp{SM} $ab, ac$ from \cref{running.example}, that result from the clause $b \vee c \leftarrow a$ and the \acl{TC} $\set{a}$. Since we intend to associate each \acl{SM} with a state of the system, $ab$ and $ac$ should be \emph{disjoint} events. So $b \vee c$ is interpreted as an \emph{exclusive disjunction} and, from that particular clause, no further relation between $b$ and $c$ is assumed. This does not prevent that other clauses may be added that entail further dependencies between $b$ and $c$, which in turn may change the \aclp{SM}. + +By not making distribution assumptions on the clauses of the program we can state such properties on the semantics of the program, as we've done in assumption \ref{assumption:smodels.disjoint}. +% +% +% +\section{Extending Probabilities}\label{sec:extending.probalilities} +% +% +% +\begin{figure}[t] + \begin{center} + \begin{tikzpicture} + \node[event] (E) {$\emptyevent$}; + \node[tchoice, above left = of E] (a) {$a$}; + \node[smodel, above left = of a] (ab) {$ab$}; + \node[smodel, above right = of a] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[event, above left = of ab] (abC) {$\co{c}ab$}; + \node[event, above right = of ac] (aBc) {$\co{b}ac$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \draw[doubt] (a) to[bend left] (ab); + \draw[doubt] (a) to[bend right] (ac); + + \draw[doubt] (ab) to[bend left] (abc); + \draw[doubt] (ab) to[bend right] (abC); + + \draw[doubt] (ac) to[bend right] (abc); + \draw[doubt] (ac) to[bend left] (aBc); + + \draw[doubt, dashed] (Ac) to (Abc); + + \draw[doubt] (A) to (Ac); + \draw[doubt] (A) to (Abc); + + \draw[doubt] (ab) to[bend right] (E); + \draw[doubt] (ac) to[bend right] (E); + \draw[doubt] (A) to[bend left] (E); + + \draw[doubt] (ab) to (b); + \draw[doubt] (ac) to (c); + % \draw[doubt] (ab) to[bend left] (a); + % \draw[doubt] (ac) to[bend right] (a); + \draw[doubt, dashed] (c) to[bend right] (bc); + \draw[doubt, dashed] (abc) to[bend left] (bc); + \draw[doubt, dashed] (bc) to (Abc); + \draw[doubt, dashed] (c) to[bend right] (Ac); + \end{tikzpicture} + \end{center} + + \caption{Some events related to the \aclp{SM} of \cref{running.example}. The circle nodes are \aclp{TC} and shaded nodes are \aclp{SM}. Solid lines represent relations with the \acp{SM} and dashed lines relations between other events. The set of events contained in all \aclp{SM}, denoted by $\emptyevent$, is empty in this example.} + \label{fig:running.example} +\end{figure} + +The diagram in \cref{fig:running.example} illustrates the problem of extending probabilities from \aclp{TC} to \aclp{SM} and then to general events in an \emph{edge-wise} process, where the value in a node is defined from the values in its neighbors. This quickly leads to coherence problems concerning probability, with no clear systematic approach. Notice that $bc$ is not directly related with any \acl{SM} therefore propagating values through edges would assign a hard to justify ($\not= 0$) value to $bc$. Instead, we propose to base the extension in the relation an event has with the \aclp{SM}. +% +% +% +\subsection{An Equivalence Relation}\label{subsec:equivalence.relation} +% +% +% +\begin{figure}[t] + \begin{center} + \begin{tikzpicture} + \node[event] (E) {$\emptyevent$}; + \node[tchoice, above left = of E] (a) {$a$}; + \node[smodel, above left = of a] (ab) {$ab$}; + \node[smodel, above right = of a] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[event, above left = of ab] (abC) {$\co{c}ab$}; + \node[event, above right = of ac] (aBc) {$\co{b}ac$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \path[draw, rounded corners, pattern=north west lines, opacity=0.2] + (ab.west) -- + (ab.north west) -- + % + (abC.south west) -- + (abC.north west) -- + (abC.north) -- + % + (abc.north east) -- + (abc.east) -- + (abc.south east) -- + % + (ab.north east) -- + (ab.east) -- + (ab.south east) -- + % + (a.north east) -- + % + (E.north east) -- + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (b.south west) -- + % + (ab.west) + ; + % ---- + \path[draw, rounded corners, pattern=north east lines, opacity=0.2] + (ac.south west) -- + (ac.west) -- + (ac.north west) -- + % + (abc.south west) -- + (abc.west) -- + (abc.north west) -- + % + (aBc.north east) -- + (aBc.east) -- + (aBc.south east) -- + % + (ac.north east) -- + % + (c.east) -- + % + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (a.south west) -- + (a.west) -- + (a.north west) -- + (a.north) -- + % + (ac.south west) + ; + % ---- + \path[draw, rounded corners, pattern=horizontal lines, opacity=0.2] + % (A.north west) -- + % + (Ac.north west) -- + % + (Abc.north west) -- + (Abc.north) -- + (Abc.north east) -- + (Abc.south east) -- + % + % (Ac.north east) -- + % (Ac.east) -- + % + % (A.east) -- + (A.south east) -- + % + (E.south east) -- + (E.south) -- + (E.south west) -- + (E.west) -- + (E.north west) -- + % + (Ac.north west) + ; + \end{tikzpicture} + \end{center} + + \caption{Classes (of consistent events) related to the \aclp{SM} of \cref{running.example} are defined through intersections and inclusions. In this picture we can see, for example, the classes $\set{\co{c}ab, ab, b}$ and $\set{a, abc}$. Different fillings correspond to different classes and, as before, the circle nodes are \aclp{TC} and shaded nodes are \aclp{SM}. Notice that $bc$ is not in a ``filled'' area.} + \label{fig:running.example.classes} +\end{figure} + +Given an ASP program, we consider a set of \emph{atoms} $ \fml{A}$, the set $\fml{L}$ of the \emph{literals} over \fml{A}, and the set of \emph{events} $\fml{E}$ such that $e \in \fml{E} \iff e \subseteq \fml{L}$. We also consider $\fml{W}$ the set of \emph{worlds} (consistent events), +%\note{Be more precise on this definition} +a set of \emph{\aclp{TC}} $\fml{T}$ such that for every $a \in \fml{A}$ we have $a \in t$ or $\neg a \in t$ +%\note{Shouldn't it be $a \in t$ or $\neg a \in t$???} +, and $\fml{S}$ the set of \emph{\aclp{SM}} such that $ \fml{S}\subset\fml{W}$. At last, the set of \aclp{SM} entailed by the \acl{TC} $t$ is denoted by $\tcgen{t}$. + +Our path to extend probabilities starts with a perspective of \aclp{SM} as playing a role similar to \emph{prime factors}. The \aclp{SM} of a program are the irreducible events entailed from that program and any event must be considered under its relation with the \aclp{SM}. + +From \cref{running.example}, consider the \acp{SM} $\co{a}, ab, ac$ and events $a, abc$ and $c$. While $a$ is related with (contained in) with both $ab, ac$, event $c$ is related only with $ac$. So, $a$ and $c$ are related with different \acp{SM}. On the other hand, both $ab, ac$ are related with $abc$. So $a$ and $abc$ are related with the same \aclp{SM}. + +\begin{definition}\label{def:stable.core} + The \emph{\ac{SC}} of the event $e\in \fml{E}$ is + \begin{equation} + \stablecore{e} := \set{s \in \fml{S} \given s \subseteq e \vee e \subseteq s}. \label{eq:stable.core} + \end{equation} + where $\fml{S}$ is the set of \aclp{SM}. +\end{definition} + +We now define an equivalence relation so that two events are related if either both are inconsistent or both are consistent and, in the latter case, with the same \acl{SC}. + +\begin{definition}\label{def:equiv.rel} + For a given program, let $u, v \in \fml{E}$. The equivalence relation $\sim$ is defined by + \begin{equation} + u \sim v :\!\iff u,v \not\in\fml{W} \vee \del{u,v \in \fml{W} \wedge \stablecore{u} = \stablecore{v}}.\label{eq:equiv.rel} + \end{equation} +\end{definition} + +Observe that the minimality of \aclp{SM} implies that, in \cref{def:stable.core}, either $e$ is a \acl{SM} or at least one of $\exists s \del{s \subseteq e}, \exists s \del{e \subseteq s}$ is false. This equivalence relation defines a partition on the set of events, where each class holds a unique relation with the \aclp{SM}. In particular we denote each class by: + +\begin{equation} + \class{e} = + \begin{cases} + \inconsistent := \fml{E} \setminus \fml{W} + & \text{if~} e \in \fml{E} \setminus \fml{W}, \\ + \set{u \in \fml{W} \given \stablecore{u} = \stablecore{e}} + & \text{if~} e \in \fml{W}. + \end{cases}\label{eq:event.class} +\end{equation} + +The combinations of the \aclp{SM}, together with the set of inconsistent events $\inconsistent$, form a set of representatives. Consider again \cref{running.example}. As previously mentioned, the \aclp{SM} are the elements of $\fml{S} = \set{\co{a}, ab, ac}$ so the quotient set of this relation is +\begin{equation} + \class{\fml{E}} = \set{ + \inconsistent, + \indepclass, + \class{\co{a}}, + \class{ab}, + \class{ac}, + \class{\co{a}, ab}, + \class{\co{a}, ac}, + \class{ab, ac}, + \class{\co{a}, ab, ac} + }, +\end{equation} +where $\indepclass$ denotes, with abuse of notation, both the class of \emph{independent} events $e$ such that $\stablecore{e} = \emptyset$ and its core and $\emptyevent$ is the set of events contained in all \acp{SM}. We have: +%\note{Remark the odd nature of $\emptyevent$.} + +\begin{equation*} + \begin{array}{l|lr} + \text{\textbf{Core}}, \stablecore{e} + & \text{\textbf{Class}}, \class{e} + & \text{\textbf{Size}}, \# \class{e} \\ + \hline + % + \inconsistent + & \co{a}a, \ldots + & 37 + \\ + % + \indepclass + & \co{b}, \co{c}, bc, \co{b}a, \co{b}c, \co{bc}, \co{c}a, \co{c}b, \co{bc}a + & 9 + \\ + % + \co{a} + & \co{a}, \co{a}b, \co{a}c, \co{ab}, \co{ac}, \co{a}bc, \co{ac}b, \co{ab}c, \co{abc} + & 9 + \\ + % + ab + & b, ab, \co{c}ab + & 3 + \\ + % + ac + & c, ac, \co{b}ac + & 3 + \\ + % + \co{a}, ab + & \emptyset + & 0 + \\ + % + \co{a}, ac + & \emptyset + & 0 + % + \\ + % + ab, ac + & a, abc + & 2 + \\ + % + \co{a}, ab, ac + & \emptyevent + & 1 + \\ + % + \hline + \class{\fml{E}} + & \fml{E} + & 64 + \end{array} +\end{equation*} + + Since all events within an equivalence class are in relation with a specific set of \aclp{SM}, \emph{measures, including probability, should be constant within classes}: + \[ + \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right). + \] + + In general, we have \emph{much more} \aclp{SM} than literals but their combinations are still \emph{much less} than events. Nevertheless, the equivalence classes allow us to propagate probabilities from \aclp{TC} to events, as explained in the next subsection. + + In this specific case, instead of dealing with $64 = 2^6$ events, we consider only the $9 = 2^3 + 1$ classes, well defined in terms of combinations of the \aclp{SM}. + +% +% +% +\subsection{From Total Choices to Events}\label{subsec:from.tchoices.to.events} +% +% +% +Our path to set a distribution on $\fml{E}$ starts with the more general problem of extending \emph{measures}, since extending \emph{probabilities} easily follows by means of a suitable normalization (done in \eqref{eq:measure.events.unconditional} and \eqref{eq:probability.event}), and has two phases: +\begin{enumerate} + \item Extension of the probabilities, \emph{as measures}, from the \aclp{TC} to events. + \item Normalization of the measures on events, recovering a probability. +\end{enumerate} + +The ``extension'' phase, traced by \cref{eq:prob.total.choice} and eqs.\ \eqref{eq:measure.tchoice} to \eqref{eq:measure.events}, starts with the measure (probability) of \aclp{TC}, $\pw{t} = \pr{T = t}$, expands it to \aclp{SM}, $\pw{s}$, and then, within the equivalence relation from \cref{eq:equiv.rel}, to (general) events, $\pw{e}$, including (consistent) worlds. + +\begin{description} + % + \item[Total Choices.] Using \cref{eq:prob.total.choice}, this case is given by + \begin{equation} + \pwc{t} := \pr{T = t}= \prod_{p\in P_t} p. + \label{eq:measure.tchoice} + \end{equation} + % + + \item[Stable Models.] Recall that each \acl{TC} $t$, together with the rules and the other facts of a program, defines the set \tcgen{t} of \aclp{SM} associated with that choice. + Given a \acl{TC} $t$, a \acl{SM} $s$, and variables or values $\theta_{s,t} \in \intcc{0, 1}$ such that $\sum_{s\in \tcgen{t}} \theta_{s,t} = 1$, we define + \begin{equation} + \pw{s, t} := \begin{cases} + \theta_{s,t} & \text{if~} s \in \tcgen{t}\cr + 0 & \text{otherwise.} + \end{cases} + \label{eq:measure.stablemodel} + \end{equation} + + % + + \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of \aclp{SM}. + \begin{description} + \item[Inconsistent Class.] The inconsistent class contains events that are logically inconsistent, thus should never be observed and have measure zero: + \begin{equation} + \pw{\inconsistent, t} := 0.\footnote{Notice that this measure being equal to zero is actually independent of the \acl{TC}.} + \label{eq:measure.class.inconsistent} + \end{equation} + \item[Independent Class.] A world that neither contains nor is contained in a \acl{SM} corresponds to a non-state, according to the program. So the respective measure is also set to zero: + \begin{equation} + \pw{\indepclass, t} := 0. + \label{eq:measure.class.independent} + \end{equation} + \item[Other Classes.] The extension must be constant within a class, its value should result from the elements in the \acl{SC}, and respects assumption \ref{assumption:smodels.disjoint} (\aclp{SM} are disjoint): + \begin{equation} + \pw{\class{e}, t} := \pw{\stablecore{e}, t} = \sum_{s\in\stablecore{e}}\pw{s, t} + \label{eq:measure.class.other} + \end{equation} + and + \begin{equation} + \pw{\class{e}} := \sum_{t \in \fml{T}} \pw{\class{e}, t}\pwc{t}. + \label{eq:measure.class.unconditional} + \end{equation} + \end{description} + % + + \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, denoting by $\# X$ the number of elements in $X$, we set: + \begin{equation} + \pw{e, t} := + \begin{cases} + \frac{\pw{\class{e}, t}}{\# \class{e}} & \text{if~}\# \class{e} > 0, \\ + 0 & \text{otherwise}. + \end{cases} + \label{eq:measure.events} + \end{equation} + and + \begin{equation} + \pw{e} := \sum_{t\in\fml{T}} \pw{e, t} \pwc{t}. + \label{eq:measure.events.unconditional} + \end{equation} +\end{description} + + + +The $\theta_{s,t}$ parameters in equation \eqref{eq:measure.stablemodel} express the \emph{program's} lack of knowledge about the measure assignment, when a single \acl{TC} entails more than one \acl{SM}. In that case, how to distribute the respective measures? Our proposal to address this problem consists in assigning an unknown measure, $\theta_{s,t}$, conditional on the \acl{TC}, $t$, to each \acl{SM} $s$. This approach allows the expression of an unknown quantity and future estimation, given observed data. +% Consider the event $bc$ from \cref{running.example}. Since $\class{bc} = \indepclass$, from \cref{eq:measure.class.independent} we get $\mu\at{bc} = 0$. data. + +% SUPERSET +Equation \eqref{eq:measure.class.other} results from assumption \ref{assumption:smodels.disjoint} and states that the measure of a class $\class{e}$ is the sum over it's \acl{SC}, $\stablecore{e}$, and \eqref{eq:measure.class.unconditional} \emph{marginalizes} the \acp{TC} on \eqref{eq:measure.class.other}. + +The \emph{normalizing factor} is: +\begin{equation*} + Z := + \sum_{e \in \fml{E}} \pw{e} = + \sum_{\class{e} \in \class{\fml{E}}} \pw{\class{e}}, +\end{equation*} + +and now equation \eqref{eq:measure.events.unconditional} provides a straightforward way to define the \emph{probability of observation of a single event}: + +\begin{equation} + \pr{E = e} := \frac{\pw{e}}{Z}.\label{eq:probability.event} +\end{equation} + +Equation \eqref{eq:measure.events.unconditional} together with external statistical knowledge, can be used to learn about the \emph{initial} probabilities of the atoms, that should not (and by \cref{prop:two.distributions} can't) be confused with the explicit $\pwcfname$ set in the program. + +It is now straightforward to check that $\pr{E}$ satisfies the Kolmogorov axioms of probability. + +Since \aclp{TC} are also events, one can ask, for an arbitrary \aclp{TC} $t$, if $\pr{T = t} = \pr{E = t}$ or, equivalently, if $\pwc{t} = \pw{t}$. However, it is easy to see that, in general, that cannot be true. While the domain of the random variable $T$ is the set of \aclp{TC}, for $E$ the domain is much larger, including all the events. Except for trivial programs, where the \acp{SM} are the \acp{TC}, some events other than \aclp{TC} have non-zero probability. + +\begin{proposition} \label{prop:two.distributions} + In a program with a \acl{SM} that is not a \acl{TC} there is at least one $t\in\fml{T}$ such that: + \begin{equation} + \pr{T = t} \not= \pr{E = t}. \label{eq:two.distributions} + \end{equation} +\end{proposition} + +\begin{proof} + Suppose towards a contradiction that $\pr{T = t} = \pr{E = t}$ for all $t \in \fml{T}$. Then + $$ + \sum_{t\in\fml{T}} \pr{E = t} = \sum_{t\in\fml{T}} \pr{T = t} = 1. + $$ + + Hence $\pr{E = x} = 0$ for all $x \in \fml{E}\setminus\fml{T}$, in contradiction with the fact that for at least one $s \in \fml{S}\setminus\fml{T}$ one has $\pr{E = s} > 0$. +\end{proof} + +The essential conclusion of \cref{prop:two.distributions} is that we are dealing with \emph{two distributions}: one, on the \acp{TC}, explicit in the annotations of the programs and another one, on the events, and entailed by the explicit annotations \emph{and the structure of the \aclp{SM}}. + +% +% +% +\section{Developed Examples}\label{sec:developed.examples} +% +% +% +Here we apply the methods from \cref{sec:extending.probalilities} to \cref{running.example} and to a well known bayesian network: the Earthquake, Burglar, Alarm problem. + +\subsection{The SBF Example}\label{subsec:sbf.example} + +We continue with the program from \cref{eq:example.1}. + +\begin{description} + % + \item[\Aclp{TC}.] The \aclp{TC}, and respective \aclp{SM}, are + % + \begin{center} + \begin{tabular}{ll|r} + \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pwc{t}$} \\ + \hline + $a$ & $ab, ac$ & $0.3$ \\ + $\co{a}$ & $\co{a}$ & $\co{0.3} = 0.7$ + \end{tabular} + \end{center} + % + + \item[\Aclp{SM}.] The $\theta_{s,t}$ parameters in this example are + $$ + \begin{array}{l|cc} + \theta_{s,t} & \co{a} & a \\ + \hline + \co{a} & 1 & 0 \\ + ab & 0 & \theta \\ + ac & 0 & \co{\theta} + \end{array} + $$ + with $\theta \in \intcc{0, 1}$. + + \item[Classes.] Following the definitions in \cref{eq:stable.core,eq:equiv.rel,eq:event.class,eq:measure.class.inconsistent,eq:measure.class.independent,eq:measure.class.other} we get the following quotient set (ignoring $\inconsistent$ and $\indepclass$), and measures: + \begin{equation*} + \begin{array}{l|ll|rr|r} + \stablecore{e} + & \pw{s, \co{a}} + & \pw{s, a} + & \pw{\class{e}, \co{a}} + & \pw{\class{e}, a} + & \pw{\class{e}} + \\[2pt] + & \co{a}, ab, ac + & \co{a}, ab, ac + & \pwcfname=0.7 + & \pwcfname=0.3 + & + \\[2pt] + \hline + \co{a} + & \boxed{1},0,0 + & \boxed{0},\theta, \co{\theta} + & 1 + & 0 + & 0.7 + \\[2pt] + % + ab + & 1,\boxed{0},0 + & 0,\boxed{\theta}, \co{\theta} + & 0 + & \theta + & 0.3\theta + \\[2pt] + % + ac + & 1,0,\boxed{0} + & 0,\theta, \boxed{\co{\theta}} + & 0 + & \co{\theta} + & 0.3\co{\theta} + \\[2pt] + % + \co{a}, ab + & \boxed{1},\boxed{0},0 + & \boxed{0},\boxed{\theta}, \co{\theta} + & 1 + & \theta + & 0.7 + 0.3\theta + \\[2pt] + % + \co{a}, ac + & \boxed{1},0,\boxed{0} + & \boxed{0},\theta, \boxed{\co{\theta}} + & 1 + & \co{\theta} + & 0.7 + 0.3\co{\theta} + \\[2pt] + % + ab, ac + & 1,\boxed{0},\boxed{0} + & 0,\boxed{\theta}, \boxed{\co{\theta}} + & 0 + & \theta + \co{\theta} = 1 + & 0.3 + \\[2pt] + % + \co{a}, ab, ac + & \boxed{1},\boxed{0},\boxed{0} + & \boxed{0},\boxed{\theta}, \boxed{\co{\theta}} + & 1 + & \theta + \co{\theta} = 1 + & 1 + \end{array} + \end{equation*} + + \item[Prior Distributions.] Following the above values (in rational form), and considering the inconsistent and independent classes (resp. $\inconsistent, \indepclass$): + \begin{equation*} + \begin{array}{lr|cc|cc} + \stablecore{e} + & \# \class{e} + & \pw{\class{e}} + & \pw{e} + & \pr{E = e} + & \pr{E \in \class{e}} + \\ + \hline + % + \inconsistent + & 37 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + \indepclass + & 9 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + \co{a} + & 9 + & \frac{7}{10} + & \frac{7}{90} + & \frac{7}{207} + & \frac{7}{23} + \\[4pt] + % + ab + & 3 + & \frac{3}{10}\theta + & \frac{1}{10}\theta + & \frac{1}{23}\theta + & \frac{3}{23}\theta + \\[4pt] + % + ac + & 3 + & \frac{3}{10}\co{\theta} + & \frac{1}{10}\co{\theta} + & \frac{1}{23}\co{\theta} + & \frac{3}{23}\co{\theta} + \\[4pt] + % + \co{a}, ab + & 0 + & \frac{7 + 3\theta}{10} + & 0 + & 0 + & 0 + \\[4pt] + % + \co{a}, ac + & 0 + & \frac{7 + 3\co{\theta}}{10} + & 0 + & 0 + & 0 + % + \\[4pt] + % + ab, ac + & 2 + & \frac{3}{10} + & \frac{3}{20} + & \frac{3}{46} + & \frac{3}{23} + \\[4pt] + % + \co{a}, ab, ac + & 1 + & 1 + & 1 + & \frac{10}{23} + & \frac{10}{23} + \\[4pt] + % + \hline + & + & + & Z = \frac{23}{10} + & + %& \Sigma = 1 + \end{array} + \end{equation*} +\end{description} + +So the prior distributions, denoted by the random variable $E$, of events and classes are: + +\begin{equation} + \begin{array}{l|ccccccccc} + \stablecore{e} & + \inconsistent & + \indepclass & + \co{a} & + ab & + ac & + \co{a}, ab & + \co{a}, ac & + ab, ac & + \co{a}, ab, ac + \\ \hline\\[-12pt] + + \pr{E = e} & + 0 & + 0 & + \frac{7}{207} & + \frac{1}{23}\theta & + \frac{1}{23}\co{\theta} & + 0 & + 0 & + \frac{3}{46} & + \frac{10}{23} + \\[4pt] + + \pr{E \in \class{e}} & + 0 & + 0 & + \frac{7}{23} & + \frac{3}{23}\theta & + \frac{3}{23}\co{\theta} & + 0 & + 0 & + \frac{3}{23} & + \frac{10}{23} + \end{array}\label{eq:sbf.prior} +\end{equation} +% +% +% +\subsubsection*{Testing the Prior Distributions} +% +% +% +These results can be \emph{tested by simulation} in a two-step process, where (1) a ``system'' is \emph{simulated}, to gather some ``observations'' and then (2) empirical distributions from those samples are \emph{related} with the prior distributions from \cref{eq:sbf.prior}. \Cref{tab:sbf.example,tab:sbf.examples.2.3} summarize some of those tests, where datasets of $n = 1000$ observations are generated and analyzed. + +\bigskip\noindent\textbf{Simulating a System.} Following some criteria, more or less related to the given program, a set of events, that represent observations, is generated. Possible simulation procedures include: +\begin{itemize} + % + \item \emph{Random.} Each sample is a \ac{RSL}. Additional sub-criteria may require, for example, consistent events, a \ac{RCE} simulation. + % + \item \emph{Model+Noise.} Gibbs' sampling \cite{geman84} tries to replicate the program model and also to add some noise. For example, let $\alpha, \beta, \gamma \in \intcc{0,1}$ be some parameters to control the sample generation. The first parameter, $\alpha$ is the ``out of model'' samples ratio; $\beta$ represents the choice $a$ or $\co{a}$ (explicit in the model) and $\gamma$ is the simulation representation of $\theta$. A single sample is then generated following the probabilistic choices below: + $$ + \begin{cases} + \alpha & \text{by \ac{RCE}} \\%[-2pt] + & + \begin{cases} + \beta & \co{a} \\%[-2pt] + & + \begin{cases} + \gamma & ab \\%[-2pt] + & ac + \end{cases} + \end{cases} + \end{cases}, + $$ + where + $$ + \begin{cases} + p & x \\%[-4pt] + & y + \end{cases} + $$ + denotes ``\emph{the value of $x$ with probability $p$, otherwise $y$}'' --- notice that $y$ might entail $x$ and \emph{vice-versa}: E.g.\ some $ab$ can be generated in the \ac{RCE}. + \item \emph{Other Processes.} Besides the two sample generations procedures above, any other processes and variations can be used. For example, requiring that one of $x, \co{x}$ literals is always in a sample or using specific distributions to guide the sampling of literals or events. +\end{itemize} + +\noindent\textbf{Relating the Empirical and the Prior Distributions.} The data from the simulated observations is used to test the prior distribution. Consider the prior, $\pr{E}$, and the empirical, $\pr{S}$, distributions and the following error function: +\begin{equation} + \err{\theta} := \sum_{e\in\fml{E}} \del{\pr{E = e} - \pr{S = e}}^2.\label{eq:err.e.s} +\end{equation} + +Since $E$ depends on $\theta$, one can ask how does the error varies with $\theta$, what is the \emph{optimal} (i.e.\ minimum) error value + \begin{equation} + \hat{\theta} := \arg\min_\theta \err{\theta}\label{eq:opt.err} + \end{equation} + and what does it tell us about the program. + + +\begin{table} + \begin{center} + $$ + \begin{array}{l|cc|c} + \stablecore{e} + & \#\set{S \in \class{e}} + & \pr{S \in \class{e}} + & \pr{E \in \class{e}} + \\ + \hline + % + \inconsistent + & 0 + & 0 + & 0 + \\[2pt] + % + \indepclass + & 24 + & \frac{24}{1000} + & 0 + \\[2pt] + % + \co{a} + & 647 + & \frac{647}{1000} + & \frac{7}{23} + \\[2pt] + % + ab + & 66 + & \frac{66}{1000} + & \frac{3}{23}\theta + \\[2pt] + % + ac + & 231 + & \frac{231}{1000} + & \frac{3}{23}\co{\theta} + \\[2pt] + % + \co{a}, ab + & 0 + & 0 + & 0 + \\[2pt] + % + \co{a}, ac + & 0 + & 0 + & 0 + % + \\[2pt] + % + ab, ac + & 7 + & \frac{7}{1000} + & \frac{3}{23} + \\[2pt] + % + \co{a}, ab, ac + & 25 + & \frac{25}{1000} + & \frac{10}{23} + \\[2pt] + \hline + & n = 1000 + \end{array} + $$ + \end{center} + + \caption{\emph{Experiment 1.} Results from an experiment where $n=1000$ samples where generated following the \emph{Model+Noise} procedure with parameters $\alpha = 0.1, \beta = 0.3, \gamma = 0.2$. The \emph{empirical} distribution is represented by the random variable $S$ while the \emph{prior}, as before, is denoted by $E$.}\label{tab:sbf.example} +\end{table} + +In order to illustrate this analysis, consider the experiment summarized in \cref{tab:sbf.example}: + +\begin{enumerate} + \item Equation \eqref{eq:err.e.s} becomes + $$ + \err{\theta} = \frac{20869963}{66125000} + \frac{477}{52900}\theta + \frac{18}{529}\theta^2. + $$ + \item The minimum of $\err{\theta}$ is at $\frac{477}{52900} + 2\frac{18}{529}\theta = 0$. Since this value is negative and $\theta \in \intcc{0,1}$, it must be $\hat{\theta} = 0$, and + $$ + \err{\hat{\theta}} = \frac{20869963}{66125000} \approx 0.31561. + $$ +\end{enumerate} + +The parameters $\alpha, \beta, \gamma$ of that experiment favour $ac$ over $ab$. In particular, setting $\gamma = 0.2$ means that in the simulation process, choices between $ab$ and $ac$ favour $ac$, 4 to 1. For completeness sake, we also describe one experiment that favours $ab$ over $ac$ (setting $\gamma=0.8$) and one balanced ($\gamma=0.5$). + +\begin{description} + \item[For $\gamma=0.8$,] the error function is + \begin{equation*} + \err{\theta} = \frac{188207311}{529000000} - \frac{21903}{264500} \theta + \frac{18}{529} \theta^{2} \approx 0.35579 - 0.08281 \theta + 0.03403 \theta ^2 + \end{equation*} + and, with $\theta\in\intcc{0, 1}$ the minimum is at $-0.08281 + 0.06805 \theta = 0$, \emph{i.e.}: + \begin{eqnarray*} + \hat{\theta} : \frac{0.08281}{0.06805} \approx 1.21683& >1. &\text{So,~} \hat{\theta} = 1, \\ + \err{\hat{\theta}} \approx 0.30699&. + \end{eqnarray*} + + \item[For $\gamma=0.5$,] the error function is + \begin{equation*} + \err{\theta} = \frac{10217413}{33062500} - \frac{2181}{66125} \theta + \frac{18}{529} \theta^{2}\approx 0.30903 - 0.03298 \theta + 0.03402 \theta ^2 + \end{equation*} + and, with $\theta\in\intcc{0, 1}$ the minimum is at $-0.03298 + 0.06804 \theta = 0$, \emph{i.e.}: + \begin{eqnarray*} + \hat{\theta} &\approx & + \frac{0.03298}{0.06804} + \approx 0.48471 + \approx \frac{1}{2}, \\ + \err{\hat{\theta}} &\approx & + 0.30104 + \end{eqnarray*} + +\end{description} + +\begin{table} + \begin{center} + $$ + \begin{array}{l|ccc} + \stablecore{e} + & \#\set{S_{0.2} \in \class{e}} + & \#\set{S_{0.8} \in \class{e}} + & \#\set{S_{0.5} \in \class{e}} + \\ + \hline + % + \inconsistent + & 0 + & 0 + & 0 + \\[2pt] + % + \indepclass + & 24 + & 28 + & 23 + \\[2pt] + % + \co{a} + & 647 + & 632 + & 614 + \\[2pt] + % + ab + & 66 + & 246 + & 165 + \\[2pt] + % + ac + & 231 + & 59 + & 169 + \\[2pt] + % + \co{a}, ab + & 0 + & 0 + & 0 + \\[2pt] + % + \co{a}, ac + & 0 + & 0 + & 0 + % + \\[2pt] + % + ab, ac + & 7 + & 8 + & 4 + \\[2pt] + % + \co{a}, ab, ac + & 25 + & 27 + & 25 + \end{array} + $$ + \end{center} + + \caption{\emph{Experiments 2 and 3.} Results from experiments, each with $n=1000$ samples generated following the \emph{Model+Noise} procedure, with parameters $\alpha = 0.1, \beta = 0.3, \gamma = 0.8$ (Experiment 2) and $\gamma=0.5$ (Experiment 3). Empirical distributions are represented by the random variables $S_{0.8}$ and $S_{0.5}$ respectively. Data from experience \cref{tab:sbf.example} is also included, and denoted by $S_{0.2}$, to provide reference.}\label{tab:sbf.examples.2.3} +\end{table} + +%\oldnote{under- and over- estimation} +These experiments show that data can indeed be used to estimate the parameters of the model. However, we observe that the estimated $\hat{\theta}$ has a tendency to over- or under- estimate the $\theta$ used to generate the samples. More precisely, in experiment \ref{tab:sbf.example} data is generated with $\gamma = 0.2$ (the surrogate of $\theta$) which is under-estimated with $\hat{\theta} = 0$ while in experiment 2, $\gamma = 0.8$ leads the over-estimation $\hat{\theta} = 1$. This suggests that we might need to refine the error estimation process. However, experiment 3 data results from $\gamma = 0.5$ and we've got $\hat{\theta} \approx 0.48471 \approx 0.5$, which is more in line with what is to be expected. +% +% +% +\subsection{An Example Involving Bayesian Networks}\label{subsec:example.bayesian.networks} +% +% +% +As it turns out, our framework is suitable to deal with more sophisticated cases, in particular cases involving bayesian networks. In order to illustrate this, in this section we see how the classical example of the Burglary, Earthquake, Alarm \cite{Judea88} works in our setting. This example is a commonly used example in bayesian networks because it illustrates reasoning under uncertainty. The gist of the example is given in \cref{Figure_Alarm}. It involves a simple network of events and conditional probabilities. + +The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) and John calls ($J$). The initial events $B$ and $E$ are assumed to be independent events that occur with probabilities $\pr{B}$ and $\pr{E}$, respectively. There is an alarm system that can be triggered by either of the initial events $B$ and $E$. The probability of the alarm going off is a conditional probability given that $B$ and $E$ have occurred. One denotes these probabilities, as per usual, by $\pr{A \given B}$, and $\pr{A \given E}$. There are two neighbors, Mary and John who have agreed to call if they hear the alarm. The probability that they do actually call is also a conditional probability denoted by $\pr{M \given A}$ and $\pr{J \given A}$, respectively. + +\begin{figure} + \begin{center} + \begin{tikzpicture}[node distance=2.5cm] + + % Nodes + \node[smodel, circle] (A) {A}; + \node[tchoice, above right of=A] (B) {B}; + \node[tchoice, above left of=A] (E) {E}; + \node[tchoice, below left of=A] (M) {M}; + \node[tchoice, below right of=A] (J) {J}; + + % Edges + \draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$\pr{B}=0.001$}} ; + \draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$\pr{E}=0.002$}} ; + \draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$\pr{M \given A}$}}; + \draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$\pr{J \given A}$}} ; + \end{tikzpicture} + \end{center} + + \begin{multicols}{3} + + \footnotesize{ + \begin{equation*} + \begin{split} + &\pr{M \given A}\\ + & \begin{array}{c|cc} + & m & \neg m \\ + \hline + a & 0.9 & 0.1 \\ + \neg a & 0.05 & 0.95 + \end{array} + \end{split} + \end{equation*} + } + + \footnotesize{ + \begin{equation*} + \begin{split} + &\pr{J \given A}\\ + & \begin{array}{c|cc} + & j & \neg j \\ + \hline + a & 0.7 & 0.3 \\ + \neg a & 0.01 & 0.99 + \end{array} + \end{split} + \end{equation*} + } + \footnotesize{ + \begin{equation*} + \begin{split} + \pr{A \given B \wedge E}\\ + \begin{array}{c|c|cc} + & & a & \neg a \\ + \hline + b & e & 0.95 & 0.05 \\ + b & \neg e & 0.94 & 0.06 \\ + \neg b & e & 0.29 & 0.71 \\ + \neg b & \neg e & 0.001 & 0.999 + \end{array} + \end{split} + \end{equation*} + } + \end{multicols} + \caption{The Earthquake, Burglary, Alarm model} + \label{Figure_Alarm} +\end{figure} + +We follow the convention of representing the (upper case) random variable $X$ by the lower case $x$. +% +Considering the probabilities given in \cref{Figure_Alarm} we obtain the following spe\-ci\-fi\-ca\-tion: + +\begin{equation*} + \begin{aligned} + \probfact{0.001}{b} & ,\cr + \probfact{0.002}{e} & ,\cr + \end{aligned} + \label{eq:not_so_simple_example} +\end{equation*} + +For the table giving the probability $\pr{M \given A}$ we obtain the program: + +\begin{equation*} + \begin{aligned} + \probfact{0.9}{\condsymb{m}{a}} & ,\cr + \probfact{0.05}{\condsymb{m}{\co{a}}} & ,\cr + m & \leftarrow a \wedge \condsymb{m}{a},\cr + m & \leftarrow \neg a \wedge \condsymb{m}{\co{a}}. + \end{aligned} +\end{equation*} + +The latter program can be simplified (abusing notation) by writing $\probrule{0.9}{m}{a}$ and $\probrule{0.05}{m}{\neg a}$. +%\note{SPA: \emph{parece-me que pode ser feito assim, mas estritamente falando já não corresponde à forma inicialmente anunciada} --- ``abusing notation''} + +Similarly, for the probability $\pr{J \given A}$ we obtain + +\begin{equation*} + \begin{aligned} + \probrule{0.7}{j}{&a}, \\ + \probrule{0.01}{j}{&\neg a}, + \end{aligned} +\end{equation*} + +Finally, for the probability $\pr{A \given B \wedge E}$ we obtain + +\begin{equation*} + \begin{aligned} + \probrule{0.95}{a}{b, e}, & & & + \probrule{0.94}{a}{b, \co{e}},\cr + \probrule{0.29}{a}{\co{b}, e}, & & & + \probrule{0.001}{a}{\co{b}, \co{e}}. + \end{aligned} +\end{equation*} + +One can then proceed as in the previous subsection and analyze this example. The details of such analysis are not given here since they are analogous, albeit admittedly more cumbersome. +% +% +% +\section{Discussion and Future Work} +% +% +% +This work is a first venture into expressing probability distributions using algebraic expressions derived from a logical program, in particular an \ac{ASP}. +We would like to point out that there is still much to explore concerning the full expressive power of logic programs and \ac{ASP} programs. So far, we have not considered recursion, logical variables or functional symbols. Also, there is still little effort to articulate with the related fields, probabilistic logical programming, machine learning, inductive programming, \emph{etc.} + +The equivalence relation from \cref{def:equiv.rel} identifies the $s \subseteq e$ and $e \subseteq s$ cases. Relations that distinguish such cases might enable better relations between the models and processes from the \aclp{SM}. + +The example from \cref{subsec:example.bayesian.networks} shows that the theory, methodology, and tools, from bayesian networks can be adapted to our approach. The connection with Markov Fields \cite{kindermann80} is left for future work. An example of a ``program selection'' application (as mentioned in \cref{item:program.selection}, \cref{sec:example.1}) is also left for future work. + +%\oldnote{under- over- estimate} +Related with the remark at the end of \cref{subsec:sbf.example}, on the tendency of $\hat{\theta}$ to under- or over- estimate $\theta$, notice that the error function in \eqref{eq:err.e.s} expresses only one of many possible ``distances'' between the empirical and prior distributions. Variations include normalizing this function by the size of $\fml{E}$ or using the \acl{KL} divergence. The key contribution of this function in this work is to find an optimal $\theta$. Moreover, further experiments, not included in this paper, with $\alpha = 0.0$, lead to $\hat{\theta} \approx \gamma$, \emph{i.e.}\ setting the prior noise to zero leads to full recovering $\theta$ from the observations. + +We decided to set the measure of inconsistent events to $0$ but, maybe, in some cases, we shouldn't. For example, since observations may be affected by noise, one can expect inconsistencies between the literals of an observation to occur. +% +% +% +\section*{Acknowledgements} +% +% +% +This work is supported by NOVALINCS (UIDB/04516/2020) with the financial support of FCT.IP. +The third author acknowledges the support of FCT - Funda\c{c}\~ao para a Ci\^{e}ncia e Tecnologia under the project UIDP/04674/2020, and the research center CIMA -- Centro de Investigação em Matemática e Aplicações. + +The authors grateful to Lígia Henriques-Rodrigues, Matthias Knorr and Ricardo Gonçalves for valuable comments on a preliminary version of this paper, and Alice Martins for contributions on software. +% +% +% +\printbibliography +% +% +% +\end{document} \ No newline at end of file diff --git a/text/paper_01/LLNCS/asp.bib b/text/paper_01/LLNCS/asp.bib new file mode 100644 index 0000000..6f1b94f --- /dev/null +++ b/text/paper_01/LLNCS/asp.bib @@ -0,0 +1,15 @@ + +@article{lifschitz_answer_2002, + title = {Answer set programming and plan generation}, + volume = {138}, + issn = {0004-3702}, + url = {https://www.sciencedirect.com/science/article/pii/S0004370202001868}, + doi = {https://doi.org/10.1016/S0004-3702(02)00186-8}, + abstract = {The idea of answer set programming is to represent a given computational problem by a logic program whose answer sets correspond to solutions, and then use an answer set solver, such as smodels or dlv, to find an answer set for this program. Applications of this method to planning are related to the line of research on the frame problem that started with the invention of formal nonmonotonic reasoning in 1980.}, + number = {1}, + journal = {Artificial Intelligence}, + author = {Lifschitz, Vladimir}, + year = {2002}, + keywords = {Answer sets, Default logic, Frame problem, Logic programming, Planning}, + pages = {39--54}, +} diff --git a/text/paper_01/LLNCS/example_annotASP2plainASP.txt b/text/paper_01/LLNCS/example_annotASP2plainASP.txt new file mode 100644 index 0000000..9b80363 --- /dev/null +++ b/text/paper_01/LLNCS/example_annotASP2plainASP.txt @@ -0,0 +1,14 @@ +a : 0.3. -a:0.7 +b : 0.5. -b:0.5 +c : 0.1. -c:0.9 + +t1 = { } => + +P(T = t1) = \sum_ + + +0.3:: a. +b ; c :- a. + +a ; -a. +b ; c :- a. diff --git a/text/paper_01/LLNCS/extended_abstract.txt b/text/paper_01/LLNCS/extended_abstract.txt new file mode 100644 index 0000000..cdc48c4 --- /dev/null +++ b/text/paper_01/LLNCS/extended_abstract.txt @@ -0,0 +1,7 @@ +We address the problem of extending probability from the total choices of an ASP program to the standard models, and from there to general events. +% +Our approach is algebraic in the sense that it relies on an equivalence relation over the set of events and uncertainty is expressed with variables and polynomial expressions. +% +We illustrate our methods with two examples, one of which shows a connection to bayesian networks. +% +Possible applications of the process described here include assigning a score to a logic program with respect to the empiric distribution of a given dataset, which in turn can be used by evolutionary algorithms searching for optimal models of that dataset. \ No newline at end of file diff --git a/text/paper_01/LLNCS/foundations.bib b/text/paper_01/LLNCS/foundations.bib new file mode 100644 index 0000000..f04a9c0 --- /dev/null +++ b/text/paper_01/LLNCS/foundations.bib @@ -0,0 +1,17 @@ + +@book{riguzzi2022foundations, + address = {New York}, + edition = {1}, + title = {Foundations of {Probabilistic} {Logic} {Programming}: {Languages}, {Semantics}, {Inference} and {Learning}}, + isbn = {978-1-00-333819-2}, + shorttitle = {Foundations of {Probabilistic} {Logic} {Programming}}, + url = {https://www.taylorfrancis.com/books/9781003338192}, + language = {en}, + urldate = {2023-03-01}, + publisher = {River Publishers}, + author = {Riguzzi, Fabrizio}, + month = sep, + year = {2022}, + doi = {10.1201/9781003338192}, + file = {Riguzzi - 2018 - Foundations of probabilistic logic programming la.pdf:/home/fc/sci/zotero/storage/X84FP7FV/Riguzzi - 2018 - Foundations of probabilistic logic programming la.pdf:application/pdf}, +} diff --git a/text/paper_01/LLNCS/llncs.cls b/text/paper_01/LLNCS/llncs.cls new file mode 100644 index 0000000..040102a --- /dev/null +++ b/text/paper_01/LLNCS/llncs.cls @@ -0,0 +1,1208 @@ +% LLNCS DOCUMENT CLASS -- version 2.19 (31-Mar-2014) +% Springer Verlag LaTeX2e support for Lecture Notes in Computer Science +% +%% +%% \CharacterTable +%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z +%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z +%% Digits \0\1\2\3\4\5\6\7\8\9 +%% Exclamation \! Double quote \" Hash (number) \# +%% Dollar \$ Percent \% Ampersand \& +%% Acute accent \' Left paren \( Right paren \) +%% Asterisk \* Plus \+ Comma \, +%% Minus \- Point \. Solidus \/ +%% Colon \: Semicolon \; Less than \< +%% Equals \= Greater than \> Question mark \? +%% Commercial at \@ Left bracket \[ Backslash \\ +%% Right bracket \] Circumflex \^ Underscore \_ +%% Grave accent \` Left brace \{ Vertical bar \| +%% Right brace \} Tilde \~} +%% +\NeedsTeXFormat{LaTeX2e}[1995/12/01] +\ProvidesClass{llncs}[2014/03/31 v2.19 +^^J LaTeX document class for Lecture Notes in Computer Science] +% Options +\let\if@envcntreset\iffalse +\DeclareOption{envcountreset}{\let\if@envcntreset\iftrue} +\DeclareOption{citeauthoryear}{\let\citeauthoryear=Y} +\DeclareOption{oribibl}{\let\oribibl=Y} +\let\if@custvec\iftrue +\DeclareOption{orivec}{\let\if@custvec\iffalse} +\let\if@envcntsame\iffalse +\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue} +\let\if@envcntsect\iffalse +\DeclareOption{envcountsect}{\let\if@envcntsect\iftrue} +\let\if@runhead\iffalse +\DeclareOption{runningheads}{\let\if@runhead\iftrue} + +\let\if@openright\iftrue +\let\if@openbib\iffalse +\DeclareOption{openbib}{\let\if@openbib\iftrue} + +% languages +\let\switcht@@therlang\relax +\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}} +\def\ds@francais{\def\switcht@@therlang{\switcht@francais}} + +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}} + +\ProcessOptions + +\LoadClass[twoside]{article} +\RequirePackage{multicol} % needed for the list of participants, index +\RequirePackage{aliascnt} + +\setlength{\textwidth}{12.2cm} +\setlength{\textheight}{19.3cm} +\renewcommand\@pnumwidth{2em} +\renewcommand\@tocrmarg{3.5em} +% +\def\@dottedtocline#1#2#3#4#5{% + \ifnum #1>\c@tocdepth \else + \vskip \z@ \@plus.2\p@ + {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by 0pt plus 2cm + \parfillskip -\rightskip \pretolerance=10000 + \parindent #2\relax\@afterindenttrue + \interlinepenalty\@M + \leavevmode + \@tempdima #3\relax + \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip + {#4}\nobreak + \leaders\hbox{$\m@th + \mkern \@dotsep mu\hbox{.}\mkern \@dotsep + mu$}\hfill + \nobreak + \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}% + \par}% + \fi} +% +\def\switcht@albion{% +\def\abstractname{Abstract.}% +\def\ackname{Acknowledgement.}% +\def\andname{and}% +\def\lastandname{\unskip, and}% +\def\appendixname{Appendix}% +\def\chaptername{Chapter}% +\def\claimname{Claim}% +\def\conjecturename{Conjecture}% +\def\contentsname{Table of Contents}% +\def\corollaryname{Corollary}% +\def\definitionname{Definition}% +\def\examplename{Example}% +\def\exercisename{Exercise}% +\def\figurename{Fig.}% +\def\keywordname{{\bf Keywords:}}% +\def\indexname{Index}% +\def\lemmaname{Lemma}% +\def\contriblistname{List of Contributors}% +\def\listfigurename{List of Figures}% +\def\listtablename{List of Tables}% +\def\mailname{{\it Correspondence to\/}:}% +\def\noteaddname{Note added in proof}% +\def\notename{Note}% +\def\partname{Part}% +\def\problemname{Problem}% +\def\proofname{Proof}% +\def\propertyname{Property}% +\def\propositionname{Proposition}% +\def\questionname{Question}% +\def\remarkname{Remark}% +\def\seename{see}% +\def\solutionname{Solution}% +\def\subclassname{{\it Subject Classifications\/}:}% +\def\tablename{Table}% +\def\theoremname{Theorem}} +\switcht@albion +% Names of theorem like environments are already defined +% but must be translated if another language is chosen +% +% French section +\def\switcht@francais{%\typeout{On parle francais.}% + \def\abstractname{R\'esum\'e.}% + \def\ackname{Remerciements.}% + \def\andname{et}% + \def\lastandname{ et}% + \def\appendixname{Appendice}% + \def\chaptername{Chapitre}% + \def\claimname{Pr\'etention}% + \def\conjecturename{Hypoth\`ese}% + \def\contentsname{Table des mati\`eres}% + \def\corollaryname{Corollaire}% + \def\definitionname{D\'efinition}% + \def\examplename{Exemple}% + \def\exercisename{Exercice}% + \def\figurename{Fig.}% + \def\keywordname{{\bf Mots-cl\'e:}}% + \def\indexname{Index}% + \def\lemmaname{Lemme}% + \def\contriblistname{Liste des contributeurs}% + \def\listfigurename{Liste des figures}% + \def\listtablename{Liste des tables}% + \def\mailname{{\it Correspondence to\/}:}% + \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}% + \def\notename{Remarque}% + \def\partname{Partie}% + \def\problemname{Probl\`eme}% + \def\proofname{Preuve}% + \def\propertyname{Caract\'eristique}% +%\def\propositionname{Proposition}% + \def\questionname{Question}% + \def\remarkname{Remarque}% + \def\seename{voir}% + \def\solutionname{Solution}% + \def\subclassname{{\it Subject Classifications\/}:}% + \def\tablename{Tableau}% + \def\theoremname{Th\'eor\`eme}% +} +% +% German section +\def\switcht@deutsch{%\typeout{Man spricht deutsch.}% + \def\abstractname{Zusammenfassung.}% + \def\ackname{Danksagung.}% + \def\andname{und}% + \def\lastandname{ und}% + \def\appendixname{Anhang}% + \def\chaptername{Kapitel}% + \def\claimname{Behauptung}% + \def\conjecturename{Hypothese}% + \def\contentsname{Inhaltsverzeichnis}% + \def\corollaryname{Korollar}% +%\def\definitionname{Definition}% + \def\examplename{Beispiel}% + \def\exercisename{\"Ubung}% + \def\figurename{Abb.}% + \def\keywordname{{\bf Schl\"usselw\"orter:}}% + \def\indexname{Index}% +%\def\lemmaname{Lemma}% + \def\contriblistname{Mitarbeiter}% + \def\listfigurename{Abbildungsverzeichnis}% + \def\listtablename{Tabellenverzeichnis}% + \def\mailname{{\it Correspondence to\/}:}% + \def\noteaddname{Nachtrag}% + \def\notename{Anmerkung}% + \def\partname{Teil}% +%\def\problemname{Problem}% + \def\proofname{Beweis}% + \def\propertyname{Eigenschaft}% +%\def\propositionname{Proposition}% + \def\questionname{Frage}% + \def\remarkname{Anmerkung}% + \def\seename{siehe}% + \def\solutionname{L\"osung}% + \def\subclassname{{\it Subject Classifications\/}:}% + \def\tablename{Tabelle}% +%\def\theoremname{Theorem}% +} + +% Ragged bottom for the actual page +\def\thisbottomragged{\def\@textbottom{\vskip\z@ plus.0001fil +\global\let\@textbottom\relax}} + +\renewcommand\small{% + \@setfontsize\small\@ixpt{11}% + \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus2\p@ + \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \parsep 0\p@ \@plus1\p@ \@minus\p@ + \topsep 8\p@ \@plus2\p@ \@minus4\p@ + \itemsep0\p@}% + \belowdisplayskip \abovedisplayskip +} + +\frenchspacing +\widowpenalty=10000 +\clubpenalty=10000 + +\setlength\oddsidemargin {63\p@} +\setlength\evensidemargin {63\p@} +\setlength\marginparwidth {90\p@} + +\setlength\headsep {16\p@} + +\setlength\footnotesep{7.7\p@} +\setlength\textfloatsep{8mm\@plus 2\p@ \@minus 4\p@} +\setlength\intextsep {8mm\@plus 2\p@ \@minus 2\p@} + +\setcounter{secnumdepth}{2} + +\newcounter {chapter} +\renewcommand\thechapter {\@arabic\c@chapter} + +\newif\if@mainmatter \@mainmattertrue +\newcommand\frontmatter{\cleardoublepage + \@mainmatterfalse\pagenumbering{Roman}} +\newcommand\mainmatter{\cleardoublepage + \@mainmattertrue\pagenumbering{arabic}} +\newcommand\backmatter{\if@openright\cleardoublepage\else\clearpage\fi + \@mainmatterfalse} + +\renewcommand\part{\cleardoublepage + \thispagestyle{empty}% + \if@twocolumn + \onecolumn + \@tempswatrue + \else + \@tempswafalse + \fi + \null\vfil + \secdef\@part\@spart} + +\def\@part[#1]#2{% + \ifnum \c@secnumdepth >-2\relax + \refstepcounter{part}% + \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}% + \else + \addcontentsline{toc}{part}{#1}% + \fi + \markboth{}{}% + {\centering + \interlinepenalty \@M + \normalfont + \ifnum \c@secnumdepth >-2\relax + \huge\bfseries \partname~\thepart + \par + \vskip 20\p@ + \fi + \Huge \bfseries #2\par}% + \@endpart} +\def\@spart#1{% + {\centering + \interlinepenalty \@M + \normalfont + \Huge \bfseries #1\par}% + \@endpart} +\def\@endpart{\vfil\newpage + \if@twoside + \null + \thispagestyle{empty}% + \newpage + \fi + \if@tempswa + \twocolumn + \fi} + +\newcommand\chapter{\clearpage + \thispagestyle{empty}% + \global\@topnum\z@ + \@afterindentfalse + \secdef\@chapter\@schapter} +\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne + \if@mainmatter + \refstepcounter{chapter}% + \typeout{\@chapapp\space\thechapter.}% + \addcontentsline{toc}{chapter}% + {\protect\numberline{\thechapter}#1}% + \else + \addcontentsline{toc}{chapter}{#1}% + \fi + \else + \addcontentsline{toc}{chapter}{#1}% + \fi + \chaptermark{#1}% + \addtocontents{lof}{\protect\addvspace{10\p@}}% + \addtocontents{lot}{\protect\addvspace{10\p@}}% + \if@twocolumn + \@topnewpage[\@makechapterhead{#2}]% + \else + \@makechapterhead{#2}% + \@afterheading + \fi} +\def\@makechapterhead#1{% +% \vspace*{50\p@}% + {\centering + \ifnum \c@secnumdepth >\m@ne + \if@mainmatter + \large\bfseries \@chapapp{} \thechapter + \par\nobreak + \vskip 20\p@ + \fi + \fi + \interlinepenalty\@M + \Large \bfseries #1\par\nobreak + \vskip 40\p@ + }} +\def\@schapter#1{\if@twocolumn + \@topnewpage[\@makeschapterhead{#1}]% + \else + \@makeschapterhead{#1}% + \@afterheading + \fi} +\def\@makeschapterhead#1{% +% \vspace*{50\p@}% + {\centering + \normalfont + \interlinepenalty\@M + \Large \bfseries #1\par\nobreak + \vskip 40\p@ + }} + +\renewcommand\section{\@startsection{section}{1}{\z@}% + {-18\p@ \@plus -4\p@ \@minus -4\p@}% + {12\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\large\bfseries\boldmath + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-18\p@ \@plus -4\p@ \@minus -4\p@}% + {8\p@ \@plus 4\p@ \@minus 4\p@}% + {\normalfont\normalsize\bfseries\boldmath + \rightskip=\z@ \@plus 8em\pretolerance=10000 }} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-18\p@ \@plus -4\p@ \@minus -4\p@}% + {-0.5em \@plus -0.22em \@minus -0.1em}% + {\normalfont\normalsize\bfseries\boldmath}} +\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% + {-12\p@ \@plus -4\p@ \@minus -4\p@}% + {-0.5em \@plus -0.22em \@minus -0.1em}% + {\normalfont\normalsize\itshape}} +\renewcommand\subparagraph[1]{\typeout{LLNCS warning: You should not use + \string\subparagraph\space with this class}\vskip0.5cm +You should not use \verb|\subparagraph| with this class.\vskip0.5cm} + +\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00} +\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01} +\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02} +\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03} +\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04} +\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05} +\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06} +\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07} +\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08} +\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09} +\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A} + +\let\footnotesize\small + +\if@custvec +\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}} +{\mbox{\boldmath$\textstyle#1$}} +{\mbox{\boldmath$\scriptstyle#1$}} +{\mbox{\boldmath$\scriptscriptstyle#1$}}} +\fi + +\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}} +\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil +\penalty50\hskip1em\null\nobreak\hfil\squareforqed +\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi} + +\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip +\halign{\hfil +$\displaystyle##$\hfil\cr\gets\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets +\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets +\cr\to\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +\gets\cr\to\cr}}}}} +\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil +$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr +\noalign{\vskip1.2pt}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr +\noalign{\vskip1pt}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +<\cr +\noalign{\vskip0.9pt}=\cr}}}}} +\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil +$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr +\noalign{\vskip1.2pt}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr +\noalign{\vskip1pt}=\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +>\cr +\noalign{\vskip0.9pt}=\cr}}}}} +\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip +\halign{\hfil +$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr +>\cr\noalign{\vskip-1pt}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr +>\cr\noalign{\vskip-0.8pt}<\cr}}} +{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr +>\cr\noalign{\vskip-0.3pt}<\cr}}}}} +\def\bbbr{{\rm I\!R}} %reelle Zahlen +\def\bbbm{{\rm I\!M}} +\def\bbbn{{\rm I\!N}} %natuerliche Zahlen +\def\bbbf{{\rm I\!F}} +\def\bbbh{{\rm I\!H}} +\def\bbbk{{\rm I\!K}} +\def\bbbp{{\rm I\!P}} +\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} +{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} +\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox +to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox +to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox +to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox +to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}} +\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm +Q$}\hbox{\raise +0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}} +\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm +T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox +to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox +to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox +to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}} +\def\bbbs{{\mathchoice +{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox +to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox +to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox +to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox +to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox +to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox +to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox +to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox +to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}} +\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}} +{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}} +{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}} +{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}} + +\let\ts\, + +\setlength\leftmargini {17\p@} +\setlength\leftmargin {\leftmargini} +\setlength\leftmarginii {\leftmargini} +\setlength\leftmarginiii {\leftmargini} +\setlength\leftmarginiv {\leftmargini} +\setlength \labelsep {.5em} +\setlength \labelwidth{\leftmargini} +\addtolength\labelwidth{-\labelsep} + +\def\@listI{\leftmargin\leftmargini + \parsep 0\p@ \@plus1\p@ \@minus\p@ + \topsep 8\p@ \@plus2\p@ \@minus4\p@ + \itemsep0\p@} +\let\@listi\@listI +\@listi +\def\@listii {\leftmargin\leftmarginii + \labelwidth\leftmarginii + \advance\labelwidth-\labelsep + \topsep 0\p@ \@plus2\p@ \@minus\p@} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii + \advance\labelwidth-\labelsep + \topsep 0\p@ \@plus\p@\@minus\p@ + \parsep \z@ + \partopsep \p@ \@plus\z@ \@minus\p@} + +\renewcommand\labelitemi{\normalfont\bfseries --} +\renewcommand\labelitemii{$\m@th\bullet$} + +\setlength\arraycolsep{1.4\p@} +\setlength\tabcolsep{1.4\p@} + +\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}% + {{\contentsname}}} + \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}} + \def\lastand{\ifnum\value{auco}=2\relax + \unskip{} \andname\ + \else + \unskip \lastandname\ + \fi}% + \def\and{\stepcounter{@auth}\relax + \ifnum\value{@auth}=\value{auco}% + \lastand + \else + \unskip, + \fi}% + \@starttoc{toc}\if@restonecol\twocolumn\fi} + +\def\l@part#1#2{\addpenalty{\@secpenalty}% + \addvspace{2em plus\p@}% % space above part line + \begingroup + \parindent \z@ + \rightskip \z@ plus 5em + \hrule\vskip5pt + \large % same size as for a contribution heading + \bfseries\boldmath % set line in boldface + \leavevmode % TeX command to enter horizontal mode. + #1\par + \vskip5pt + \hrule + \vskip1pt + \nobreak % Never break after part entry + \endgroup} + +\def\@dotsep{2} + +\let\phantomsection=\relax + +\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else +{}\fi} + +\def\addnumcontentsmark#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline + {\thechapter}#3}{\thepage}\hyperhrefextend}}% +\def\addcontentsmark#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}% +\def\addcontentsmarkwop#1#2#3{% +\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}% + +\def\@adcmk[#1]{\ifcase #1 \or +\def\@gtempa{\addnumcontentsmark}% + \or \def\@gtempa{\addcontentsmark}% + \or \def\@gtempa{\addcontentsmarkwop}% + \fi\@gtempa{toc}{chapter}% +} +\def\addtocmark{% +\phantomsection +\@ifnextchar[{\@adcmk}{\@adcmk[3]}% +} + +\def\l@chapter#1#2{\addpenalty{-\@highpenalty} + \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup + \parindent \z@ \rightskip \@tocrmarg + \advance\rightskip by 0pt plus 2cm + \parfillskip -\rightskip \pretolerance=10000 + \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip + {\large\bfseries\boldmath#1}\ifx0#2\hfil\null + \else + \nobreak + \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern + \@dotsep mu$}\hfill + \nobreak\hbox to\@pnumwidth{\hss #2}% + \fi\par + \penalty\@highpenalty \endgroup} + +\def\l@title#1#2{\addpenalty{-\@highpenalty} + \addvspace{8pt plus 1pt} + \@tempdima \z@ + \begingroup + \parindent \z@ \rightskip \@tocrmarg + \advance\rightskip by 0pt plus 2cm + \parfillskip -\rightskip \pretolerance=10000 + \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip + #1\nobreak + \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern + \@dotsep mu$}\hfill + \nobreak\hbox to\@pnumwidth{\hss #2}\par + \penalty\@highpenalty \endgroup} + +\def\l@author#1#2{\addpenalty{\@highpenalty} + \@tempdima=15\p@ %\z@ + \begingroup + \parindent \z@ \rightskip \@tocrmarg + \advance\rightskip by 0pt plus 2cm + \pretolerance=10000 + \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip + \textit{#1}\par + \penalty\@highpenalty \endgroup} + +\setcounter{tocdepth}{0} +\newdimen\tocchpnum +\newdimen\tocsecnum +\newdimen\tocsectotal +\newdimen\tocsubsecnum +\newdimen\tocsubsectotal +\newdimen\tocsubsubsecnum +\newdimen\tocsubsubsectotal +\newdimen\tocparanum +\newdimen\tocparatotal +\newdimen\tocsubparanum +\tocchpnum=\z@ % no chapter numbers +\tocsecnum=15\p@ % section 88. plus 2.222pt +\tocsubsecnum=23\p@ % subsection 88.8 plus 2.222pt +\tocsubsubsecnum=27\p@ % subsubsection 88.8.8 plus 1.444pt +\tocparanum=35\p@ % paragraph 88.8.8.8 plus 1.666pt +\tocsubparanum=43\p@ % subparagraph 88.8.8.8.8 plus 1.888pt +\def\calctocindent{% +\tocsectotal=\tocchpnum +\advance\tocsectotal by\tocsecnum +\tocsubsectotal=\tocsectotal +\advance\tocsubsectotal by\tocsubsecnum +\tocsubsubsectotal=\tocsubsectotal +\advance\tocsubsubsectotal by\tocsubsubsecnum +\tocparatotal=\tocsubsubsectotal +\advance\tocparatotal by\tocparanum} +\calctocindent + +\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}} +\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}} +\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}} +\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}} +\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}} + +\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn + \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}} + \@starttoc{lof}\if@restonecol\twocolumn\fi} +\def\l@figure{\@dottedtocline{1}{0em}{1.5em}} + +\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn + \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}} + \@starttoc{lot}\if@restonecol\twocolumn\fi} +\let\l@table\l@figure + +\renewcommand\listoffigures{% + \section*{\listfigurename + \@mkboth{\listfigurename}{\listfigurename}}% + \@starttoc{lof}% + } + +\renewcommand\listoftables{% + \section*{\listtablename + \@mkboth{\listtablename}{\listtablename}}% + \@starttoc{lot}% + } + +\ifx\oribibl\undefined +\ifx\citeauthoryear\undefined +\renewenvironment{thebibliography}[1] + {\section*{\refname} + \def\@biblabel##1{##1.} + \small + \list{\@biblabel{\@arabic\c@enumiv}}% + {\settowidth\labelwidth{\@biblabel{#1}}% + \leftmargin\labelwidth + \advance\leftmargin\labelsep + \if@openbib + \advance\leftmargin\bibindent + \itemindent -\bibindent + \listparindent \itemindent + \parsep \z@ + \fi + \usecounter{enumiv}% + \let\p@enumiv\@empty + \renewcommand\theenumiv{\@arabic\c@enumiv}}% + \if@openbib + \renewcommand\newblock{\par}% + \else + \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}% + \fi + \sloppy\clubpenalty4000\widowpenalty4000% + \sfcode`\.=\@m} + {\def\@noitemerr + {\@latex@warning{Empty `thebibliography' environment}}% + \endlist} +\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw + {\let\protect\noexpand\immediate + \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces} +\newcount\@tempcntc +\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi + \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do + {\@ifundefined + {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries + ?}\@warning + {Citation `\@citeb' on page \thepage \space undefined}}% + {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}% + \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne + \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}% + \else + \advance\@tempcntb\@ne + \ifnum\@tempcntb=\@tempcntc + \else\advance\@tempcntb\m@ne\@citeo + \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}} +\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else + \@citea\def\@citea{,\,\hskip\z@skip}% + \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else + {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else + \def\@citea{--}\fi + \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi} +\else +\renewenvironment{thebibliography}[1] + {\section*{\refname} + \small + \list{}% + {\settowidth\labelwidth{}% + \leftmargin\parindent + \itemindent=-\parindent + \labelsep=\z@ + \if@openbib + \advance\leftmargin\bibindent + \itemindent -\bibindent + \listparindent \itemindent + \parsep \z@ + \fi + \usecounter{enumiv}% + \let\p@enumiv\@empty + \renewcommand\theenumiv{}}% + \if@openbib + \renewcommand\newblock{\par}% + \else + \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}% + \fi + \sloppy\clubpenalty4000\widowpenalty4000% + \sfcode`\.=\@m} + {\def\@noitemerr + {\@latex@warning{Empty `thebibliography' environment}}% + \endlist} + \def\@cite#1{#1}% + \def\@lbibitem[#1]#2{\item[]\if@filesw + {\def\protect##1{\string ##1\space}\immediate + \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces} + \fi +\else +\@cons\@openbib@code{\noexpand\small} +\fi + +\def\idxquad{\hskip 10\p@}% space that divides entry from number + +\def\@idxitem{\par\hangindent 10\p@} + +\def\subitem{\par\setbox0=\hbox{--\enspace}% second order + \noindent\hangindent\wd0\box0}% index entry + +\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third + \noindent\hangindent\wd0\box0}% order index entry + +\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax} + +\renewenvironment{theindex} + {\@mkboth{\indexname}{\indexname}% + \thispagestyle{empty}\parindent\z@ + \parskip\z@ \@plus .3\p@\relax + \let\item\par + \def\,{\relax\ifmmode\mskip\thinmuskip + \else\hskip0.2em\ignorespaces\fi}% + \normalfont\small + \begin{multicols}{2}[\@makeschapterhead{\indexname}]% + } + {\end{multicols}} + +\renewcommand\footnoterule{% + \kern-3\p@ + \hrule\@width 2truecm + \kern2.6\p@} + \newdimen\fnindent + \fnindent1em +\long\def\@makefntext#1{% + \parindent \fnindent% + \leftskip \fnindent% + \noindent + \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1} + +\long\def\@makecaption#1#2{% + \small + \vskip\abovecaptionskip + \sbox\@tempboxa{{\bfseries #1.} #2}% + \ifdim \wd\@tempboxa >\hsize + {\bfseries #1.} #2\par + \else + \global \@minipagefalse + \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}% + \fi + \vskip\belowcaptionskip} + +\def\fps@figure{htbp} +\def\fnum@figure{\figurename\thinspace\thefigure} +\def \@floatboxreset {% + \reset@font + \small + \@setnobreak + \@setminipage +} +\def\fps@table{htbp} +\def\fnum@table{\tablename~\thetable} +\renewenvironment{table} + {\setlength\abovecaptionskip{0\p@}% + \setlength\belowcaptionskip{10\p@}% + \@float{table}} + {\end@float} +\renewenvironment{table*} + {\setlength\abovecaptionskip{0\p@}% + \setlength\belowcaptionskip{10\p@}% + \@dblfloat{table}} + {\end@dblfloat} + +\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname + ext@#1\endcsname}{#1}{\protect\numberline{\csname + the#1\endcsname}{\ignorespaces #2}}\begingroup + \@parboxrestore + \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par + \endgroup} + +% LaTeX does not provide a command to enter the authors institute +% addresses. The \institute command is defined here. + +\newcounter{@inst} +\newcounter{@auth} +\newcounter{auco} +\newdimen\instindent +\newbox\authrun +\newtoks\authorrunning +\newtoks\tocauthor +\newbox\titrun +\newtoks\titlerunning +\newtoks\toctitle + +\def\clearheadinfo{\gdef\@author{No Author Given}% + \gdef\@title{No Title Given}% + \gdef\@subtitle{}% + \gdef\@institute{No Institute Given}% + \gdef\@thanks{}% + \global\titlerunning={}\global\authorrunning={}% + \global\toctitle={}\global\tocauthor={}} + +\def\institute#1{\gdef\@institute{#1}} + +\def\institutename{\par + \begingroup + \parskip=\z@ + \parindent=\z@ + \setcounter{@inst}{1}% + \def\and{\par\stepcounter{@inst}% + \noindent$^{\the@inst}$\enspace\ignorespaces}% + \setbox0=\vbox{\def\thanks##1{}\@institute}% + \ifnum\c@@inst=1\relax + \gdef\fnnstart{0}% + \else + \xdef\fnnstart{\c@@inst}% + \setcounter{@inst}{1}% + \noindent$^{\the@inst}$\enspace + \fi + \ignorespaces + \@institute\par + \endgroup} + +\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or + {\star\star\star}\or \dagger\or \ddagger\or + \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger + \or \ddagger\ddagger \else\@ctrerr\fi}} + +\def\inst#1{\unskip$^{#1}$} +\def\fnmsep{\unskip$^,$} +\def\email#1{{\tt#1}} +\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}% +\@ifpackageloaded{babel}{% +\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}% +\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}% +\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}% +\@ifundefined{extrasngerman}{}{\addto\extrasngerman{\switcht@deutsch}}% +}{\switcht@@therlang}% +\providecommand{\keywords}[1]{\par\addvspace\baselineskip +\noindent\keywordname\enspace\ignorespaces#1}% +} +\def\homedir{\~{ }} + +\def\subtitle#1{\gdef\@subtitle{#1}} +\clearheadinfo +% +%%% to avoid hyperref warnings +\providecommand*{\toclevel@author}{999} +%%% to make title-entry parent of section-entries +\providecommand*{\toclevel@title}{0} +% +\renewcommand\maketitle{\newpage +\phantomsection + \refstepcounter{chapter}% + \stepcounter{section}% + \setcounter{section}{0}% + \setcounter{subsection}{0}% + \setcounter{figure}{0} + \setcounter{table}{0} + \setcounter{equation}{0} + \setcounter{footnote}{0}% + \begingroup + \parindent=\z@ + \renewcommand\thefootnote{\@fnsymbol\c@footnote}% + \if@twocolumn + \ifnum \col@number=\@ne + \@maketitle + \else + \twocolumn[\@maketitle]% + \fi + \else + \newpage + \global\@topnum\z@ % Prevents figures from going at top of page. + \@maketitle + \fi + \thispagestyle{empty}\@thanks +% + \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}% + \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}% + \instindent=\hsize + \advance\instindent by-\headlineindent + \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else + \addcontentsline{toc}{title}{\the\toctitle}\fi + \if@runhead + \if!\the\titlerunning!\else + \edef\@title{\the\titlerunning}% + \fi + \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}% + \ifdim\wd\titrun>\instindent + \typeout{Title too long for running head. Please supply}% + \typeout{a shorter form with \string\titlerunning\space prior to + \string\maketitle}% + \global\setbox\titrun=\hbox{\small\rm + Title Suppressed Due to Excessive Length}% + \fi + \xdef\@title{\copy\titrun}% + \fi +% + \if!\the\tocauthor!\relax + {\def\and{\noexpand\protect\noexpand\and}% + \protected@xdef\toc@uthor{\@author}}% + \else + \def\\{\noexpand\protect\noexpand\newline}% + \protected@xdef\scratch{\the\tocauthor}% + \protected@xdef\toc@uthor{\scratch}% + \fi + \addtocontents{toc}{\noexpand\protect\noexpand\authcount{\the\c@auco}}% + \addcontentsline{toc}{author}{\toc@uthor}% + \if@runhead + \if!\the\authorrunning! + \value{@inst}=\value{@auth}% + \setcounter{@auth}{1}% + \else + \edef\@author{\the\authorrunning}% + \fi + \global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}% + \ifdim\wd\authrun>\instindent + \typeout{Names of authors too long for running head. Please supply}% + \typeout{a shorter form with \string\authorrunning\space prior to + \string\maketitle}% + \global\setbox\authrun=\hbox{\small\rm + Authors Suppressed Due to Excessive Length}% + \fi + \xdef\@author{\copy\authrun}% + \markboth{\@author}{\@title}% + \fi + \endgroup + \setcounter{footnote}{\fnnstart}% + \clearheadinfo} +% +\def\@maketitle{\newpage + \markboth{}{}% + \def\lastand{\ifnum\value{@inst}=2\relax + \unskip{} \andname\ + \else + \unskip \lastandname\ + \fi}% + \def\and{\stepcounter{@auth}\relax + \ifnum\value{@auth}=\value{@inst}% + \lastand + \else + \unskip, + \fi}% + \begin{center}% + \let\newline\\ + {\Large \bfseries\boldmath + \pretolerance=10000 + \@title \par}\vskip .8cm +\if!\@subtitle!\else {\large \bfseries\boldmath + \vskip -.65cm + \pretolerance=10000 + \@subtitle \par}\vskip .8cm\fi + \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}% + \def\thanks##1{}\@author}% + \global\value{@inst}=\value{@auth}% + \global\value{auco}=\value{@auth}% + \setcounter{@auth}{1}% +{\lineskip .5em +\noindent\ignorespaces +\@author\vskip.35cm} + {\small\institutename} + \end{center}% + } + +% definition of the "\spnewtheorem" command. +% +% Usage: +% +% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font} +% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font} +% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font} +% +% New is "cap_font" and "body_font". It stands for +% fontdefinition of the caption and the text itself. +% +% "\spnewtheorem*" gives a theorem without number. +% +% A defined spnewthoerem environment is used as described +% by Lamport. +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\def\@thmcountersep{} +\def\@thmcounterend{.} + +\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}} + +% definition of \spnewtheorem with number + +\def\@spnthm#1#2{% + \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}} +\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}} + +\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}\@addtoreset{#1}{#3}% + \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand + \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname + {\@definecounter{#1}% + \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@spothm#1[#2]#3#4#5{% + \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}% + {\expandafter\@ifdefinable\csname #1\endcsname + {\newaliascnt{#1}{#2}% + \expandafter\xdef\csname #1name\endcsname{#3}% + \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}% + \global\@namedef{end#1}{\@endtheorem}}}} + +\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@ +\refstepcounter{#1}% +\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}} + +\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}% + \ignorespaces} + +\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname + the#1\endcsname}{#5}{#3}{#4}\ignorespaces} + +\def\@spbegintheorem#1#2#3#4{\trivlist + \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4} + +\def\@spopargbegintheorem#1#2#3#4#5{\trivlist + \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5} + +% definition of \spnewtheorem* without number + +\def\@sthm#1#2{\@Ynthm{#1}{#2}} + +\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname + {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}% + \expandafter\xdef\csname #1name\endcsname{#2}% + \global\@namedef{end#1}{\@endtheorem}}} + +\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@ +\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}} + +\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces} + +\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1} + {#4}{#2}{#3}\ignorespaces} + +\def\@Begintheorem#1#2#3{#3\trivlist + \item[\hskip\labelsep{#2#1\@thmcounterend}]} + +\def\@Opargbegintheorem#1#2#3#4{#4\trivlist + \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }} + +\if@envcntsect + \def\@thmcountersep{.} + \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape} +\else + \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape} + \if@envcntreset + \@addtoreset{theorem}{section} + \else + \@addtoreset{theorem}{chapter} + \fi +\fi + +%definition of divers theorem environments +\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily} +\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily} +\if@envcntsame % alle Umgebungen wie Theorem. + \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}} +\else % alle Umgebungen mit eigenem Zaehler + \if@envcntsect % mit section numeriert + \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}} + \else % nicht mit section numeriert + \if@envcntreset + \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4} + \@addtoreset{#1}{section}} + \else + \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4} + \@addtoreset{#1}{chapter}}% + \fi + \fi +\fi +\spn@wtheorem{case}{Case}{\itshape}{\rmfamily} +\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily} +\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape} +\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape} +\spn@wtheorem{example}{Example}{\itshape}{\rmfamily} +\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily} +\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape} +\spn@wtheorem{note}{Note}{\itshape}{\rmfamily} +\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily} +\spn@wtheorem{property}{Property}{\itshape}{\rmfamily} +\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape} +\spn@wtheorem{question}{Question}{\itshape}{\rmfamily} +\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily} +\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily} + +\def\@takefromreset#1#2{% + \def\@tempa{#1}% + \let\@tempd\@elt + \def\@elt##1{% + \def\@tempb{##1}% + \ifx\@tempa\@tempb\else + \@addtoreset{##1}{#2}% + \fi}% + \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname + \expandafter\def\csname cl@#2\endcsname{}% + \@tempc + \let\@elt\@tempd} + +\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist + \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5} + \def\@Opargbegintheorem##1##2##3##4{##4\trivlist + \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }} + } + +\renewenvironment{abstract}{% + \list{}{\advance\topsep by0.35cm\relax\small + \leftmargin=1cm + \labelwidth=\z@ + \listparindent=\z@ + \itemindent\listparindent + \rightmargin\leftmargin}\item[\hskip\labelsep + \bfseries\abstractname]} + {\endlist} + +\newdimen\headlineindent % dimension for space between +\headlineindent=1.166cm % number and text of headings. + +\def\ps@headings{\let\@mkboth\@gobbletwo + \let\@oddfoot\@empty\let\@evenfoot\@empty + \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}% + \leftmark\hfil} + \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}% + \llap{\thepage}} + \def\chaptermark##1{}% + \def\sectionmark##1{}% + \def\subsectionmark##1{}} + +\def\ps@titlepage{\let\@mkboth\@gobbletwo + \let\@oddfoot\@empty\let\@evenfoot\@empty + \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}% + \hfil} + \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}% + \llap{\thepage}} + \def\chaptermark##1{}% + \def\sectionmark##1{}% + \def\subsectionmark##1{}} + +\if@runhead\ps@headings\else +\ps@empty\fi + +\setlength\arraycolsep{1.4\p@} +\setlength\tabcolsep{1.4\p@} + +\endinput +%end of file llncs.cls diff --git a/text/paper_01/LLNCS/zugz-paper_01.code-workspace b/text/paper_01/LLNCS/zugz-paper_01.code-workspace new file mode 100644 index 0000000..975da5d --- /dev/null +++ b/text/paper_01/LLNCS/zugz-paper_01.code-workspace @@ -0,0 +1,11 @@ +{ + "folders": [ + { + "path": "." + }, + { + "path": "../.." + } + ], + "settings": {} +} \ No newline at end of file diff --git a/text/paper_01/LLNCS/zugzwang.bib b/text/paper_01/LLNCS/zugzwang.bib new file mode 100644 index 0000000..ea300c8 --- /dev/null +++ b/text/paper_01/LLNCS/zugzwang.bib @@ -0,0 +1,251 @@ +@article {Teugels90, + AUTHOR = {Teugels, Jozef L.}, + TITLE = {Some representations of the multivariate {B}ernoulli and + binomial distributions}, + JOURNAL = {J. Multivariate Anal.}, + FJOURNAL = {Journal of Multivariate Analysis}, + VOLUME = {32}, + YEAR = {1990}, + NUMBER = {2}, + PAGES = {256--268}, + ISSN = {0047-259X,1095-7243}, + MRCLASS = {62H17 (62H20)}, + MRNUMBER = {1046768}, +MRREVIEWER = {Friedrich\ Liese}, + DOI = {10.1016/0047-259X(90)90084-U}, + URL = {https://doi.org/10.1016/0047-259X(90)90084-U}, +} + + +@book{kindermann80, + author = {Kindermann, Ross and Snell, J. Laurie}, + title = {Markov random fields and their applications}, + series = {Contemporary Mathematics}, + volume = {1}, + publisher = {American Mathematical Society, Providence, RI}, + year = {1980}, + pages = {ix+142}, + isbn = {0-8218-5001-6}, + mrclass = {60K35 (60G60 82A42 82A67 94A05)}, + mrnumber = {620955}, + mrreviewer = {J.\ Theodore\ Cox} +} + +@article{geman84, + author = {Geman, Stuart and Geman, Donald}, + journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, + title = {Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images}, + year = {1984}, + volume = {PAMI-6}, + number = {6}, + pages = {721-741}, + doi = {10.1109/TPAMI.1984.4767596} +} + + +@book{Judea88, + author = {Pearl, Judea}, + title = {Probabilistic reasoning in intelligent systems: networks of + plausible inference}, + series = {The Morgan Kaufmann Series in Representation and Reasoning}, + publisher = {Morgan Kaufmann, San Mateo, CA}, + year = {1988}, + pages = {xx+552}, + isbn = {0-934613-73-7}, + mrclass = {68-02 (68T01 92A25)}, + mrnumber = {965765}, + mrreviewer = {V. Yu. Trakhtman} +} + + @article{sympy, + title = {SymPy: symbolic computing in Python}, + author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony}, + year = 2017, + month = jan, + keywords = {Python, Computer algebra system, Symbolics}, + abstract = { + SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy. + }, + volume = 3, + pages = {e103}, + journal = {PeerJ Computer Science}, + issn = {2376-5992}, + url = {https://doi.org/10.7717/peerj-cs.103}, + doi = {10.7717/peerj-cs.103} +} + +@inproceedings{verreet2022inference, + title = {Inference and learning with model uncertainty in probabilistic logic programs}, + author = {Verreet, Victor and Derkinderen, Vincent and Dos Martires, Pedro Zuidberg and De Raedt, Luc}, + booktitle = {Proceedings of the AAAI Conference on Artificial Intelligence}, + volume = {36}, + number = {9}, + pages = {10060--10069}, + year = {2022} +} + +@article{cropper2022inductive, + title = {Inductive logic programming at 30}, + author = {Cropper, Andrew and Duman{\v{c}}i{\'c}, Sebastijan and Evans, Richard and Muggleton, Stephen H}, + journal = {Machine Learning}, + volume = {111}, + number = {1}, + pages = {147--172}, + year = {2022}, + publisher = {Springer} +} + +@article{cozman2020joy, + title = {The joy of probabilistic answer set programming: semantics, complexity, expressivity, inference}, + author = {Cozman, Fabio Gagliardi and Mau{\'a}, Denis Deratani}, + journal = {International Journal of Approximate Reasoning}, + volume = {125}, + pages = {218--239}, + year = {2020}, + publisher = {Elsevier} +} + +@article{gebser2012answer, + title = {Answer set solving in practice}, + author = {Gebser, Martin and Kaminski, Roland and Kaufmann, Benjamin and Schaub, Torsten}, + journal = {Synthesis lectures on artificial intelligence and machine learning}, + volume = {6}, + number = {3}, + pages = {1--238}, + year = {2012}, + publisher = {Morgan \& Claypool Publishers} +} + + +@article{bezanson2017julia, + author = {Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and Shah, Viral B.}, + title = {Julia: A Fresh Approach to Numerical Computing}, + journal = {SIAM Review}, + volume = {59}, + number = {1}, + pages = {65-98}, + year = {2017}, + doi = {10.1137/141000671}, + %url = {https://doi.org/10.1137/141000671}, + %eprint = {https://doi.org/10.1137/141000671}, + abstract = { Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is designed to be easy and fast and questions notions generally held to be “laws of nature" by practitioners of numerical computing: \beginlist \item High-level dynamic programs have to be slow. \item One must prototype in one language and then rewrite in another language for speed or deployment. \item There are parts of a system appropriate for the programmer, and other parts that are best left untouched as they have been built by the experts. \endlist We introduce the Julia programming language and its design---a dance between specialization and abstraction. Specialization allows for custom treatment. Multiple dispatch, a technique from computer science, picks the right algorithm for the right circumstance. Abstraction, which is what good computation is really about, recognizes what remains the same after differences are stripped away. Abstractions in mathematics are captured as code through another technique from computer science, generic programming. Julia shows that one can achieve machine performance without sacrificing human convenience. } +} + +@article{gowda2021high, + title={High-performance symbolic-numerics via multiple dispatch}, + author={Gowda, Shashi and Ma, Yingbo and Cheli, Alessandro and Gwozdz, Maja and Shah, Viral B and Edelman, Alan and Rackauckas, Christopher}, + journal={arXiv preprint arXiv:2105.03949}, + year={2021} +} + +@article{bouchetvalat2023dataframes, + title={DataFrames.jl: Flexible and Fast Tabular Data in Julia}, + volume={107}, + %url={https://www.jstatsoft.org/index.php/jss/article/view/v107i04}, + doi={10.18637/jss.v107.i04}, + abstract={DataFrames.jl is a package written for and in the Julia language offering flexible and efficient handling of tabular data sets in memory. Thanks to Julia’s unique strengths, it provides an appealing set of features: Rich support for standard data processing tasks and excellent flexibility and efficiency for more advanced and non-standard operations. We present the fundamental design of the package and how it compares with implementations of data frames in other languages, its main features, performance, and possible extensions. We conclude with a practical illustration of typical data processing operations.}, + number={4}, + journal={Journal of Statistical Software}, + author={Bouchet-Valat, Milan and Kamiński, Bogumił}, + year={2023}, + pages={1--32} +} + +@book{riguzzi2022foundations, + address = {New York}, + edition = {1}, + title = {Foundations of {Probabilistic} {Logic} {Programming}: {Languages}, {Semantics}, {Inference} and {Learning}}, + isbn = {978-1-00-333819-2}, + shorttitle = {Foundations of {Probabilistic} {Logic} {Programming}}, + %url = {https://www.taylorfrancis.com/books/9781003338192}, + language = {en}, + urldate = {2023-03-01}, + publisher = {River Publishers}, + author = {Riguzzi, Fabrizio}, + month = sep, + year = {2022}, + doi = {10.1201/9781003338192}, +} + +@inproceedings{sato1995statistical, + title={A Statistical Learning Method for Logic Programs with Distribution Semantics}, + author={Taisuke Sato}, + booktitle={International Conference on Logic Programming}, + year={1995}, + %url={https://api.semanticscholar.org/CorpusID:10424169} +} + + +@article{lifschitz2002answer, + title = {Answer set programming and plan generation}, + volume = {138}, + issn = {0004-3702}, + %url = {https://www.sciencedirect.com/science/article/pii/S0004370202001868}, + doi = {https://doi.org/10.1016/S0004-3702(02)00186-8}, + abstract = {The idea of answer set programming is to represent a given computational problem by a logic program whose answer sets correspond to solutions, and then use an answer set solver, such as smodels or dlv, to find an answer set for this program. Applications of this method to planning are related to the line of research on the frame problem that started with the invention of formal nonmonotonic reasoning in 1980.}, + number = {1}, + journal = {Artificial Intelligence}, + author = {Lifschitz, Vladimir}, + year = {2002}, + keywords = {Answer sets, Default logic, Frame problem, Logic programming, Planning}, + pages = {39--54}, +} + +@inproceedings{lee2016weighted, + title={Weighted rules under the stable model semantics}, + author={Lee, Joohyung and Wang, Yi}, + booktitle={Fifteenth international conference on the principles of knowledge representation and reasoning}, + year={2016} +} + +@article{baral2009probabilistic, + title={Probabilistic reasoning with {A}nswer {S}ets}, + author={Baral, Chitta and Gelfond, Michael and Rushton, Nelson}, + journal={Theory and Practice of Logic Programming}, + volume={9}, + number={1}, + pages={57--144}, + year={2009}, + publisher={Cambridge University Press} +} + +@inproceedings{de2007problog, + title={ProbLog: A probabilistic {P}rolog and its application in link discovery}, + author={De Raedt, Luc and Kimmig, Angelika and Toivonen, Hannu and Veloso, M}, + booktitle={IJCAI 2007, Proceedings of the 20th international joint conference on artificial intelligence}, + pages={2462--2467}, + year={2007}, + organization={IJCAI-INT JOINT CONF ARTIF INTELL} +} + +@inproceedings{lee2017lpmln, + title={LPMLN, {W}eak {C}onstraints, and {P}-log}, + author={Lee, Joohyung and Yang, Zhun}, + booktitle={Proceedings of the AAAI Conference on Artificial Intelligence}, + volume={31}, + number={1}, + year={2017} +} + +@article{pajunen2021solution, + title={Solution enumeration by optimality in {A}nswer {S}et {P}rogramming}, + author={Pajunen, Jukka and Janhunen, Tomi}, + journal={Theory and Practice of Logic Programming}, + volume={21}, + number={6}, + pages={750--767}, + year={2021}, + publisher={Cambridge University Press} +} + + +@article{alberti2017cplint, + title={cplint on SWISH: Probabilistic logical inference with a web browser}, + author={Alberti, Marco and Bellodi, Elena and Cota, Giuseppe and Riguzzi, Fabrizio and Zese, Riccardo}, + journal={Intelligenza Artificiale}, + volume={11}, + number={1}, + pages={47--64}, + year={2017}, + publisher={IOS Press} +} diff --git a/text/paper_01/asp.bib b/text/paper_01/asp.bib new file mode 100644 index 0000000..6f1b94f --- /dev/null +++ b/text/paper_01/asp.bib @@ -0,0 +1,15 @@ + +@article{lifschitz_answer_2002, + title = {Answer set programming and plan generation}, + volume = {138}, + issn = {0004-3702}, + url = {https://www.sciencedirect.com/science/article/pii/S0004370202001868}, + doi = {https://doi.org/10.1016/S0004-3702(02)00186-8}, + abstract = {The idea of answer set programming is to represent a given computational problem by a logic program whose answer sets correspond to solutions, and then use an answer set solver, such as smodels or dlv, to find an answer set for this program. Applications of this method to planning are related to the line of research on the frame problem that started with the invention of formal nonmonotonic reasoning in 1980.}, + number = {1}, + journal = {Artificial Intelligence}, + author = {Lifschitz, Vladimir}, + year = {2002}, + keywords = {Answer sets, Default logic, Frame problem, Logic programming, Planning}, + pages = {39--54}, +} diff --git a/text/paper_01/example_annotASP2plainASP.txt b/text/paper_01/example_annotASP2plainASP.txt new file mode 100644 index 0000000..9b80363 --- /dev/null +++ b/text/paper_01/example_annotASP2plainASP.txt @@ -0,0 +1,14 @@ +a : 0.3. -a:0.7 +b : 0.5. -b:0.5 +c : 0.1. -c:0.9 + +t1 = { } => + +P(T = t1) = \sum_ + + +0.3:: a. +b ; c :- a. + +a ; -a. +b ; c :- a. diff --git a/text/paper_01/foundations.bib b/text/paper_01/foundations.bib new file mode 100644 index 0000000..f04a9c0 --- /dev/null +++ b/text/paper_01/foundations.bib @@ -0,0 +1,17 @@ + +@book{riguzzi2022foundations, + address = {New York}, + edition = {1}, + title = {Foundations of {Probabilistic} {Logic} {Programming}: {Languages}, {Semantics}, {Inference} and {Learning}}, + isbn = {978-1-00-333819-2}, + shorttitle = {Foundations of {Probabilistic} {Logic} {Programming}}, + url = {https://www.taylorfrancis.com/books/9781003338192}, + language = {en}, + urldate = {2023-03-01}, + publisher = {River Publishers}, + author = {Riguzzi, Fabrizio}, + month = sep, + year = {2022}, + doi = {10.1201/9781003338192}, + file = {Riguzzi - 2018 - Foundations of probabilistic logic programming la.pdf:/home/fc/sci/zotero/storage/X84FP7FV/Riguzzi - 2018 - Foundations of probabilistic logic programming la.pdf:application/pdf}, +} diff --git a/text/paper_01/pre-paper.pdf b/text/paper_01/pre-paper.pdf new file mode 100644 index 0000000..a39d0e4 Binary files /dev/null and b/text/paper_01/pre-paper.pdf differ diff --git a/text/paper_01/pre-paper.tex b/text/paper_01/pre-paper.tex new file mode 100644 index 0000000..32aaa16 --- /dev/null +++ b/text/paper_01/pre-paper.tex @@ -0,0 +1,1342 @@ +\documentclass[a4paper, 10pt]{article} +%\documentclass{llncs} + +\usepackage[ +bibstyle=numeric, +citestyle=numeric +]{biblatex} %Imports biblatex package +\addbibresource{zugzwang.bib} %Import the bibliography file + +\usepackage[x11colors]{xcolor} + +\usepackage{tikz} +\tikzset{ +event/.style={}, +smodel/.style={fill=gray!25}, +tchoice/.style={draw, circle}, +indep/.style={},%{draw, dashed}, +proptc/.style = {-latex, dashed}, +propsm/.style = {-latex, thick}, +doubt/.style = {gray} +} +\usetikzlibrary{calc, positioning, patterns} + +\usepackage{hyperref} +\hypersetup{ +colorlinks=true, +linkcolor=blue, +citecolor=blue, +urlcolor=blue, +} + +\usepackage{commath} +\usepackage{amsthm} +\newtheorem{assumption}{Assumption} +\newtheorem{definition}{Definition} +\newtheorem{proposition}{Proposition} +\newtheorem{example}{Example} +\newtheorem{theorem}{Theorem} +\usepackage{amssymb} +\usepackage[normalem]{ulem} +\usepackage[nice]{nicefrac} +\usepackage{stmaryrd} +\usepackage{acronym} +\usepackage{multicol} +\usepackage{cleveref} +% +% Local commands +% +\newcommand{\naf}{\ensuremath{\sim\!}} +\newcommand{\larr}{\ensuremath{\leftarrow}} +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} +\newcommand{\deft}[1]{\textbf{#1}} +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} +\newcommand{\ent}{\ensuremath{\lhd}} +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} +\newcommand{\uset}[1]{\ensuremath{#1^{\ast}}} +\newcommand{\lset}[1]{\ensuremath{#1_{\ast}}} +\newcommand{\yset}[1]{\ensuremath{\left\langle #1 \right\rangle}} +\newcommand{\stablecore}[1]{\ensuremath{\left\llbracket #1 \right\rrbracket}} +\newcommand{\uclass}[1]{\ensuremath{\intco{#1}}} +\newcommand{\lclass}[1]{\ensuremath{\intoc{#1}}} +\newcommand{\smclass}[1]{\ensuremath{\intcc{#1}}} +\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}} +\newcommand{\err}[1]{\ensuremath{\mathrm{err}\at{#1}}} +\newcommand{\pw}[1]{\ensuremath{\mu\at{#1}}} +\newcommand{\pwcfname}{\ensuremath{\mu_{\textrm{TC}}}} +\newcommand{\pwc}[1]{\ensuremath{\pwcfname\at{#1}}} +\newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}} +\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}} +\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}} +\newcommand{\rep}[2]{\ensuremath{\left\langle #1 \middle| #2 \right\rangle}} +\newcommand{\inconsistent}{\bot} +\newcommand{\given}{\ensuremath{~\middle|~}} +\newcommand{\emptyevent}{\ensuremath{\vartriangle}} +\newcommand{\indepclass}{\ensuremath{\Diamond}} +\newcommand{\probfact}[2]{\ensuremath{#2\mkern-4mu:\mkern-4mu#1}} +\newcommand{\probrule}[3]{\probfact{#1}{#2} \leftarrow #3} +%\newcommand{\tcgen}[1]{\ensuremath{\widehat{#1}}} +\newcommand{\tcgen}[1]{\ensuremath{\left<#1\right>}} +\newcommand{\lfrac}[2]{\ensuremath{{#1}/{#2}}} +\newcommand{\condsymb}[2]{\ensuremath{p_{#1|#2}}} +% +%\newcommand{\oldnote}[1]{\marginpar{\scriptsize #1}} +\newcommand{\oldnote}[1]{\note{#1}} +\newcommand{\todo}[1]{{\color{red!50!black}(\emph{#1})}} +% \newcommand{\oldremark}[2]{\uwave{#1}~{\color{green!40!black}(\emph{#2})}} +\newcommand{\oldremark}[2]{\remark{#1}{#2}} +\newcommand{\oldreplace}[2]{\sout{#1}/{\color{green!20!black}#2}} +\newcommand{\delete}[1]{\xout{#1}} +\newcommand{\franc}[1]{{\color{orange!60!black}#1}} +\newcommand{\bruno}{\color{red!60!blue}} +% +\newcounter{remark} +\newcommand{\remark}[2]{% + \stepcounter{remark}% + \!{\color{red}/\!}% + #1% + {\!\color{red}/}\footnotemark[\arabic{remark}]% + \footnotetext[\arabic{remark}]{{\color{red}/}#2}% + } +\newcommand{\note}[1]{ + \stepcounter{remark}% + {\!\!\color{red}/}\footnotemark[\arabic{remark}]\!\!% + \footnotetext[\arabic{remark}]{{\color{red}/}#1} +} +% +% Acronyms +% +\acrodef{BK}[BK]{background knowledge} +\acrodef{ASP}[ASP]{answer set programming} +\acrodef{NP}[NP]{normal program} +\acrodef{DS}[DS]{distribution semantics} +\acrodef{PF}[PF]{probabilistic fact} +\acrodef{TC}[TC]{total choice} +\acrodef{SM}[SM]{stable model} +\acrodef{SC}[SC]{stable core} +\acrodef{KL}[KL]{Kullback-Leibler} +\acrodef{SBF}[SBF]{Simple But Fruitful} +\acrodef{RSL}[RSL]{Random Set of Literals} +\acrodef{RCE}[RCE]{Random Consistent Event} +% +% +% +\title{An Algebraic Approach to Stochastic ASP} +% +% +% +\author{ +\begin{tabular}{ccc} + Salvador Abreu + \footnote{Universidade de Évora, NOVALINCS} + & + Francisco Coelho + \footnote{Universidade de Évora, NOVALINCS, High Performance Computing Chair} + & Bruno Dinis + \footnote{Universidade de Évora, CIMA, CMAFcIO} + \\ + \texttt{spa@uevora.pt} + & \texttt{fc@uevora.pt} + & \texttt{bruno.dinis@uevora.pt} +\end{tabular} +} +% +% +% +\begin{document} +% +% +% +\maketitle +% +% +% +\begin{abstract} + We address the problem of extending probability from the total choices of an \acs{ASP} program to the \aclp{SM}, and from there to general events. + % + Our approach is algebraic in the sense that it relies on an equivalence relation over the set of events and uncertainty is expressed with variables and polynomial expressions. + % + We illustrate our methods with two examples, one of which shows a connection to bayesian networks. +\end{abstract} +% +% +% +\section{Introduction and Motivation} +% +% +% +A major limitation of logical representations in real world applications is the implicit assumption that the \acl{BK} is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how \acl{ASP} programs with probabilistic facts can lead to characterizations of probability functions on the program's domain, which is not straightforward in the context of \acl{ASP}, as explained below (see also \cite{cozman2020joy,verreet2022inference,baral2009probabilistic,pajunen2021solution}). Unlike current systems such as ProbLog \cite{de2007problog}, P-log \cite{baral2009probabilistic}, LP\textsuperscript{MLN} \cite{lee2016weighted}, or cplint \cite{alberti2017cplint}, that derive a probability distribution from a program, in our system some choices are represented by a parameter that can be later estimated from further information, \emph{e.g.}\ observations. This approach enables later refinement and scoring of a partial program of a model from additional evidence. + +\Ac{ASP} \cite{lifschitz2002answer} is a logic programming paradigm based on the \ac{SM} semantics of \acp{NP} that can be implemented using the latest advances in SAT solving technology. Unlike ProLog, \ac{ASP} is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and both hard and weak constraints. + +The \ac{DS} \cite{sato1995statistical,riguzzi2022foundations} is a key approach to extend logical representations with probabilistic reasoning. +% +Let $\fml{A}$ be a finite set of atoms. A \emph{pre-total choice} is a subset $t^{\ast}$ of \fml{A}. The \emph{\acl{TC}} (TC) associated to $t^{\ast}$ is the set $t := t^{\ast} \cup \set{\co{a} \given a \in \fml{A} \setminus t^{\ast}}$. \Acp{PF} are the most basic \ac{DS} stochastic primitives and take the form $\probfact{p}{a}$ where $a\in\fml{A}$ is associated to some $p\in\intcc{0, 1}$. Each \ac{PF} then represents a boolean random variable that is true with probability $p$ and false with probability $\co{p} = 1 - p$. + +\note{revisit this part.}Let $F = \set{\probfact{p}{a} \given a \in \fml{A}, p \in \intcc{0, 1}}$. For a \acl{TC} $t$ over $\fml{A}$, define +$$ +P_t := \set{ p \given a \in t^{\ast} \wedge \probfact{p}{a} \in F} \cup + \set{\co{p} \given a \in t \setminus t^{\ast} \wedge \probfact{p}{a} \in F} +$$ + +and + +\begin{equation} + \pr{T = t} = \prod_{p \in P_t} p, + \label{eq:prob.total.choice} +\end{equation} + +where $T$ is a random variable whose values are \aclp{TC}. + +Our goal is to extend this probability (which is, indeed, a product of Bernoulli distributions), from \aclp{TC}, to cover the program domain. We use the term ``program'' as a set of rules and facts, plain and probabilistic. We can foresee two key applications of this extended probability: + +\begin{enumerate} + \item Support probabilistic reasoning/tasks on the program domain. + \item Also, given a dataset and a divergence measure, the program can be scored (by the divergence w.r.t.\ the \emph{empiric} distribution of the dataset), and weighted or sorted amongst other programs. These are key ingredients in algorithms searching, for example, optimal models of a dataset. +\end{enumerate} + +To extend probabilities from \aclp{TC} we start with the stance that \emph{a program describes an observable system}, that \emph{the \aclp{SM} are all the possible states} of that system and that \emph{observations (i.e.\ events) are stochastic} --- one observation can be sub-complete or super-complete, and might not determine the real state of the system. From here, probabilities must be extended from \acp{TC} to \acp{SM} and then to any event. +% +This extension process starts with a critical problem, illustrated by the example in \cref{sec:example.1}, concerning situations where multiple \acp{SM}, $ab$ and $ac$, result from a single \ac{TC}, $a$, but there is not enough information (in the program) to assign a single probability to each \ac{SM}. We propose to address this issue by using algebraic variables to describe that lack of information and then estimate the value of those variables from empirical data. This lack of uniqueness is also addressed in \cite{cozman2020joy} along a different approach, using credal sets. + +In another related work \cite{verreet2022inference} epistemic uncertainty (or model uncertainty) is considered as a lack of knowledge about the underlying model, that may be mitigated via further observations. This seems to presuppose a bayesian approach to imperfect knowledge in the sense that having further observations allows to improve/correct the model. Indeed, that approach uses Beta distributions on the total choices in order to be able to learn a distribution on the \remark{events}{Check this: do they learn distributions on the events?}. This approach seems to be specially fitted to being able to tell when some probability lies beneath some given value. Our approach seems to be similar in spirit, while remaining algebraic in the way that the extension of probabilities is addressed. + +The example in \cref{sec:example.1} uses the code available in the project's repository\footnote{\url{https://git.xdi.uevora.pt/fc/sasp}}, developed with the \textit{Julia} programming language \cite{bezanson2017julia}, and the \textit{Symbolics} \cite{gowda2021high}, and \textit{DataFrames} \cite{bouchetvalat2023dataframes} libraries. +% +% +% +\section{A Simple but Fruitful Example}\label{sec:example.1} +% +% +% +In this section we consider a somewhat simple case, which we call the \ac{SBF} example, that showcases the problem of extending probabilities from \aclp{TC} to \aclp{SM} and then to events. As mentioned before, the main issue arises from the lack of information in the program to assign a single probability to each stable model. This becomes a crucial problem in situations where multiple \aclp{SM} result from a single \acl{TC}. We will come back to this example in \cref{subsec:sbf.example}, after we present our proposal for extending probabilities from \aclp{TC} to \aclp{SM} in \cref{sec:extending.probalilities}. + + +\begin{example}\label{running.example} + Consider $\fml{A} = \set{a, b, c}$ and the following program + %\note{Introduce the notation $\probfact{p}{a}$ and what is the underlying ASP program.} + + \begin{equation} + \begin{aligned} + \probfact{0.3}{a} & ,\cr + b \vee c & \leftarrow a. + \end{aligned} + \label{eq:example.1} + \end{equation} + + %\note{Explain how the SM are defined.} + %\note{Explain our position about negation and be clear about $\co{a} = \neg a$ and not $\co{a} =\,\sim\!\! a$.} + %\note{Introduce the parameterization $\theta_{s,t}$.} + The \emph{standard form} of this program results from replacing annotated facts, such as $\probfact{0.3}{a}$, by the associated disjunctions, $a \vee \neg a$. The \aclp{SM} of the annotated program are the same as the ones from the standard form: $\co{a}, ab$ and $ac$, where $\co{a}$ stands for $\neg a$ (see \cref{fig:running.example}). While it is straightforward to assume $\pr{\co{a}}=0.7$, there is no obvious, explicit, way to assign values to $\pr{ab}$ and $\pr{ac}$. For instance, we can use a parameter $\theta$ as in + $$ + \begin{aligned} + \pr{ab} & = 0.3 \theta,\cr + \pr{ac} & = 0.3 (1 - \theta) + \end{aligned} + $$ + to express our knowledge that $ab,ac$ are events related in a certain way and, simultaneously, our uncertainty about that relation. The pa\-ra\-me\-ter $\theta=\theta_{s,t}$ depends on both the \acl{SM} $s$ and the \acl{TC} $t$. This uncertainty can then be addressed with the help of adequate distributions, such as empirical distributions from a dataset. +\end{example} + +If an \ac{ASP} program is intended to describe some system then: + +\begin{enumerate} + + \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the program domain. + + \item In the case where some statistical knowledge is available, for example, in the form of a distribution, we consider it as ``external'' knowledge about the parameters, that doesn't affect the extension procedure described below. + + \item Statistical knowledge can be used to estimate parameters and to ``score'' the program. + + \item\label{item:program.selection} If that program is only but one of many possible candidates then that score can be used, \emph{e.g.} as fitness, by algorithms searching (optimal) programs of a dataset of observations. + + \item If observations are not consistent with the program, then we ought to conclude that the program is wrong and must be changed accordingly. +\end{enumerate} + +Currently, we are addressing the problem of extending a probability function (possibly using parameters such as $\theta$), defined on the \acp{SM} of a program, to all the events of that program. This extension must satisfy the Kolmogorov axioms of probability (see \cref{prop:kolmogorov}) so that probabilistic reasoning is consistent with the \ac{ASP} program and follow our interpretation of \aclp{SM} as the states of an observable system. + +As sets, the \acp{SM} can have non-empty intersection. But, as states of a system, we assume that \acp{SM} are disjoint events, in the following sense: + +\begin{assumption}\label{assumption:smodels.disjoint} + \Aclp{SM} are disjoint events: For any set $X$ of \aclp{SM}, + \begin{equation} + \pr{X} = \sum_{s\in X}\pr{s} + \end{equation} +\end{assumption} + +Consider the \aclp{SM} $ab, ac$ from \cref{running.example}, that result from the clause $b \vee c \leftarrow a$ and the \acl{TC} $\set{a}$. Since we intend to associate each \acl{SM} with a state of the system, $ab$ and $ac$ should be \emph{disjoint} events. So $b \vee c$ is interpreted as an \emph{exclusive disjunction} and, from that particular clause, no further relation between $b$ and $c$ is assumed. This does not prevent that other clauses may be added that entail further dependencies between $b$ and $c$, which in turn may change the \aclp{SM}. + +By not making distribution assumptions on the clauses of the program we can state such properties on the semantics of the program, as we've done in assumption \ref{assumption:smodels.disjoint}. +% +% +% +\section{Extending Probabilities}\label{sec:extending.probalilities} +% +% +% +\begin{figure}[t] + \begin{center} + \begin{tikzpicture} + \node[event] (E) {$\emptyevent$}; + \node[tchoice, above left = of E] (a) {$a$}; + \node[smodel, above left = of a] (ab) {$ab$}; + \node[smodel, above right = of a] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[event, above left = of ab] (abC) {$\co{c}ab$}; + \node[event, above right = of ac] (aBc) {$\co{b}ac$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \draw[doubt] (a) to[bend left] (ab); + \draw[doubt] (a) to[bend right] (ac); + + \draw[doubt] (ab) to[bend left] (abc); + \draw[doubt] (ab) to[bend right] (abC); + + \draw[doubt] (ac) to[bend right] (abc); + \draw[doubt] (ac) to[bend left] (aBc); + + \draw[doubt, dashed] (Ac) to (Abc); + + \draw[doubt] (A) to (Ac); + \draw[doubt] (A) to (Abc); + + \draw[doubt] (ab) to[bend right] (E); + \draw[doubt] (ac) to[bend right] (E); + \draw[doubt] (A) to[bend left] (E); + + \draw[doubt] (ab) to (b); + \draw[doubt] (ac) to (c); + % \draw[doubt] (ab) to[bend left] (a); + % \draw[doubt] (ac) to[bend right] (a); + \draw[doubt, dashed] (c) to[bend right] (bc); + \draw[doubt, dashed] (abc) to[bend left] (bc); + \draw[doubt, dashed] (bc) to (Abc); + \draw[doubt, dashed] (c) to[bend right] (Ac); + \end{tikzpicture} + \end{center} + + \caption{Some events related to the \aclp{SM} of \cref{running.example}. The circle nodes are \aclp{TC} and shaded nodes are \aclp{SM}. Solid lines represent relations with the \acp{SM} and dashed lines relations between other events. The set of events contained in all \aclp{SM}, denoted by $\emptyevent$, is empty in this example.} + \label{fig:running.example} +\end{figure} + +The diagram in \cref{fig:running.example} illustrates the problem of extending probabilities from \aclp{TC} to \aclp{SM} and then to general events in an \emph{edge-wise} process, where the value in a node is defined from the values in its neighbors. This quickly leads to coherence problems concerning probability, with no clear systematic approach. Notice that $bc$ is not directly related with any \acl{SM} therefore propagating values through edges would assign a hard to justify ($\not= 0$) value to $bc$. Instead, we propose to base the extension in the relation an event has with the \aclp{SM}. +% +% +% +\subsection{An Equivalence Relation}\label{subsec:equivalence.relation} +% +% +% +\begin{figure}[t] + \begin{center} + \begin{tikzpicture} + \node[event] (E) {$\emptyevent$}; + \node[tchoice, above left = of E] (a) {$a$}; + \node[smodel, above left = of a] (ab) {$ab$}; + \node[smodel, above right = of a] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[event, above left = of ab] (abC) {$\co{c}ab$}; + \node[event, above right = of ac] (aBc) {$\co{b}ac$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \path[draw, rounded corners, pattern=north west lines, opacity=0.2] + (ab.west) -- + (ab.north west) -- + % + (abC.south west) -- + (abC.north west) -- + (abC.north) -- + % + (abc.north east) -- + (abc.east) -- + (abc.south east) -- + % + (ab.north east) -- + (ab.east) -- + (ab.south east) -- + % + (a.north east) -- + % + (E.north east) -- + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (b.south west) -- + % + (ab.west) + ; + % ---- + \path[draw, rounded corners, pattern=north east lines, opacity=0.2] + (ac.south west) -- + (ac.west) -- + (ac.north west) -- + % + (abc.south west) -- + (abc.west) -- + (abc.north west) -- + % + (aBc.north east) -- + (aBc.east) -- + (aBc.south east) -- + % + (ac.north east) -- + % + (c.east) -- + % + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (a.south west) -- + (a.west) -- + (a.north west) -- + (a.north) -- + % + (ac.south west) + ; + % ---- + \path[draw, rounded corners, pattern=horizontal lines, opacity=0.2] + % (A.north west) -- + % + (Ac.north west) -- + % + (Abc.north west) -- + (Abc.north) -- + (Abc.north east) -- + (Abc.south east) -- + % + % (Ac.north east) -- + % (Ac.east) -- + % + % (A.east) -- + (A.south east) -- + % + (E.south east) -- + (E.south) -- + (E.south west) -- + (E.west) -- + (E.north west) -- + % + (Ac.north west) + ; + \end{tikzpicture} + \end{center} + + \caption{Classes (of consistent events) related to the \aclp{SM} of \cref{running.example} are defined through intersections and inclusions. In this picture we can see, for example, the classes $\set{\co{c}ab, ab, b}$ and $\set{a, abc}$. Different fillings are different classes and, as before, the circle nodes are \aclp{TC} and shaded nodes are \aclp{SM}. Notice that $bc$ is not in a ``filled'' area.} + \label{fig:running.example.classes} +\end{figure} + +Given an ASP program, we consider a set of \emph{atoms} $ \fml{A}$, the set $\fml{L}$ of the \emph{literals} over \fml{A}, and the set of \emph{events} $\fml{E}$ such that $e \in \fml{E} \iff e \subseteq \fml{L}$. We also consider $\fml{W}$ the set of \emph{worlds} (consistent events), \note{Be more precise on this definition} a set of \emph{\aclp{TC}} $\fml{T}$ such that for every $a \in \fml{A}$ we have $t = a$ or $t = \neg a$, and $\fml{S}$ the set of \emph{\aclp{SM}} such that $ \fml{S}\subset\fml{W}$. At last, the set of \aclp{SM} entailed by the \acl{TC} $t$ is denoted by $\tcgen{t}$. + +Our path to extend probabilities starts with a perspective of \aclp{SM} as playing a role similar to \emph{prime factors}. The \aclp{SM} of a program are the irreducible events entailed from that program and any event must be considered under its relation with the \aclp{SM}. + +From \cref{running.example}, consider the \acp{SM} $\co{a}, ab, ac$ and events $a, abc$ and $c$. While $a$ is related with (contained in) with both $ab, ac$, event $c$ is related only with $ac$. So, $a$ and $c$ are related with different \acp{SM}. On the other hand, both $ab, ac$ are related with $abc$. So $a$ and $abc$ are related with the same \aclp{SM}. + +\begin{definition}\label{def:stable.core} + The \emph{\ac{SC}} of the event $e\in \fml{E}$ is + \begin{equation} + \stablecore{e} := \set{s \in \fml{S} \given s \subseteq e \vee e \subseteq s}. \label{eq:stable.core} + \end{equation} + where $\fml{S}$ is the set of \aclp{SM}. +\end{definition} + +We now define an equivalence relation so that two events are related if either both are inconsistent or both are consistent and, in the latter case, with the same \acl{SC}. + +\begin{definition}\label{def:equiv.rel} + For a given program, let $u, v \in \fml{E}$. The equivalence relation $\sim$ is defined by + \begin{equation} + u \sim v :\!\iff u,v \not\in\fml{W} \vee \del{u,v \in \fml{W} \wedge \stablecore{u} = \stablecore{v}}.\label{eq:equiv.rel} + \end{equation} +\end{definition} + +Observe that the minimality of \aclp{SM} implies that, in \cref{def:stable.core}, either $e$ is a \acl{SM} or at least one of $\exists s \del{s \subseteq e}, \exists s \del{e \subseteq s}$ is false. This equivalence relation defines a partition on the set of events, where each class holds a unique relation with the \aclp{SM}. In particular we denote each class by: + +\begin{equation} + \class{e} = + \begin{cases} + \inconsistent := \fml{E} \setminus \fml{W} + & \text{if~} e \in \fml{E} \setminus \fml{W}, \\ + \set{u \in \fml{W} \given \stablecore{u} = \stablecore{e}} + & \text{if~} e \in \fml{W}. + \end{cases}\label{eq:event.class} +\end{equation} + +The combinations of the \aclp{SM}, together with the set $\inconsistent$, form a set of representatives. Consider again \cref{running.example}. As previously mentioned, the \aclp{SM} are the elements of $\fml{S} = \set{\co{a}, ab, ac}$ so the quotient set of this relation, with abuse of notation, is: +\begin{equation} + \class{\fml{E}} = \set{ + \inconsistent, + \indepclass, + \class{\co{a}}, + \class{ab}, + \class{ac}, + \class{\co{a}, ab}, + \class{\co{a}, ac}, + \class{ab, ac}, + \class{\co{a}, ab, ac} + }, +\end{equation} +where $\indepclass$ denotes, with abuse of notation, both the class of \emph{independent} events $e$ such that $\stablecore{e} = \emptyset$ and its core. We have\note{Remark the odd nature of $\emptyevent$.}: + +\begin{equation*} + \begin{array}{l|lr} + \text{\textbf{Core}}, \stablecore{e} + & \text{\textbf{Class}}, \class{e} + & \text{\textbf{Size}}, \# \class{e} \\ + \hline + % + \inconsistent + & \co{a}a, \ldots + & 37 + \\ + % + \indepclass + & \co{b}, \co{c}, bc, \co{b}a, \co{b}c, \co{bc}, \co{c}a, \co{c}b, \co{bc}a + & 9 + \\ + % + \co{a} + & \co{a}, \co{a}b, \co{a}c, \co{ab}, \co{ac}, \co{a}bc, \co{ac}b, \co{ab}c, \co{abc} + & 9 + \\ + % + ab + & b, ab, \co{c}ab + & 3 + \\ + % + ac + & c, ac, \co{b}ac + & 3 + \\ + % + \co{a}, ab + & \emptyset + & 0 + \\ + % + \co{a}, ac + & \emptyset + & 0 + % + \\ + % + ab, ac + & a, abc + & 2 + \\ + % + \co{a}, ab, ac + & \emptyevent + & 1 + \\ + % + \hline + \class{\fml{E}} + & \fml{E} + & 64 + \end{array} +\end{equation*} + +\begin{itemize} + % + \item Since all events within an equivalence class are in relation with a specific set of \aclp{SM}, \emph{measures, including probability, should be constant within classes}: + \[ + \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right). + \] + % + \item In general, we have \emph{much more} \aclp{SM} than literals but their combinations are still \emph{much less} than events. Nevertheless, the equivalence classes allow us to propagate probabilities from \aclp{TC} to events, as explained in the next subsection. + % + \item In this specific case, instead of dealing with $64 = 2^6$ events, we consider only the $9 = 2^3 + 1$ classes, well defined in terms of combinations of the \aclp{SM}. +\end{itemize} +% +% +% +\subsection{From Total Choices to Events}\label{subsec:from.tchoices.to.events} +% +% +% +Our path to set a distribution on $\fml{E}$ starts with the more general problem of extending \emph{measures}, since extending \emph{probabilities} easily follows by means of a suitable normalization (see \eqref{eq:measure.events.unconditional} and \eqref{eq:probability.event}), and has two phases: +\begin{enumerate} + \item Extension of the probabilities, \emph{as measures}, from the \aclp{TC} to events. + \item Normalization of the measures on events, recovering a probability. +\end{enumerate} + +The ``extension'' phase, traced by \cref{eq:prob.total.choice} and eqs.\ \eqref{eq:measure.tchoice} to \eqref{eq:measure.events}, starts with the measure (probability) of \aclp{TC}, $\pw{t} = \pr{T = t}$, expands it to \aclp{SM}, $\pw{s}$, and then, within the equivalence relation from \cref{eq:equiv.rel}, to (general) events, $\pw{e}$, including (consistent) worlds. + +\begin{description} + % + \item[Total Choices.] Using \cref{eq:prob.total.choice}, this case is given by + \begin{equation} + \pwc{t} := \pr{T = t}= \prod_{p\in P_t} p. + \label{eq:measure.tchoice} + \end{equation} + % + + \item[Stable Models.] Recall that each \acl{TC} $t$, together with the rules and the other facts of a program, defines the set \tcgen{t} of \aclp{SM} associated with that choice. \note{Remark that we want to associate a number or an unknown to each SM of each TC.} \note{Recall what was said in Example 1 (?)} + Given a \acl{TC} $t$, a \acl{SM} $s$, and variables or values $\theta_{s,t} \in \intcc{0, 1}$, we define + \begin{equation} + \pw{s, t} := \begin{cases} + \theta_{s,t} & \text{if~} s \in \tcgen{t}\cr + 0 & \text{otherwise} + \end{cases} + \label{eq:measure.stablemodel} + \end{equation} + such that $\sum_{s\in \tcgen{t}} \theta_{s,t} = 1$. + % + + \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of \aclp{SM}. + \begin{description} + \item[Inconsistent Class.] The inconsistent class contains events that are logically inconsistent, thus should never be observed and have measure zero: + \begin{equation} + \pw{\inconsistent, t} := 0.\footnote{Notice that this measure being equal to zero is actually independent of the \acl{TC}.} + \label{eq:measure.class.inconsistent} + \end{equation} + \item[Independent Class.] A world that neither contains nor is contained in a \acl{SM} corresponds to a non-state, according to the program. So the respective measure is also set to zero: + \begin{equation} + \pw{\indepclass, t} := 0. + \label{eq:measure.class.independent} + \end{equation} + \item[Other Classes.] The extension must be constant within a class, its value should result from the elements in the \acl{SC}, and respects assumption \ref{assumption:smodels.disjoint} (\aclp{SM} are disjoint): + \begin{equation} + \pw{\class{e}, t} := \pw{\stablecore{e}, t} = \sum_{s\in\stablecore{e}}\pw{s, t} + \label{eq:measure.class.other} + \end{equation} + and + \begin{equation} + \pw{\class{e}} := \sum_{t \in \fml{T}} \pw{\class{e}, t}\pwc{t}. + \label{eq:measure.class.unconditional} + \end{equation} + \end{description} + % + + \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, denoting by $\# X$ the number of elements in $X$, we set: + \begin{equation} + \pw{e, t} := + \begin{cases} + \frac{\pw{\class{e}, t}}{\# \class{e}} & \text{if~}\# \class{e} > 0, \\ + 0 & \text{otherwise}. + \end{cases} + \label{eq:measure.events} + \end{equation} + and + \begin{equation} + \pw{e} := \sum_{t\in\fml{T}} \pw{e, t} \pwc{t}. + \label{eq:measure.events.unconditional} + \end{equation} +\end{description} + + + +The $\theta_{s,t}$ parameters in equation \eqref{eq:measure.stablemodel} express the \emph{program's} lack of knowledge about the measure assignment, when a single \acl{TC} entails more than one \acl{SM}. In that case, how to distribute the respective measures? Our proposal to address this problem consists in assigning an unknown measure, $\theta_{s,t}$, conditional on the \acl{TC}, $t$, to each \acl{SM} $s$. This approach allows the expression of an unknown quantity and future estimation, given observed + +% Consider the event $bc$ from \cref{running.example}. Since $\class{bc} = \indepclass$, from \cref{eq:measure.class.independent} we get $\mu\at{bc} = 0$. data. + +% SUPERSET +Equation \eqref{eq:measure.class.other} results from assumption \ref{assumption:smodels.disjoint} and states that the measure of a class $\class{e}$ is the sum over it's \acl{SC}, $\stablecore{e}$, and \eqref{eq:measure.class.unconditional} \emph{marginalizes} the \acp{TC} on \eqref{eq:measure.class.other}. + +The \emph{normalizing factor} is: +\begin{equation*} + Z := + \sum_{e \in \fml{E}} \pw{e} = + \sum_{\class{e} \in \class{\fml{E}}} \pw{\class{e}}, +\end{equation*} + +and now equation \eqref{eq:measure.events.unconditional} provides a straightforward way to define the \emph{probability of observation of a single event}: + +\begin{equation} + \pr{E = e} := \frac{\pw{e}}{Z}.\label{eq:probability.event} +\end{equation} + +Equation \eqref{eq:measure.events.unconditional} together with external statistical knowledge, can be used to learn about the \emph{initial} probabilities of the atoms, that should not (and by \cref{prop:two.distributions} can't) be confused with the explicit $\pwcfname$ set in the program. + +It is now straightforward to check that $\pr{E}$ satisfies the Kolmogorov axioms of probability. + +Since \aclp{TC} are also events, one can ask, for an arbitrary \aclp{TC} $t$, if $\pr{T = t} = \pr{E = t}$ or, equivalently, if $\pwc{t} = \pw{t}$. However, it is easy to see that, in general, that cannot be true. While the domain of the random variable $T$ is the set of \aclp{TC}, for $E$ the domain is much larger, including all the events. Except for trivial programs, where the \acp{SM} are the \acp{TC}, some events other than \aclp{TC} have non-zero probability. + +\begin{proposition} \label{prop:two.distributions} + In a program with a \acl{SM} that is not a \acl{TC} there is at least one $t\in\fml{T}$ such that: + \begin{equation} + \pr{T = t} \not= \pr{E = t}. \label{eq:two.distributions} + \end{equation} +\end{proposition} + +\begin{proof} + Supposing towards a contradiction that $\pr{T = t} = \pr{E = t}$ for all $t \in \fml{T}$. Then + $$ + \sum_{t\in\fml{T}} \pr{E = t} = \sum_{t\in\fml{T}} \pr{T = t} = 1. + $$ + + Hence $\pr{E = x} = 0$ for all $x \in \fml{E}\setminus\fml{T}$, in contradiction with the fact that for at least one $s \in \fml{S}\setminus\fml{T}$ one has $\pr{E = s} > 0$. +\end{proof} + +The essential conclusion of \cref{prop:two.distributions} is that we are dealing with \emph{two distributions}: one, on the \acp{TC}, explicit in the annotations of the programs and another one, on the events, and entailed by the explicit annotations \emph{and the structure of the \aclp{SM}}. + +% +% +% +\section{Developed Examples}\label{sec:developed.examples} +% +% +% +Here we apply the methods from \cref{sec:extending.probalilities} to the SBF example and to a well known bayesian network, the Earthquake, Burglar, Alarm problem. + +\subsection{The SBF Example}\label{subsec:sbf.example} + +We continue with the program from \cref{eq:example.1}. + +\begin{description} + % + \item[\Aclp{TC}.] The \aclp{TC}, and respective \aclp{SM}, are + % + \begin{center} + \begin{tabular}{ll|r} + \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pwc{t}$} \\ + \hline + $a$ & $ab, ac$ & $0.3$ \\ + $\co{a}$ & $\co{a}$ & $\co{0.3} = 0.7$ + \end{tabular} + \end{center} + % + + \item[\Aclp{SM}.] The $\theta_{s,t}$ parameters in this example are + $$ + \begin{array}{l|cc} + \theta_{s,t} & \co{a} & a \\ + \hline + \co{a} & 1 & 0 \\ + ab & 0 & \theta \\ + ac & 0 & \co{\theta} + \end{array} + $$ + with $\theta \in \intcc{0, 1}$. + + \item[Classes.] Following the definitions in \cref{eq:stable.core,eq:equiv.rel,eq:event.class,eq:measure.class.inconsistent,eq:measure.class.independent,eq:measure.class.other} we get the following quotient set (ignoring $\inconsistent$ and $\indepclass$), and measures: + \begin{equation*} + \begin{array}{l|ll|rr|r} + \stablecore{e} + & \pw{s, \co{a}} + & \pw{s, a} + & \pw{\class{e}, \co{a}} + & \pw{\class{e}, a} + & \pw{\class{e}} + \\[2pt] + & \co{a}, ab, ac + & \co{a}, ab, ac + & \pwcfname=0.7 + & \pwcfname=0.3 + & + \\[2pt] + \hline + \co{a} + & \boxed{1},0,0 + & \boxed{0},\theta, \co{\theta} + & 1 + & 0 + & 0.7 + \\[2pt] + % + ab + & 1,\boxed{0},0 + & 0,\boxed{\theta}, \co{\theta} + & 0 + & \theta + & 0.3\theta + \\[2pt] + % + ac + & 1,0,\boxed{0} + & 0,\theta, \boxed{\co{\theta}} + & 0 + & \co{\theta} + & 0.3\co{\theta} + \\[2pt] + % + \co{a}, ab + & \boxed{1},\boxed{0},0 + & \boxed{0},\boxed{\theta}, \co{\theta} + & 1 + & \theta + & 0.7 + 0.3\theta + \\[2pt] + % + \co{a}, ac + & \boxed{1},0,\boxed{0} + & \boxed{0},\theta, \boxed{\co{\theta}} + & 1 + & \co{\theta} + & 0.7 + 0.3\co{\theta} + \\[2pt] + % + ab, ac + & 1,\boxed{0},\boxed{0} + & 0,\boxed{\theta}, \boxed{\co{\theta}} + & 0 + & \theta + \co{\theta} = 1 + & 0.3 + \\[2pt] + % + \co{a}, ab, ac + & \boxed{1},\boxed{0},\boxed{0} + & \boxed{0},\boxed{\theta}, \boxed{\co{\theta}} + & 1 + & \theta + \co{\theta} = 1 + & 1 + \end{array} + \end{equation*} + + \item[Prior Distributions.] Following the above values (in rational form), and considering the inconsistent and independent classes (resp. $\inconsistent, \indepclass$): + \begin{equation*} + \begin{array}{lr|cc|cc} + \stablecore{e} + & \# \class{e} + & \pw{\class{e}} + & \pw{e} + & \pr{E = e} + & \pr{E \in \class{e}} + \\ + \hline + % + \inconsistent + & 37 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + \indepclass + & 9 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + \co{a} + & 9 + & \frac{7}{10} + & \frac{7}{90} + & \frac{7}{207} + & \frac{7}{23} + \\[4pt] + % + ab + & 3 + & \frac{3}{10}\theta + & \frac{1}{10}\theta + & \frac{1}{23}\theta + & \frac{3}{23}\theta + \\[4pt] + % + ac + & 3 + & \frac{3}{10}\co{\theta} + & \frac{1}{10}\co{\theta} + & \frac{1}{23}\co{\theta} + & \frac{3}{23}\co{\theta} + \\[4pt] + % + \co{a}, ab + & 0 + & \frac{7 + 3\theta}{10} + & 0 + & 0 + & 0 + \\[4pt] + % + \co{a}, ac + & 0 + & \frac{7 + 3\co{\theta}}{10} + & 0 + & 0 + & 0 + % + \\[4pt] + % + ab, ac + & 2 + & \frac{3}{10} + & \frac{3}{20} + & \frac{3}{46} + & \frac{3}{23} + \\[4pt] + % + \co{a}, ab, ac + & 1 + & 1 + & 1 + & \frac{10}{23} + & \frac{10}{23} + \\[4pt] + % + \hline + & + & + & Z = \frac{23}{10} + & + %& \Sigma = 1 + \end{array} + \end{equation*} +\end{description} + +So the prior distributions, denoted by the random variable $E$, of events and classes are: + +\begin{equation} + \begin{array}{l|ccccccccc} + \stablecore{e} & + \inconsistent & + \indepclass & + \co{a} & + ab & + ac & + \co{a}, ab & + \co{a}, ac & + ab, ac & + \co{a}, ab, ac + \\ \hline\\[-12pt] + + \pr{E = e} & + 0 & + 0 & + \frac{7}{207} & + \frac{1}{23}\theta & + \frac{1}{23}\co{\theta} & + 0 & + 0 & + \frac{3}{46} & + \frac{10}{23} + \\[4pt] + + \pr{E \in \class{e}} & + 0 & + 0 & + \frac{7}{23} & + \frac{3}{23}\theta & + \frac{3}{23}\co{\theta} & + 0 & + 0 & + \frac{3}{23} & + \frac{10}{23} + \end{array}\label{eq:sbf.prior} +\end{equation} +% +% +% +\subsubsection*{Testing the Prior Distributions} +% +% +% +These results can be \emph{tested by simulation} in a two-step process, where (1) a ``system'' is \emph{simulated}, to gather some ``observations'' and then (2) empirical distributions from those samples are \emph{related} with the prior distributions from \cref{eq:sbf.prior}. \Cref{tab:sbf.example,tab:sbf.examples.2.3} summarize some of those tests, where datasets of $n = 1000$ observations are generated and analyzed. + +\bigskip\noindent\textbf{Simulating a System.} Following some criteria, more or less related to the given program, a set of events, that represent observations, is generated. Possible simulation procedures include: +\begin{itemize} + % + \item \emph{Random.} Each sample is a \ac{RSL}. Additional sub-criteria may require, for example, consistent events, a \ac{RCE} simulation. + % + \item \emph{Model+Noise.} Gibbs' sampling \cite{geman84} tries to replicate the program model and also to add some noise. For example, let $\alpha, \beta, \gamma \in \intcc{0,1}$ be some parameters to control the sample generation. The first parameter, $\alpha$ is the ``out of model'' samples ratio; $\beta$ represents the total choice $a$ or $\co{a}$ (explicit in the model) and $\gamma$ is the simulation representation of $\theta$. A single sample is then generated following the probabilistic choices below: + $$ + \begin{cases} + \alpha & \text{by \ac{RCE}} \\%[-2pt] + & + \begin{cases} + \beta & \co{a} \\%[-2pt] + & + \begin{cases} + \gamma & ab \\%[-2pt] + & ac + \end{cases} + \end{cases} + \end{cases}, + $$ + where + $$ + \begin{cases} + p & x \\%[-4pt] + & y + \end{cases} + $$ + denotes ``\emph{the value of $x$ with probability $p$, otherwise $y$}'' --- notice that $y$ might entail $x$ and \emph{vice-versa}: E.g.\ some $ab$ can be generated in the \ac{RCE}. + \item \emph{Other Processes.} Besides the two sample generations procedures above, any other processes and variations can be used. For example, requiring that one of $x, \co{x}$ literals is always in a sample or using specific distributions to guide the sampling of literals or events. +\end{itemize} + +\noindent\textbf{Relating the Empirical and the Prior Distributions.} The data from the simulated observations is used to test the prior distribution. Consider the prior, $\pr{E}$, and the empirical, $\pr{S}$, distributions and the following error function: +\begin{equation} + \err{\theta} := \sum_{e\in\fml{E}} \del{\pr{E = e} - \pr{S = e}}^2.\label{eq:err.e.s} +\end{equation} + +\begin{itemize} + \item Since $E$ depends on $\theta$, one can ask how does the error varies with $\theta$. + \item What is the \emph{optimal} (i.e.\ minimum) error value + \begin{equation} + \hat{\theta} := \arg\min_\theta \err{\theta}\label{eq:opt.err} + \end{equation} + and what does it tell us about the program. +\end{itemize} + +\begin{table} + \begin{center} + $$ + \begin{array}{l|cc|c} + \stablecore{e} + & \#\set{S \in \class{e}} + & \pr{S \in \class{e}} + & \pr{E \in \class{e}} + \\ + \hline + % + \inconsistent + & 0 + & 0 + & 0 + \\[2pt] + % + \indepclass + & 24 + & \frac{24}{1000} + & 0 + \\[2pt] + % + \co{a} + & 647 + & \frac{647}{1000} + & \frac{7}{23} + \\[2pt] + % + ab + & 66 + & \frac{66}{1000} + & \frac{3}{23}\theta + \\[2pt] + % + ac + & 231 + & \frac{231}{1000} + & \frac{3}{23}\co{\theta} + \\[2pt] + % + \co{a}, ab + & 0 + & 0 + & 0 + \\[2pt] + % + \co{a}, ac + & 0 + & 0 + & 0 + % + \\[2pt] + % + ab, ac + & 7 + & \frac{7}{1000} + & \frac{3}{23} + \\[2pt] + % + \co{a}, ab, ac + & 25 + & \frac{25}{1000} + & \frac{10}{23} + \\[2pt] + \hline + & n = 1000 + \end{array} + $$ + \end{center} + + \caption{\textbf{Experiment 1.} Results from an experiment where $n=1000$ samples where generated following the \emph{Model+Noise} procedure with parameters $\alpha = 0.1, \beta = 0.3, \gamma = 0.2$. The \emph{empirical} distribution is represented by the random variable $S$ while the \emph{prior}, as before, is denoted by $E$.}\label{tab:sbf.example} +\end{table} + +In order to illustrate this analysis, consider the experiment summarized in \cref{tab:sbf.example}: + +\begin{enumerate} + \item Equation \eqref{eq:err.e.s} becomes + $$ + \err{\theta} = \frac{20869963}{66125000} + \frac{477}{52900}\theta + \frac{18}{529}\theta^2. + $$ + \item The minimum of $\err{\theta}$ is at $\frac{477}{52900} + 2\frac{18}{529}\theta = 0$. Since this value is negative and $\theta \in \intcc{0,1}$, it must be $\hat{\theta} = 0$, and + $$ + \err{\hat{\theta}} = \frac{20869963}{66125000} \approx 0.31561. + $$ +\end{enumerate} + +The parameters $\alpha, \beta, \gamma$ of that experiment favour $ac$ over $ab$. In particular, setting $\gamma = 0.2$ means that in the simulation process, choices between $ab$ and $ac$ favour $ac$, 4 to 1. For completeness sake, we also describe one experiment that favours $ab$ over $ac$ (setting $\gamma=0.8$) and one balanced ($\gamma=0.5$). + +\begin{description} + \item[For $\gamma=0.8$,] the error function is + \begin{equation*} + \err{\theta} = \frac{188207311}{529000000} - \frac{21903}{264500} \theta + \frac{18}{529} \theta^{2} \approx 0.35579 - 0.08281 \theta + 0.03403 \theta ^2 + \end{equation*} + and, with $\theta\in\intcc{0, 1}$ the minimum is at $-0.08281 + 0.06805 \theta = 0$, \emph{i.e.}: + \begin{eqnarray*} + \hat{\theta} : \frac{0.08281}{0.06805} \approx 1.21683& >1. &\text{So,~} \hat{\theta} = 1, \\ + \err{\hat{\theta}} \approx 0.30699&. + \end{eqnarray*} + + \item[For $\gamma=0.5$,] the error function is + \begin{equation*} + \err{\theta} = \frac{10217413}{33062500} - \frac{2181}{66125} \theta + \frac{18}{529} \theta^{2}\approx 0.30903 - 0.03298 \theta + 0.03402 \theta ^2 + \end{equation*} + and, with $\theta\in\intcc{0, 1}$ the minimum is at $-0.03298 + 0.06804 \theta = 0$, \emph{i.e.}: + \begin{eqnarray*} + \hat{\theta} &\approx & + \frac{0.03298}{0.06804} + \approx 0.48471 + \approx \frac{1}{2}, \\ + \err{\hat{\theta}} &\approx & + 0.30104 + \end{eqnarray*} + +\end{description} + +\begin{table} + \begin{center} + $$ + \begin{array}{l|ccc} + \stablecore{e} + & \#\set{S_{0.2} \in \class{e}} + & \#\set{S_{0.8} \in \class{e}} + & \#\set{S_{0.5} \in \class{e}} + \\ + \hline + % + \inconsistent + & 0 + & 0 + & 0 + \\[2pt] + % + \indepclass + & 24 + & 28 + & 23 + \\[2pt] + % + \co{a} + & 647 + & 632 + & 614 + \\[2pt] + % + ab + & 66 + & 246 + & 165 + \\[2pt] + % + ac + & 231 + & 59 + & 169 + \\[2pt] + % + \co{a}, ab + & 0 + & 0 + & 0 + \\[2pt] + % + \co{a}, ac + & 0 + & 0 + & 0 + % + \\[2pt] + % + ab, ac + & 7 + & 8 + & 4 + \\[2pt] + % + \co{a}, ab, ac + & 25 + & 27 + & 25 + \end{array} + $$ + \end{center} + + \caption{\textbf{Experiments 2 and 3.} Results from experiments where, in each, $n=1000$ samples are generated following the \emph{Model+Noise} procedure with parameters $\alpha = 0.1, \beta = 0.3, \gamma = 0.8$ (Experiment 2) and $\gamma=0.5$ (Experiment 3). Empirical distributions are represented by the random variables $S_{0.8}$ and $S_{0.5}$ respectively. Data from experience \cref{tab:sbf.example} is also included, and denoted by $S_{0.2}$, to provide reference.}\label{tab:sbf.examples.2.3} +\end{table} + +%\oldnote{under- and over- estimation} +These experiments show that data can indeed be used to estimate the parameters of the model. However, we observe that the estimated $\hat{\theta}$ has a tendency to over- or under- estimate the $\theta$ used to generate the samples. More precisely, in experiment \ref{tab:sbf.example} data is generated with $\gamma = 0.2$ (the surrogate of $\theta$) which is under-estimated with $\hat{\theta} = 0$ while in experiment 2, $\gamma = 0.8$ leads the over-estimation $\hat{\theta} = 1$. This suggests that we might need to refine the error estimation process. However, experiment 3 data results from $\gamma = 0.5$ and we've got $\hat{\theta} \approx 0.48471 \approx 0.5$, which is more in line with what is to be expected. +% +% +% +\subsection{An Example Involving Bayesian Networks}\label{subsec:example.bayesian.networks} +% +% +% +As it turns out, our framework is suitable to deal with more sophisticated cases, in particular cases involving Bayesian networks. In order to illustrate this, in this section we see how the classical example of the Burglary, Earthquake, Alarm \cite{Judea88} works in our setting. This example is a commonly used example in Bayesian networks because it illustrates reasoning under uncertainty. The gist of the example is given in \cref{Figure_Alarm}. It involves a simple network of events and conditional probabilities. + +The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) and John calls ($J$). The initial events $B$ and $E$ are assumed to be independent events that occur with probabilities $\pr{B}$ and $\pr{E}$, respectively. There is an alarm system that can be triggered by either of the initial events $B$ and $E$. The probability of the alarm going off is a conditional probability given that $B$ and $E$ have occurred. One denotes these probabilities, as per usual, by $\pr{A \given B}$, and $\pr{A \given E}$. There are two neighbors, Mary and John who have agreed to call if they hear the alarm. The probability that they do actually call is also a conditional probability denoted by $\pr{M \given A}$ and $\pr{J \given A}$, respectively. + +\begin{figure} + \begin{center} + \begin{tikzpicture}[node distance=2.5cm] + + % Nodes + \node[smodel, circle] (A) {A}; + \node[tchoice, above right of=A] (B) {B}; + \node[tchoice, above left of=A] (E) {E}; + \node[tchoice, below left of=A] (M) {M}; + \node[tchoice, below right of=A] (J) {J}; + + % Edges + \draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$\pr{B}=0.001$}} ; + \draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$\pr{E}=0.002$}} ; + \draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$\pr{M \given A}$}}; + \draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$\pr{J \given A}$}} ; + \end{tikzpicture} + \end{center} + + \begin{multicols}{3} + + \footnotesize{ + \begin{equation*} + \begin{split} + &\pr{M \given A}\\ + & \begin{array}{c|cc} + & m & \neg m \\ + \hline + a & 0.9 & 0.1 \\ + \neg a & 0.05 & 0.95 + \end{array} + \end{split} + \end{equation*} + } + + \footnotesize{ + \begin{equation*} + \begin{split} + &\pr{J \given A}\\ + & \begin{array}{c|cc} + & j & \neg j \\ + \hline + a & 0.7 & 0.3 \\ + \neg a & 0.01 & 0.99 + \end{array} + \end{split} + \end{equation*} + } + \footnotesize{ + \begin{equation*} + \begin{split} + \pr{A \given B \wedge E}\\ + \begin{array}{c|c|cc} + & & a & \neg a \\ + \hline + b & e & 0.95 & 0.05 \\ + b & \neg e & 0.94 & 0.06 \\ + \neg b & e & 0.29 & 0.71 \\ + \neg b & \neg e & 0.001 & 0.999 + \end{array} + \end{split} + \end{equation*} + } + \end{multicols} + \caption{The Earthquake, Burglary, Alarm model} + \label{Figure_Alarm} +\end{figure} + +We follow the convention of representing the (upper case) random variable $X$ by the lower case $x$. +% +Considering the probabilities given in \cref{Figure_Alarm} we obtain the following spe\-ci\-fi\-ca\-tion: + +\begin{equation*} + \begin{aligned} + \probfact{0.001}{b} & ,\cr + \probfact{0.002}{e} & ,\cr + \end{aligned} + \label{eq:not_so_simple_example} +\end{equation*} + +For the table giving the probability $\pr{M \given A}$ we obtain the program: + +\begin{equation*} + \begin{aligned} + \probfact{0.9}{\condsymb{m}{a}} & ,\cr + \probfact{0.05}{\condsymb{m}{\co{a}}} & ,\cr + m & \leftarrow a \wedge \condsymb{m}{a},\cr + m & \leftarrow \neg a \wedge \condsymb{m}{\co{a}}. + \end{aligned} +\end{equation*} + +This latter program can be simplified by writing $\probrule{0.9}{m}{a}$ and $\probrule{0.05}{m}{\neg a}$. +\note{SPA: \emph{parece-me que pode ser feito assim, mas estritamente falando já não corresponde à forma inicialmente anunciada}} + +Similarly, for the probability $\pr{J \given A}$ we obtain + +\begin{equation*} + \begin{aligned} + \probrule{0.7}{j}{&a}, \\ + \probrule{0.01}{j}{&\neg a}, + \end{aligned} +\end{equation*} + +Finally, for the probability $\pr{A \given B \wedge E}$ we obtain + +\begin{equation*} + \begin{aligned} + \probrule{0.95}{a}{b, e}, & & & + \probrule{0.94}{a}{b, \co{e}},\cr + \probrule{0.29}{a}{\co{b}, e}, & & & + \probrule{0.001}{a}{\co{b}, \co{e}}. + \end{aligned} +\end{equation*} + +One can then proceed as in the previous subsection and analyze this example. The details of such analysis are not given here since they are analogous, albeit admittedly more cumbersome. +% +% +% +\section{Discussion and Future Work} +% +% +% +This work is a first venture into expressing probability distributions using algebraic expressions derived from a logical program. +We would like to point out that there is still much to explore concerning the full expressive power of logic programs and \ac{ASP} programs. So far, we have not considered recursion, logical variables or functional symbols. Also, there is still little effort to articulate with the related fields, probabilistic logical programming, machine learning, inductive programming, \emph{etc.} + +The equivalence relation from \cref{def:equiv.rel} identifies the $s \subseteq e$ and $e \subseteq s$ cases. Relations that distinguish such cases might enable better relations between the models and processes from the \aclp{SM}. + +The example from \cref{subsec:example.bayesian.networks} shows that the theory, methodology, and tools, from bayesian networks can be adapted to our approach. The connection with Markov Fields \cite{kindermann80} is left for future work. An example of a ``program selection'' application (as mentioned in \cref{item:program.selection}, \cref{sec:example.1}) is also left for future work. + +%\oldnote{under- over- estimate} +Related with the remark at the end of \cref{subsec:sbf.example}, on the tendency of $\hat{\theta}$ to under- or over- estimate $\theta$, notice that the error function in \eqref{eq:err.e.s} expresses only one of many possible ``distances'' between the empirical and prior distributions. Variations include normalizing this function by the size of $\fml{E}$ or using the \acl{KL} divergence. The key contribution of this function in this work is to find an optimal $\theta$. Moreover, further experiments, not included in this paper, with $\alpha = 0.0$, lead to $\hat{\theta} \approx \gamma$, \emph{i.e.}\ setting the prior noise to zero leads to full recovering $\theta$ from the observations. + +We decided to set the measure of inconsistent events to $0$ but, maybe, in some cases, we shouldn't. For example, since observations may be affected by noise, one can expect inconsistencies between the literals of an observation to occur. +% +% +% +\section*{Acknowledgements} +% +% +% +This work is supported by NOVALINCS (UIDB/04516/2020) with the financial support of FCT.IP. +The third author acknowledges the support of FCT - Funda\c{c}\~ao para a Ci\^{e}ncia e Tecnologia under the projects: UIDP/04561/2020 and UIDP/04674/2020, and the research centers CMAFcIO -- Centro de Matem\'{a}tica, Aplica\c{c}\~{o}es Fundamentais e Investiga\c{c}\~{a}o Operacional and CIMA -- Centro de Investigação em Matemática e Aplicações. + +The authors thank Lígia Henriques-Rodrigues, Matthias Knorr and Ricardo Gonçalves for valuable comments on a preliminary version of this paper, and Alice Martins for contributions on software. +% +% +% +\printbibliography +% +% +% +\end{document} \ No newline at end of file diff --git a/text/paper_01/related_papers_plingo.md b/text/paper_01/related_papers_plingo.md new file mode 100644 index 0000000..ac7a239 --- /dev/null +++ b/text/paper_01/related_papers_plingo.md @@ -0,0 +1,9 @@ +[1] J. Lee and Y. Wang. (2016). Weighted Rules under the Stable Model Semantics + +[2] C. Baral and M. Gelfond and J.N. Rushton. (2009), Probabilistic Reasoning with Answer Sets + +[3] L. De Raedt and A. Kimmig and H. Toivonen ProbLog: A Probabilistic Prolog and its Applications in Link Discovery + +[4] J. Lee and Z. Yang (2017). LPMLN, Weak Constraints and P-log + +[5] J. Pajunen and T. Janhunen. (2021). Solution Enumeration by Optimality in Answer Set Programming. Theory and Practice of Logic Programming, 21(6), 750-767. \ No newline at end of file diff --git a/text/paper_01/reviews/pre-paperComments.pdf b/text/paper_01/reviews/pre-paperComments.pdf new file mode 100644 index 0000000..d32ce86 Binary files /dev/null and b/text/paper_01/reviews/pre-paperComments.pdf differ diff --git a/text/paper_01/temp.fc/LecBayesNetsAndInference.pdf b/text/paper_01/temp.fc/LecBayesNetsAndInference.pdf new file mode 100644 index 0000000..067233e Binary files /dev/null and b/text/paper_01/temp.fc/LecBayesNetsAndInference.pdf differ diff --git a/text/paper_01/temp.fc/hj18.pdf b/text/paper_01/temp.fc/hj18.pdf new file mode 100644 index 0000000..6f78ce1 Binary files /dev/null and b/text/paper_01/temp.fc/hj18.pdf differ diff --git a/text/paper_01/temp.fc/lec15.pdf b/text/paper_01/temp.fc/lec15.pdf new file mode 100644 index 0000000..0fbb6ff Binary files /dev/null and b/text/paper_01/temp.fc/lec15.pdf differ diff --git a/text/paper_01/zugz-paper_01.code-workspace b/text/paper_01/zugz-paper_01.code-workspace new file mode 100644 index 0000000..975da5d --- /dev/null +++ b/text/paper_01/zugz-paper_01.code-workspace @@ -0,0 +1,11 @@ +{ + "folders": [ + { + "path": "." + }, + { + "path": "../.." + } + ], + "settings": {} +} \ No newline at end of file diff --git a/text/paper_01/zugzwang.bib b/text/paper_01/zugzwang.bib new file mode 100644 index 0000000..342f020 --- /dev/null +++ b/text/paper_01/zugzwang.bib @@ -0,0 +1,232 @@ +@book{kindermann80, + author = {Kindermann, Ross and Snell, J. Laurie}, + title = {Markov random fields and their applications}, + series = {Contemporary Mathematics}, + volume = {1}, + publisher = {American Mathematical Society, Providence, RI}, + year = {1980}, + pages = {ix+142}, + isbn = {0-8218-5001-6}, + mrclass = {60K35 (60G60 82A42 82A67 94A05)}, + mrnumber = {620955}, + mrreviewer = {J.\ Theodore\ Cox} +} + +@article{geman84, + author = {Geman, Stuart and Geman, Donald}, + journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, + title = {Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images}, + year = {1984}, + volume = {PAMI-6}, + number = {6}, + pages = {721-741}, + doi = {10.1109/TPAMI.1984.4767596} +} + + +@book{Judea88, + author = {Pearl, Judea}, + title = {Probabilistic reasoning in intelligent systems: networks of + plausible inference}, + series = {The Morgan Kaufmann Series in Representation and Reasoning}, + publisher = {Morgan Kaufmann, San Mateo, CA}, + year = {1988}, + pages = {xx+552}, + isbn = {0-934613-73-7}, + mrclass = {68-02 (68T01 92A25)}, + mrnumber = {965765}, + mrreviewer = {V. Yu. Trakhtman} +} + + @article{sympy, + title = {SymPy: symbolic computing in Python}, + author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony}, + year = 2017, + month = jan, + keywords = {Python, Computer algebra system, Symbolics}, + abstract = { + SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy. + }, + volume = 3, + pages = {e103}, + journal = {PeerJ Computer Science}, + issn = {2376-5992}, + url = {https://doi.org/10.7717/peerj-cs.103}, + doi = {10.7717/peerj-cs.103} +} + +@inproceedings{verreet2022inference, + title = {Inference and learning with model uncertainty in probabilistic logic programs}, + author = {Verreet, Victor and Derkinderen, Vincent and Dos Martires, Pedro Zuidberg and De Raedt, Luc}, + booktitle = {Proceedings of the AAAI Conference on Artificial Intelligence}, + volume = {36}, + number = {9}, + pages = {10060--10069}, + year = {2022} +} + +@article{cropper2022inductive, + title = {Inductive logic programming at 30}, + author = {Cropper, Andrew and Duman{\v{c}}i{\'c}, Sebastijan and Evans, Richard and Muggleton, Stephen H}, + journal = {Machine Learning}, + volume = {111}, + number = {1}, + pages = {147--172}, + year = {2022}, + publisher = {Springer} +} + +@article{cozman2020joy, + title = {The joy of probabilistic answer set programming: semantics, complexity, expressivity, inference}, + author = {Cozman, Fabio Gagliardi and Mau{\'a}, Denis Deratani}, + journal = {International Journal of Approximate Reasoning}, + volume = {125}, + pages = {218--239}, + year = {2020}, + publisher = {Elsevier} +} + +@article{gebser2012answer, + title = {Answer set solving in practice}, + author = {Gebser, Martin and Kaminski, Roland and Kaufmann, Benjamin and Schaub, Torsten}, + journal = {Synthesis lectures on artificial intelligence and machine learning}, + volume = {6}, + number = {3}, + pages = {1--238}, + year = {2012}, + publisher = {Morgan \& Claypool Publishers} +} + + +@article{bezanson2017julia, + author = {Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and Shah, Viral B.}, + title = {Julia: A Fresh Approach to Numerical Computing}, + journal = {SIAM Review}, + volume = {59}, + number = {1}, + pages = {65-98}, + year = {2017}, + doi = {10.1137/141000671}, + %url = {https://doi.org/10.1137/141000671}, + %eprint = {https://doi.org/10.1137/141000671}, + abstract = { Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is designed to be easy and fast and questions notions generally held to be “laws of nature" by practitioners of numerical computing: \beginlist \item High-level dynamic programs have to be slow. \item One must prototype in one language and then rewrite in another language for speed or deployment. \item There are parts of a system appropriate for the programmer, and other parts that are best left untouched as they have been built by the experts. \endlist We introduce the Julia programming language and its design---a dance between specialization and abstraction. Specialization allows for custom treatment. Multiple dispatch, a technique from computer science, picks the right algorithm for the right circumstance. Abstraction, which is what good computation is really about, recognizes what remains the same after differences are stripped away. Abstractions in mathematics are captured as code through another technique from computer science, generic programming. Julia shows that one can achieve machine performance without sacrificing human convenience. } +} + +@article{gowda2021high, + title={High-performance symbolic-numerics via multiple dispatch}, + author={Gowda, Shashi and Ma, Yingbo and Cheli, Alessandro and Gwozdz, Maja and Shah, Viral B and Edelman, Alan and Rackauckas, Christopher}, + journal={arXiv preprint arXiv:2105.03949}, + year={2021} +} + +@article{bouchetvalat2023dataframes, + title={DataFrames.jl: Flexible and Fast Tabular Data in Julia}, + volume={107}, + %url={https://www.jstatsoft.org/index.php/jss/article/view/v107i04}, + doi={10.18637/jss.v107.i04}, + abstract={DataFrames.jl is a package written for and in the Julia language offering flexible and efficient handling of tabular data sets in memory. Thanks to Julia’s unique strengths, it provides an appealing set of features: Rich support for standard data processing tasks and excellent flexibility and efficiency for more advanced and non-standard operations. We present the fundamental design of the package and how it compares with implementations of data frames in other languages, its main features, performance, and possible extensions. We conclude with a practical illustration of typical data processing operations.}, + number={4}, + journal={Journal of Statistical Software}, + author={Bouchet-Valat, Milan and Kamiński, Bogumił}, + year={2023}, + pages={1--32} +} + +@book{riguzzi2022foundations, + address = {New York}, + edition = {1}, + title = {Foundations of {Probabilistic} {Logic} {Programming}: {Languages}, {Semantics}, {Inference} and {Learning}}, + isbn = {978-1-00-333819-2}, + shorttitle = {Foundations of {Probabilistic} {Logic} {Programming}}, + %url = {https://www.taylorfrancis.com/books/9781003338192}, + language = {en}, + urldate = {2023-03-01}, + publisher = {River Publishers}, + author = {Riguzzi, Fabrizio}, + month = sep, + year = {2022}, + doi = {10.1201/9781003338192}, +} + +@inproceedings{sato1995statistical, + title={A Statistical Learning Method for Logic Programs with Distribution Semantics}, + author={Taisuke Sato}, + booktitle={International Conference on Logic Programming}, + year={1995}, + %url={https://api.semanticscholar.org/CorpusID:10424169} +} + + +@article{lifschitz2002answer, + title = {Answer set programming and plan generation}, + volume = {138}, + issn = {0004-3702}, + %url = {https://www.sciencedirect.com/science/article/pii/S0004370202001868}, + doi = {https://doi.org/10.1016/S0004-3702(02)00186-8}, + abstract = {The idea of answer set programming is to represent a given computational problem by a logic program whose answer sets correspond to solutions, and then use an answer set solver, such as smodels or dlv, to find an answer set for this program. Applications of this method to planning are related to the line of research on the frame problem that started with the invention of formal nonmonotonic reasoning in 1980.}, + number = {1}, + journal = {Artificial Intelligence}, + author = {Lifschitz, Vladimir}, + year = {2002}, + keywords = {Answer sets, Default logic, Frame problem, Logic programming, Planning}, + pages = {39--54}, +} + +@inproceedings{lee2016weighted, + title={Weighted rules under the stable model semantics}, + author={Lee, Joohyung and Wang, Yi}, + booktitle={Fifteenth international conference on the principles of knowledge representation and reasoning}, + year={2016} +} + +@article{baral2009probabilistic, + title={Probabilistic reasoning with {A}nswer {S}ets}, + author={Baral, Chitta and Gelfond, Michael and Rushton, Nelson}, + journal={Theory and Practice of Logic Programming}, + volume={9}, + number={1}, + pages={57--144}, + year={2009}, + publisher={Cambridge University Press} +} + +@inproceedings{de2007problog, + title={ProbLog: A probabilistic {P}rolog and its application in link discovery}, + author={De Raedt, Luc and Kimmig, Angelika and Toivonen, Hannu and Veloso, M}, + booktitle={IJCAI 2007, Proceedings of the 20th international joint conference on artificial intelligence}, + pages={2462--2467}, + year={2007}, + organization={IJCAI-INT JOINT CONF ARTIF INTELL} +} + +@inproceedings{lee2017lpmln, + title={LPMLN, {W}eak {C}onstraints, and {P}-log}, + author={Lee, Joohyung and Yang, Zhun}, + booktitle={Proceedings of the AAAI Conference on Artificial Intelligence}, + volume={31}, + number={1}, + year={2017} +} + +@article{pajunen2021solution, + title={Solution enumeration by optimality in {A}nswer {S}et {P}rogramming}, + author={Pajunen, Jukka and Janhunen, Tomi}, + journal={Theory and Practice of Logic Programming}, + volume={21}, + number={6}, + pages={750--767}, + year={2021}, + publisher={Cambridge University Press} +} + + +@article{alberti2017cplint, + title={cplint on SWISH: Probabilistic logical inference with a web browser}, + author={Alberti, Marco and Bellodi, Elena and Cota, Giuseppe and Riguzzi, Fabrizio and Zese, Riccardo}, + journal={Intelligenza Artificiale}, + volume={11}, + number={1}, + pages={47--64}, + year={2017}, + publisher={IOS Press} +} diff --git a/text/presentation_01/00base/abstract.md b/text/presentation_01/00base/abstract.md new file mode 100644 index 0000000..1ffc91a --- /dev/null +++ b/text/presentation_01/00base/abstract.md @@ -0,0 +1,13 @@ +We address the problem of extending probability from the total choices of an ASP specification to the stable models and, from there, to general events. + +Our approach is algebraic in the sense that it relies on an equivalence relation over the set of events and uncertainty is expressed with variables and polynomial expressions. + +We frame our work in the context of machine learning and induction of logic problems, the two (current) forms of artificial intelligence. + +References: + +- https://arxiv.org/abs/1801.00631, Gary Marcus, Deep Learning: A Critical Appraisal, 2018. +- https://arxiv.org/abs/1911.01547, François Chollet, On the Measure of Intelligence, 2019. +- https://arxiv.org/abs/1801.00631, Bengio et al., A Meta-Transfer Objective for Learning to Disentangle Causal Mechanisms, 2019. +- https://arxiv.org/abs/1801.00631, Cropper et al., Turning 30: New Ideas in Inductive Logic Programming, 2020. +- https://doi.org/10.1201/9781003427421, Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming, 2023. \ No newline at end of file diff --git a/text/presentation_01/00base/color.lp b/text/presentation_01/00base/color.lp new file mode 100644 index 0000000..11a6225 --- /dev/null +++ b/text/presentation_01/00base/color.lp @@ -0,0 +1,15 @@ +node(1..6). + +edge(1,2). edge(2,4). edge(3,1). +edge(4,1). edge(5,3). edge(6,2). +edge(1,3). edge(2,5). edge(3,4). +edge(4,2). edge(5,4). edge(6,3). +edge(1,4). edge(2,6). edge(3,5). +edge(5,6). edge(6,5). + +color(r). color(b). color(g). + +1 { color(X,C) : color(C) } 1 :- node(X). +:- edge(X,Y), color(X,C), color(Y,C). + +#show color/2. \ No newline at end of file diff --git a/text/presentation_01/00base/drafts.ipynb b/text/presentation_01/00base/drafts.ipynb new file mode 100644 index 0000000..b6c2d3d --- /dev/null +++ b/text/presentation_01/00base/drafts.ipynb @@ -0,0 +1,11322 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "# using Plots" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using RDatasets" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "iris = dataset(\"datasets\", \"iris\");\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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1.6 & 0.2 & setosa & magenta \\\\\n", + "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n", + "\\end{tabular}\n" + ], + "text/plain": [ + "\u001b[1m150×6 DataFrame\u001b[0m\n", + "\u001b[1m Row \u001b[0m│\u001b[1m SepalLength \u001b[0m\u001b[1m SepalWidth \u001b[0m\u001b[1m PetalLength \u001b[0m\u001b[1m PetalWidth \u001b[0m\u001b[1m Species \u001b[0m\u001b[1m Color \u001b[0m\n", + " │\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Cat… \u001b[0m\u001b[90m String \u001b[0m\n", + "─────┼──────────────────────────────────────────────────────────────────────\n", + " 1 │ 5.1 3.5 1.4 0.2 setosa magenta\n", + " 2 │ 4.9 3.0 1.4 0.2 setosa magenta\n", + " 3 │ 4.7 3.2 1.3 0.2 setosa magenta\n", + " 4 │ 4.6 3.1 1.5 0.2 setosa magenta\n", + " 5 │ 5.0 3.6 1.4 0.2 setosa magenta\n", + " 6 │ 5.4 3.9 1.7 0.4 setosa magenta\n", + " 7 │ 4.6 3.4 1.4 0.3 setosa magenta\n", + " 8 │ 5.0 3.4 1.5 0.2 setosa magenta\n", + " 9 │ 4.4 2.9 1.4 0.2 setosa magenta\n", + " 10 │ 4.9 3.1 1.5 0.1 setosa magenta\n", + " 11 │ 5.4 3.7 1.5 0.2 setosa magenta\n", + " ⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮\n", + " 141 │ 6.7 3.1 5.6 2.4 virginica cyan\n", + " 142 │ 6.9 3.1 5.1 2.3 virginica cyan\n", + " 143 │ 5.8 2.7 5.1 1.9 virginica cyan\n", + " 144 │ 6.8 3.2 5.9 2.3 virginica cyan\n", + " 145 │ 6.7 3.3 5.7 2.5 virginica cyan\n", + " 146 │ 6.7 3.0 5.2 2.3 virginica cyan\n", + " 147 │ 6.3 2.5 5.0 1.9 virginica cyan\n", + " 148 │ 6.5 3.0 5.2 2.0 virginica cyan\n", + " 149 │ 6.2 3.4 5.4 2.3 virginica cyan\n", + " 150 │ 5.9 3.0 5.1 1.8 virginica cyan\n", + "\u001b[36m 129 rows omitted\u001b[0m" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "unique(iris.Species)\n", + "species_color = Dict(\"virginica\" => \"cyan\", \"setosa\" => \"magenta\", \"versicolor\" => \"yellow\")\n", + "transform!(iris, :Species => ByRow(s -> species_color[s]) => :Color)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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"cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "WARNING: both Gadfly and Plots export \"plot\"; uses of it in module Main must be qualified\n" + ] + }, + { + "ename": "LoadError", + "evalue": "UndefVarError: `plot` not defined", + "output_type": "error", + "traceback": [ + "UndefVarError: `plot` not defined", + "", + "Stacktrace:", + " [1] top-level scope", + " @ In[8]:1" + ] + } + ], + "source": [ + "plot(dataset(\"datasets\",\"iris\"), x=\"SepalWidth\", y=\"SepalLength\", color=\"Species\")" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + " \n", + " \n", + " \n", + " SepalLength\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " 4\n", + " \n", + " \n", + " \n", + " \n", + " 5\n", + " \n", + " \n", + " \n", + " \n", + " 6\n", + " 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" \n", + " \n", + " \n", + " 2\n", + " \n", + " \n", + " \n", + " \n", + " 3\n", + " \n", + " \n", + " \n", + " \n", + " 4\n", + " \n", + " \n", + " \n", + " \n", + " 5\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " SepalWidth\n", + " \n", + " \n", + " \n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "Plot(...)" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using RDatasets, Gadfly, Compose\n", + "set_default_plot_size(131.25mm, 90mm)\n", + "iris = dataset(\"datasets\",\"iris\")\n", + "p = plot(iris, \n", + " layer(x= :SepalLength, y= :SepalWidth, color= :Species),\n", + " layer(x -> 0.65 * x - 0.4, 4, 7.5, color=[colorant\"red\"], style(line_width=1mm)),\n", + " Guide.colorkey(title=\"Iris\", pos=[0.05w,-0.28h]))\n" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "import Cairo,\n", + "Fontconfig\n", + "draw(PDF(\"iris_plot.pdf\", 131.25mm, 90mm), p)" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2-element Vector{Float64}:\n", + " 131.25\n", + " 90.0" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "1.25 .* [105, 72]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.9.3", + "language": "julia", + "name": "julia-1.9" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.9.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/text/presentation_01/00base/iris_plot.pdf b/text/presentation_01/00base/iris_plot.pdf new file mode 100644 index 0000000..ec92c9b Binary files /dev/null and b/text/presentation_01/00base/iris_plot.pdf differ diff --git a/text/presentation_01/00base/iris_plot.svg b/text/presentation_01/00base/iris_plot.svg new file mode 100644 index 0000000..b4f9bec --- /dev/null +++ b/text/presentation_01/00base/iris_plot.svg @@ -0,0 +1,2295 @@ + + + + + + + + + + + + SepalLength + + + + + + + 4 + + + + + 5 + + + + + 6 + + + + + 7 + + + + + 8 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 5.9,3.0 + + + + + + + + + + + + + + 6.2,3.4 + + + + + + + + + + + + + + 6.5,3.0 + + + + + + + + + + + + + + 6.3,2.5 + + + + + + + + + + + + + + 6.7,3.0 + + + + + + + + + + + + + + 6.7,3.3 + + + + + + + + + + + + + + 6.8,3.2 + + + + + + + + + + + + + + 5.8,2.7 + + + + + + + + + + + + + + 6.9,3.1 + + + + + + + + + + + + + + 6.7,3.1 + + + + + + + + + + + + + + 6.9,3.1 + + + + + + + + + + + + + + 6.0,3.0 + + + + + + + + + + + + + + 6.4,3.1 + + + + + + + + + + + + + + 6.3,3.4 + + + + + 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+ + + + + + + + + 2.0 + + + + + 2.5 + + + + + 3.0 + + + + + 3.5 + + + + + 4.0 + + + + + 4.5 + + + + + + + SepalWidth + + + + + + + + + + diff --git a/text/presentation_01/00base/pqueens.lp b/text/presentation_01/00base/pqueens.lp new file mode 100644 index 0000000..18a3d70 --- /dev/null +++ b/text/presentation_01/00base/pqueens.lp @@ -0,0 +1,21 @@ +% Column and row domains. +col(1 .. n). +row(1 .. n). + +% One queen on some row per colum. +1 { queen(I, J) : row(J) } 1 :- col(I). + +% One queen on some columns per row. +1 { queen(I, J) : col(I) } 1 :- row(J). + +% Number descending diagonals. +diag1(I, J, I - J + n) :- col(I), row(J). +% Number ascending diagonals. +diag2(I, J, I + J - 1) :- col(I), row(J). +% +% Negative Restrictions +:- D = 1 .. 2 * n - 1, not { queen(I, J) : diag1(I, J, D) } 1. +:- D = 1 .. 2 * n - 1, not { queen(I, J) : diag2(I, J, D) } 1. +% +% Output this predicate. +#show queen/2. \ No newline at end of file diff --git a/text/presentation_01/00base/probasp.pdf b/text/presentation_01/00base/probasp.pdf new file mode 100644 index 0000000..ac26505 Binary files /dev/null and b/text/presentation_01/00base/probasp.pdf differ diff --git a/text/presentation_01/00base/probasp.tex b/text/presentation_01/00base/probasp.tex new file mode 100644 index 0000000..c50e518 --- /dev/null +++ b/text/presentation_01/00base/probasp.tex @@ -0,0 +1,1203 @@ +% !TeX program=xelatex +\documentclass[bigger,xcolor={x11names,svgnames}]{beamer} + +\useinnertheme{circles} +\usefonttheme[onlymath]{serif} +\usefonttheme{structurebold} +\setbeamertemplate{navigation symbols}{} +\setbeamertemplate{footline}[frame number] +%\usepackage[x11names,usenames]{xcolor} +\setbeamercolor{highlight block}{bg=gray} +\setbeamercolor{alerted text}{fg=Crimson} +\usepackage{tikz} +\usetikzlibrary{ +automata,% +positioning,% +calc,% +patterns, +decorations,% +decorations.pathmorphing,% +arrows.meta, +shapes.misc +} +\tikzset{ + vert/.style={draw, circle, inner sep=2pt}, + event/.style={}, + smodel/.style={fill=gray!25}, + tchoice/.style={draw, circle}, + indep/.style={},%{draw, dashed}, + proptc/.style = {-latex, dashed}, + propsm/.style = {-latex, thick}, + doubt/.style = {gray} +} + +\usepackage{tkz-graph} +\newcommand{\qlr}[2]{\ensuremath{\begin{matrix}#1\cr\begin{aligned}\hline #2\end{aligned}\end{matrix}}} + +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[normalem]{ulem} % To strikeout +\usepackage{commath} +\usepackage{amssymb} +\usepackage[normalem]{ulem} +\usepackage[nice]{nicefrac} +\usepackage{stmaryrd} +\usepackage{acronym} +\usepackage{multicol} +\usepackage{cleveref} + +\usepackage{listings} + +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} +\newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}} +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} +\newcommand{\condsymb}[2]{\ensuremath{p_{#1|#2}}} +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} +\newcommand{\deft}[1]{\textbf{#1}} +\newcommand{\emptyevent}{\ensuremath{\vartriangle}} +\newcommand{\ent}{\ensuremath{\lhd}} +\newcommand{\err}[1]{\ensuremath{\mathrm{err}\at{#1}}} +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} +\newcommand{\given}{\ensuremath{~\middle|~}} +\newcommand{\inconsistent}{\bot} +\newcommand{\indepclass}{\ensuremath{\Diamond}} +\newcommand{\isep}{~,~} +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} +\newcommand{\larr}{\ensuremath{\leftarrow}} +\newcommand{\lclass}[1]{\ensuremath{\intoc{#1}}} +\newcommand{\lfrac}[2]{\ensuremath{{#1}/{#2}}} +\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}} +\newcommand{\lset}[1]{\ensuremath{#1_{\ast}}} +\newcommand{\naf}{\ensuremath{\sim\!}} +\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}} +\newcommand{\probfact}[2]{\ensuremath{#2:#1}} +\newcommand{\probrule}[3]{\probfact{#2}{#1} \leftarrow #3} +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} +\newcommand{\pw}[1]{\ensuremath{\mu\at{#1}}} +\newcommand{\pwc}[1]{\ensuremath{\pwcfname\at{#1}}} +\newcommand{\pwcfname}{\ensuremath{\mu_{\textrm{TC}}}} +\newcommand{\q}[1]{\mathbf{#1}} +\newcommand{\rep}[2]{\ensuremath{\left\langle #1 \middle| #2 \right\rangle}} +\newcommand{\smclass}[1]{\ensuremath{\intcc{#1}}} +\newcommand{\stablecore}[1]{\ensuremath{\left\llbracket #1 \right\rrbracket}} +\newcommand{\tcgen}[1]{\ensuremath{\left<#1\right>}} +\newcommand{\uclass}[1]{\ensuremath{\intco{#1}}} +\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}} +\newcommand{\uset}[1]{\ensuremath{#1^{\ast}}} +\newcommand{\yset}[1]{\ensuremath{\left\langle #1 \right\rangle}} +%\newcommand{\tcgen}[1]{\ensuremath{\widehat{#1}}} + +\title{Stochastic Answer Set Programming} +\subtitle{A Research Program} +\author{Francisco Coelho} +\date{November 15, 2023} +\institute[\texttt{fc@uevora.pt}]{ +NOVA LINCS\\ +High Performance Computing Chair\\ +Departamento de Informática, Universidade de Évora +} +% +%=============================================================== +% +\begin{document} +% +\lstset{language=Prolog} +% +%=============================================================== +% +\begin{frame}[plain] + \titlepage + + \begin{center} + \footnotesize This is a join work with \textbf{Salvador Abreu}@DInf and \textbf{Bruno Dinis}@DMat. + \end{center} +\end{frame} +% +%=============================================================== +% +\section*{Motivation} +% +%=============================================================== +% +\begin{frame} + \frametitle{In Short} + + + \begin{itemize} + \item About \textbf{Machine Learning}: + \begin{itemize} + \item Vector or matrix based models lack ``structure''. + \item Large models don't \emph{explain} data. + \end{itemize} + \item About \textbf{Logic Programs}: + \begin{itemize} + \item Logic programs formalize knowledge. + \item Logic doesn't \emph{capture} uncertainty and is \emph{fragile} to noise. + \end{itemize} + \item \textbf{Probabilistic Logic Programs} extend formal knowledge with probabilities. + \begin{itemize} + \item How to propagate probabilities through rules? + \end{itemize} + \end{itemize} + \vfill + \begin{center} + \alert{\bf Goal:} Combine Logic and Statistics. + \end{center} +\end{frame} +% +%=============================================================== +% \begin{frame} +% %------------------------------------------------------------- +% %[fragile] +% %------------------------------------------------------------- +% \frametitle{Statistics and Machine Learning} +% %------------------------------------------------------------- +% \vfill +% \begin{itemize} +% \item \textbf{Data Analysis:} understand and summarize. +% \item \textbf{Model Building:} tools and techniques. +% \item \textbf{Model Evaluation:} performance. +% \item [] +% \item To calculate the accuracy/precision/recall of a model. +% \item To choose the right model for a problem. +% \item To tune the hyperparameters of a model. +% \end{itemize} +% \vfill +% \begin{center}\footnotesize +% Highlights of \texttt{Bard}'s (Google's LLM) reply to ``Explain what is the role of statistics in machine learning''. +% \end{center} +% \end{frame} +% +%=============================================================== +% +\section{Machine Learning} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\bf Machine Learning} + \end{center} + \vfill + \begin{itemize} + \item Standard Example --- Iris Classification + \item Assumptions of Machine Learning + \item Where Machine Learning Fails + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{The Standard Example --- Iris Classification} + %------------------------------------------------------------- + \small + \begin{center} + Learning Functions: \href{https://en.wikipedia.org/wiki/Iris_flower_data_set}{The famous Iris database} + \end{center} + \begin{columns} + \column{0.4\textwidth} + \begin{itemize}\setlength{\itemsep}{-0.5em} + \item[$x_1$] sepal length. + \item[$x_2$] sepal width. + \item[$x_3$] petal length. + \item[$x_4$] petal width. + \item[$y$] species (one of \emph{setosa}, \emph{versicolor}, \emph{virginica}). + \end{itemize} + \column{0.6\textwidth} + \begin{center} + \includegraphics[width=\textwidth]{iris_plot.pdf} + \end{center} + \end{columns} + \vfill + \begin{itemize} + \item A \emph{setosa} model: $ - 0.40 -0.65x_1 + 1.00x_2 > 0.00$. + \item A general \textbf{model template}: + $$ + f_\theta(\vec{x}) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_3 + \theta_4 x_4> 0 + $$ + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Assumptions of Machine Learning} + %------------------------------------------------------------- + \begin{itemize} + \item Each instance is described in a \textbf{single row} by a \textbf{fixed set of features} + $$ + \begin{array}{cccc|c} + \mathbf{x_1} & \mathbf{x_2} & \ldots & \mathbf{x_n} & \mathbf{y} \\ + \hline + x_{11} & x_{21} & \ldots & x_{n1} & y_1 \\ + & & \vdots \\ + x_{1m} & x_{2m} & \ldots & x_{nm} & y_m \\ + \end{array} + .$$ + \item Instances are \textbf{independent} of one another, \textbf{given the model} + $$ + y = f_\theta(\vec{x}). + $$ + \item Parameters \textbf{minimize estimation error} \emph{e.g.} + $$ + \hat{\theta} = \arg\min_\theta \sum_i \norm{y_i - f_\theta(\vec{x_{i}})}. + $$ + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Failing Assumptions\hfill~1/2} + %------------------------------------------------------------- + \vfill + \begin{columns}[c] + \column{0.5\textwidth} + $$ + \begin{array}{ccc} + \text{student} & \text{course} & \text{grade} \\ + \hline + s_1 & c_1 & a \\ + s_2 & c_1 & c \\ + s_1 & c_2 & b \\ + s_2 & c_3 & b \\ + s_3 & c_2 & b \\ + s_4 & c_3 & b \\ + s_3 & c_4 & X \\ + s_4 & c_4 & Y \\ + \end{array} + $$ + \column{0.5\textwidth} + \emph{Infer $X \succ_g Y$.} + + \bigskip + ``\emph{Proof.}'' + + \bigskip + $s_1 \succ_s s_2$ (rows 1 \& 2)\\ so $c_2 \succ_c c_3$ (3 \& 4). + + \bigskip + Then $s_3 \succ_s s_4$ (5 \& 6)\\ therefore $X \succ_g Y\qed$. + \end{columns} + \vfill + \begin{itemize} + \item Instances are not independent. + \item Small dataset, so \emph{no statistics}. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Failing Assumptions\hfill~2/2} + %------------------------------------------------------------- + \footnotesize + \begin{center} + \textbf{Prompt:} Given \texttt{f(machine) = e}, \texttt{f(learning) = g} and \texttt{f(algorithm) = m}, what is the value of \texttt{f(ksadbf)}? + \end{center} + + \vfill + \begin{columns}[t] + \scriptsize + \column{0.35\textwidth} + $$ + \begin{array}{r|l} + \textbf{input} & \textbf{output} \\ + \hline + machine & e \\ + learning & g \\ + algorithm & m \\ + ksadbf & ? + \end{array} + $$ + + \bigskip + \textbf{Bard:} \tt f(ksadbf) = 11. + + We can see that the value of f(x) is simply the first letter of x minus 96. + + \column{0.65\textwidth} + + \textbf{ChatGPT:} \tt The information provided seems to define a function $f$ where certain inputs are associated with specific outputs. Based on the given information: + \begin{itemize} + \item $f(\text{machine}) = e$ + \item $f(\text{learning}) = g$ + \item $f(\text{algorithm}) = m$ + \end{itemize} + However, there is no information given about \( f(\text{ksadbf}) \). Without additional information or a specific rule that determines the output for \( f(\text{ksadbf}) \), \emph{it is not possible to determine its value}. + % + If there is a pattern or rule for how the outputs are assigned based on the inputs, you would need to provide that information for me to help determine the value of \( f(\text{ksadbf}) \). + \end{columns} + + \vfill + \begin{center} + No help from state-of-the-art Natural Language Processing or LLMs. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Where Machine Learning Fails} + %------------------------------------------------------------- + \begin{itemize} + \item No \emph{interpretability}: large models ($\dim \theta \sim \text{millions}$) are \alert{not understandable by humans}. + \item Need for \emph{large number of training} examples. + \item Unfit for \emph{multiple relations}, \emph{intersample dependencies} and \emph{variable features}: The \textit{Students, Courses and Results} and \emph{Last Letter} examples. + \item Poor \emph{generalization}: A model trained in the iris dataset is useless in any other domain. + \item \alert{No use of background knowledge}. + \end{itemize} +\end{frame} +% +%=============================================================== +% +\section{Logic Programming} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\bf Logic Programming} + \end{center} + \vfill + \begin{itemize} + \item An Example of Logic Programming. + \item Inductive Logic Programming. + \item Where ILP Fails. + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + [fragile] + %------------------------------------------------------------- + \frametitle{An Example of Logic Programming} + %------------------------------------------------------------- + \begin{columns}[c] + \column{0.4\textwidth} + \begin{tikzpicture}[>=Latex] + \node[vert] (v6) {6}; + \node[vert, above left = of v6] (v3) {3}; + \node[vert, above right = of v6] (v5) {5}; + \node[vert, below left = of v6] (v1) {1}; + \node[vert, below right = of v6] (v2) {2}; + \node[vert, below right = of v1] (v4) {4}; + \draw[->] (v6) to (v3); + \draw[<->] (v6) to (v5); + \draw[<->] (v6) to (v2); + \draw[<->] (v3) to (v5); + \draw[<->] (v3) to (v1); + \draw[->] (v3) to[bend right, out=225, in =180,relative=false] (v4); + \draw[->] (v1) to (v2); + \draw[<->] (v1) to (v4); + \draw[->] (v2) to (v5); + \draw[<->] (v2) to (v4); + \draw[->] (v5) to[bend left, out=-45, in=0,relative=false] (v4); + \end{tikzpicture} + \column{0.6\textwidth} + \scriptsize + \begin{lstlisting} +node(1..6). + +edge(1,2). edge(2,4). edge(3,1). +edge(4,1). edge(5,3). edge(6,2). +edge(1,3). edge(2,5). edge(3,4). +edge(4,2). edge(5,4). edge(6,3). +edge(1,4). edge(2,6). edge(3,5). +edge(5,6). edge(6,5). + +col(r). col(b). col(g). + +1 { color(X,C) : col(C) } 1 :- node(X). +:- edge(X,Y), color(X,C), color(Y,C). + +#show color/2. + \end{lstlisting} + \end{columns} + \vfill + \scriptsize + \begin{lstlisting} +color(2,b) color(1,g) color(4,r) color(3,b) color(5,g) color(6,r) +color(1,r) color(2,b) color(4,g) color(3,b) color(5,r) color(6,g) +color(1,r) color(2,g) color(4,b) color(3,g) color(5,r) color(6,b) +color(1,b) color(2,g) color(4,r) color(3,g) color(5,b) color(6,r) +color(2,r) color(1,g) color(4,b) color(3,r) color(5,g) color(6,b) +color(2,r) color(1,b) color(4,g) color(3,r) color(5,b) color(6,g) + \end{lstlisting} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Inductive Logic Programming} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + Learning Logic Programs from Examples. + \end{center} + + \vfill + \alert{Generate rules} that\ldots + \begin{itemize} + \item use \textbf{background knowledge} + $$ + \begin{aligned} + & parent(john,mary), & & parent(david,steve), \\ + & parent(kathy,mary), & & female(kathy), \\ + & male(john), & & male(david). + \end{aligned} + $$ + \item to entail all the \textbf{positive examples}, $father(john,mary), father(david,steve)$, + \item but none of the \textbf{negative examples}. $father(kathy,mary), father(john,steve)$, + + \end{itemize} + + \vfill + A \textbf{solution} is $$father(X,Y) \leftarrow parent(X,Y) \wedge male(X).$$ + +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Where Logic Programming Fails} + %------------------------------------------------------------- + \begin{center} + Meanwhile, in the \textbf{real world}, samples are \emph{incomplete} and come with \emph{noise}. + \end{center} + \vfill + \textbf{Logic inference is \alert{fragile}}: a mistake in the transcription of a fact is dramatic to the consequences: + \begin{itemize} + \item $parent(david,mary)$. + \item $parent(jonh,mary)$. + \end{itemize} + \vfill + \begin{center} + The statistic essence of machine learning provides \alert{robustness}. + \end{center} +\end{frame} +% +%=============================================================== +% +\section{Probabilistic Logic Programming} +% +%=============================================================== +% +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \vfill + \begin{center} + {\huge\bf Probabilistic Logic Programming} + \end{center} + \vfill + \begin{itemize} + \item Define distributions from logic programs. + \item Stochastic ASP: Specifying distributions. + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilistic Logic Programs (PLPs)} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + Logic programs \textbf{annotated} with probabilities. + \end{center} + + \vfill + \begin{columns}[c] + + \column{0.4\textwidth} + \begin{tikzpicture}[>=Latex] + \node[draw, rounded rectangle] (A) {$Alarm$}; + \node[draw, rounded rectangle, below = of A] (J) {$Johncalls$}; + \draw[->] (A) to (J); + \end{tikzpicture} + + \column{0.6\textwidth} + $ + \begin{aligned} + alarm:0.00251, & \\ + johncalls:0.9 & \leftarrow alarm, \\ + johncalls:0.05 & \leftarrow \neg alarm + \end{aligned} + $ + \end{columns} + + \vfill + \begin{itemize} + \item \alert{$alarm:0.00251$} is $alarm \vee \neg alarm$ plus $P(Alarm = true) = 0.00251$. + \item \alert{$johncalls:0.9 \leftarrow alarm$} is + $$P\del{Johncalls = true \middle| Alarm = true} = 0.9$$ + \end{itemize} + + \vfill + \begin{center} + Any bayesian network can be represented by a PLP. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Distributions from Logic Programs} + %------------------------------------------------------------- + \vfill + The program + $$ + \begin{aligned} + alarm:0.00251, & \\ + johncalls:0.9 & \leftarrow alarm, \\ + johncalls:0.05 & \leftarrow \neg alarm + \end{aligned} + $$ + entails four possible models (or worlds): + $$ + \begin{array}{r|r} + \text{model} & \text{probability} \\ + \hline + alarm, johncalls & 0.002259 \\ + alarm, \neg johncalls & 0.000251 \\ + \neg alarm, johncalls & 0.049874 \\ + \neg alarm, \neg johncalls & 0.947616 \\ + \end{array} + $$ + \begin{itemize} + \item \alert{\textbf{Models}} are special sets of \emph{literals} \textbf{entailed} from the program. + \item Probabilities \emph{propagate} from facts, through rules. + \end{itemize} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{There's a Problem\ldots} + %------------------------------------------------------------- + \vfill + The program + $$ + \begin{aligned} + alarm:0.00251, & \\ + johncalls \vee marycalls & \leftarrow alarm + \end{aligned} + $$ + entails three \alert{stable} (\emph{i.e.}\ minimal) models + $$ + \begin{array}{r|l} + \text{model} & \text{probability} \\ + \hline + alarm, johncalls & x \\ + alarm, marycalls & y \\ + \neg alarm & 0.99749 + \end{array} + $$ + but \textbf{no single way to set $x,y$}. + \vfill + \begin{center} + Some \emph{Probabilistic Logic Programs} define more than one joint distribution. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{\ldots and an Oportunity} + %------------------------------------------------------------- + \begin{center} + Some \emph{PLP}s define more than one joint distribution. + \end{center} + \vfill + \begin{itemize} + \item There is \textbf{no single probability assignment} from the facts stable models: $x,y \in \intcc{0,1}$. + \item But any assignment is bound by Kolmogorov's axioms, and \textbf{forms equations} such as: + $$x + y = P\del{alarm}.$$ + \item Existing \textbf{data can be used to estimate the unknowns} in those equations. + \end{itemize} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Stable Models, Events and Probabilities} + %------------------------------------------------------------- + \vfill + \begin{center} + What are we talking about? + \end{center} + \vfill + \begin{itemize} + \item A logic program has \textbf{atoms} (and \textbf{literals}) and \textbf{rules}: + $$ + \begin{aligned} + &male(john), \neg parent(kathy,mary), \\ + &father(X, Y) \leftarrow parent(X, Y) \wedge male(X). + \end{aligned} + $$ + \item A \alert{\textbf{stable model}} is a \textbf{minimal} model that contains: + \begin{itemize} + \item program's \emph{facts}: $parent(john,mary),~male(john)$. + \item consequences, by the \emph{rules}: $father(john,mary)$. + \end{itemize} + \item Some programs have more than one model: + \begin{tabular}{c|c} + \textbf{Logic Program} & \textbf{Stable Models} \\ + \hline + $a \vee \neg a, b \vee c \leftarrow a$ + & + $\set{\neg a}, \set{a, b}, \set{a, c}$ + \end{tabular} + \end{itemize} + \vfill + \begin{center} + How to propagate probability from annotated facts to other \emph{events}? + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Logic Programs and Probabilities} + %------------------------------------------------------------- + % \vfill + % \begin{center} + % The \textbf{space of events}, $\Omega$, is the set of all sets of literals. + % \end{center} + \vfill + \begin{itemize} + \item Consider the literals of a logic program $$L = \set{a_1, \ldots a_n, \neg a_1, \ldots \neg a_n}.$$ + \item Any model of that program is a (consistent) subset of $L$. + \item Let $\Omega = \mathbf{P}\del{L}$, \emph{i.e.} an \alert{event} $e$ is a subset of $L$, $e \subseteq L$. + \begin{itemize} + \item Setting a probability for some events seems straightforward: $P\del{\neg alarm} = 0.997483558$. + \item For others, not so much: + \begin{itemize} + \item $P\del{alarm, johncalls}$, $P\del{johncalls, marycalls, alarm}$, $P\del{marycalls}$? + \item $P\del{alarm, \neg alarm}$, $P\del{\neg marycalls}$? + \end{itemize} + \end{itemize} + \end{itemize} + \vfill + \begin{center} + How to \alert{propagate} probability from \emph{facts} to \emph{consequences} or other \emph{events}? + \end{center} +\end{frame} +% +\newcommand{\diagram}{ + \resizebox{!}{24ex}{ + \begin{tikzpicture}%[scale=0.6, every node/.style={scale=0.6}] + \node[event] (E) {$\emptyevent$}; + \node[tchoice, above left = of E, pin={[red!50!black]175:$0.3$}] (a) {$a$}; + \node[smodel, above left = of a, pin={[red!50!black]175:$0.3\theta$}] (ab) {$ab$}; + \node[smodel, above right = of a, pin={[red!50!black]175:$0.3\co{\theta}$}] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[event, above left = of ab] (abC) {$\co{c}ab$}; + \node[event, above right = of ac] (aBc) {$\co{b}ac$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc, pin={[red!50!black]175:$0.7$}] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \path[draw, rounded corners, fill=cyan, opacity=0.1] + (ab.west) -- + (ab.north west) -- + % + (abC.south west) -- + (abC.north west) -- + (abC.north) -- + % + (abc.north east) -- + (abc.east) -- + (abc.south east) -- + % + (ab.north east) -- + (ab.east) -- + (ab.south east) -- + % + (a.north east) -- + % + (E.north east) -- + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (b.south west) -- + % + (ab.west) + ; + % ---- + \path[draw, rounded corners, fill=yellow, opacity=0.1] + (ac.south west) -- + (ac.west) -- + (ac.north west) -- + % + (abc.south west) -- + (abc.west) -- + (abc.north west) -- + % + (aBc.north east) -- + (aBc.east) -- + (aBc.south east) -- + % + (ac.north east) -- + % + (c.east) -- + % + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (a.south west) -- + (a.west) -- + (a.north west) -- + (a.north) -- + % + (ac.south west) + ; + % ---- + \path[draw, rounded corners, fill=magenta, opacity=0.1] + % (A.north west) -- + % + (Ac.north west) -- + % + (Abc.north west) -- + (Abc.north) -- + (Abc.north east) -- + (Abc.south east) -- + % + % (Ac.north east) -- + % (Ac.east) -- + % + % (A.east) -- + (A.south east) -- + % + (E.south east) -- + (E.south) -- + (E.south west) -- + (E.west) -- + (E.north west) -- + % + (Ac.north west) + ; + \end{tikzpicture} + } +} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Classes of Events} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + \diagram + \end{center} + + \vfill + \begin{center} + \begin{tabular}{lr} + $\begin{aligned} + \probfact{0.3}{a} \\ + b \vee c & \leftarrow a + \end{aligned}$ + & + $\co{a} = \set{\neg a}, ab = \set{a, b}, ac = \set{a, c}$ + \end{tabular} + \end{center} + + \vfill + \begin{itemize} + \item Define \alert{equivalence classes} for all events, based on $\subseteq, \supseteq$ relations with the \textbf{stable models}. + \item This example shows $6$ out of $2^3 + 1$ classes. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilities for all Events} + %------------------------------------------------------------- + \vfill + \begin{center} + \diagram + \end{center} + \vfill + \footnotesize + \begin{enumerate} + \item Set \alert{weights} in the stable models (shaded nodes), using parameters when needed: $\mu\del{\co{a}} = 0.7; \mu\del{ab} = 0.3\theta; \mu\del{ac}=0.3\del{1 - \theta}$ + \item Assume that the stable models are \alert{disjoint events}. + \item Define \alert{weight of an event} as the sum of the weights of the related stable models. + \item Normalize weights to get a (probability) \alert{distribution}. + \end{enumerate} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilities for all Events} + %------------------------------------------------------------- + \vfill + % \begin{center} + % \diagram + % \end{center} + % \vfill + % \scriptsize + + \begin{equation*} + \begin{array}{clr|cc|cc} + & \stablecore{e} + & \# \class{e} + & \pw{\class{e}} + & \pw{e} + & \pr{E = e} + & \pr{E \in \class{e}} + \\ + \hline + % + & \inconsistent + & 37 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + \square + & \indepclass + & 9 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + {\color{magenta!20}\blacksquare } + & \co{a} + & 9 + & \frac{7}{10} + & \frac{7}{90} + & \frac{7}{207} + & \frac{7}{23} + \\[4pt] + % + {\color{cyan!20}\blacksquare } + & ab + & 3 + & \frac{3}{10}\theta + & \frac{1}{10}\theta + & \frac{1}{23}\alert{\theta} + & \frac{3}{23}\theta + \\[4pt] + % + {\color{yellow!20}\blacksquare } + & ac + & 3 + & \frac{3}{10}\co{\theta} + & \frac{1}{10}\co{\theta} + & \frac{1}{23}\alert{\co{\theta}} + & \frac{3}{23}\co{\theta} + \\[4pt] + % + & \co{a}, ab + & 0 + & \frac{7 + 3\theta}{10} + & 0 + & 0 + & 0 + \\[4pt] + % + & \co{a}, ac + & 0 + & \frac{7 + 3\co{\theta}}{10} + & 0 + & 0 + & 0 + % + \\[4pt] + % + {\color{green!20}\blacksquare } + & ab, ac + & 2 + & \frac{3}{10} + & \frac{3}{20} + & \frac{3}{46} + & \frac{3}{23} + \\[4pt] + % + {\color{gray!20}\blacksquare } + & \co{a}, ab, ac + & 1 + & 1 + & 1 + & \frac{10}{23} + & \frac{10}{23} + \\[4pt] + % + \hline + & + & 64 + & + & Z = \frac{23}{10} + & + %& \Sigma = 1 + \end{array} + \end{equation*} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Estimating the Parameters} + %------------------------------------------------------------- + A \alert{sample} can be used to estimate the parameters $\theta$, by minimizing + \begin{equation*} + \err{\theta} := \sum_{e\in\fml{E}} \del{\pr{E = e\given \Theta = \theta} - \pr{S = e}}^2.\label{eq:err.e.s} + \end{equation*} + where + \begin{itemize} + \item $\fml{E}$ is the set of all events, + \item $\pr{E\given \Theta}$ the \textbf{model+parameters} based distribution, + \item $\pr{S}$ is the \textbf{empiric} distribution from the given sample. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Behind Parameter Estimation} + %------------------------------------------------------------- + \vfill + So, we can derive a distribution $\pr{E\given \Theta = \hat{\theta}}$ from a program $P$ and a sample $S$. + \begin{itemize} + \item The sample defines an empiric distribution $\pr{S}$\ldots + \item \ldots that is used to estimate $\theta$ in $\pr{E\given \Theta}$\ldots + \item \ldots and \alert{score the program} $P$ w.r.t.\ that sample using, \emph{e.g.} the $\err{}$ function. + \end{itemize} + \vfill + +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Back to Inductive Logic Programming} + %------------------------------------------------------------- + \vfill + Recall the \emph{Learning Logic Programs from Examples} setting: + \begin{itemize} + \item Given \textbf{positive} and \textbf{negative} examples, and \textbf{background knowledge}\ldots + \item find a \textbf{program}\ldots + \begin{itemize} + \item \ldots using the facts and relations from the \textbf{BK}\ldots + \item \ldots such that \textbf{all the PE} and \textbf{none the NE} examples are entailed. + \end{itemize} + \end{itemize} + \vfill + \begin{quotation} + Given a sample of events, and a set of programs, \alert{the score} of those programs (w.r.t. the sample) \alert{can be used in evolutionary algorithms} while searching for better solutions. + \end{quotation} +\end{frame} +% +%=============================================================== +% +\section{In Conclusion} +% +%=============================================================== +% +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \vfill + \begin{center} + {\huge\bf In Conclusion} + \end{center} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + \begin{itemize} + \item \textbf{Machine Learning} has limitations. + \item As does \textbf{Inductive Logic Programming}. + \item But, distributions can be defined by \textbf{Stochastic Logic Programs}. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + Distributions can be defined by \textbf{Stochastic Logic Programs}. Here we: + \begin{enumerate} + \item Look at the program's \textbf{stable models} and + \item Use them to partition the \textbf{events} and then + \item Using annotated probabilities, define: + \begin{enumerate} + \item a finite \textbf{measure}\ldots + \item that, normalized, is a \textbf{distribution} on all events. + \end{enumerate} + \end{enumerate} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + Distributions can be defined by \textbf{Stochastic Logic Programs}. + \begin{itemize} + \item These distributions might have some \textbf{parameters}, due to indeterminism in the program. + \item A \textbf{sample} can be used to estimate those parameters\ldots + \item \ldots and \textbf{score} programs concurring to describe it. + \item This score a key ingredient in \textbf{evolutionary algorithms}. + \end{itemize} + \begin{quotation} + \ldots and a step towards the \alert{induction of stochastic logic programs} using \textbf{data} and \textbf{background knowledge}. + \end{quotation} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Future Work} + %------------------------------------------------------------- + \vfill + \begin{center} + Induction of Stochastic Logic (ASP) Programs. + \end{center} + \vfill + \begin{enumerate} + \item \textbf{Meta-programming:} formal rules for rule generation. + \item \textbf{Generation}, \textbf{Combination} and \textbf{Mutation} operators. + \item \textbf{Complexity.} + \item \textbf{Applications.} + \item \textbf{Profit.} + \end{enumerate} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\alert{\bf Thank You!}} + \vfill + Questions? + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{References} + \begin{itemize} + \item \href{https://arxiv.org/abs/1801.00631}{Gary Marcus, \emph{Deep Learning: A Critical Appraisal}, 2018}. + \item \href{https://arxiv.org/abs/1911.01547}{François Chollet, \emph{On the Measure of Intelligence}, 2019}. + \item \href{https://arxiv.org/abs/1801.00631}{Bengio \emph{et al.}, \emph{A Meta-Transfer Objective for Learning to Disentangle Causal Mechanisms}, 2019}. + \item \href{https://arxiv.org/abs/1801.00631}{Cropper \emph{et al.}, \emph{Turning 30: New Ideas in Inductive Logic Programming}, 2020}. + \item \href{https://doi.org/10.1201/9781003427421}{Fabrizio Riguzzi, \emph{Foundations of Probabilistic Logic Programming }, 2023}. + \end{itemize} + %------------------------------------------------------------- +\end{frame} +% +\end{document} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{TITLE} + %------------------------------------------------------------- +\end{frame} +% + + diff --git a/text/presentation_01/00base/scg.lp b/text/presentation_01/00base/scg.lp new file mode 100644 index 0000000..f45d374 --- /dev/null +++ b/text/presentation_01/00base/scg.lp @@ -0,0 +1,40 @@ +r(s1, c1, a). +r(s2, c1, c). +r(s1, c2, b). +r(s2, c3, b). +r(s3, c2, b). +r(s4, c3, b). +1 { r(s3, c4, G) : grade(G) } 1. +1 { r(s4, c4, G) : grade(G) } 1. + +student(s1). +student(s2). +student(s3). +student(s4). + +course(c1). +course(c2). +course(c3). +course(c4). + +grade(a). +grade(b). +grade(c). + +better(a, b). +better(b, c). +better(G1, G2) :- better(G1, G3), better(G3, G2). + +harder(C1, C2) :- r(S, C1, G1), r(S, C2, G2), better(G2, G1). +harder(C1, C2) :- harder(C1, C3), harder(C3, C2). + +smarter(S1, S2) :- r(S1, C, G1), r(S2, C, G2), better(G1, G2). +smarter(S1, S2) :- r(S1, C1, G), r(S2, C2, G), harder(C1, C2). +smarter(S1, S2) :- smarter(S1, S3), smarter(S3, S2). + +:- r(S1, C, G1), r(S2, C, G2), smarter(S1, S2), better(G2, G1). +:- r(S, C1, G1), r(S, C2, G2), harder(C1, C2), better(G1, G2). + +query(G1, G2) :- r(s3, c4, G1), r(s4, c4, G2). + +#show query/2. \ No newline at end of file diff --git a/text/presentation_01/bits_goa/bits_goa-2024-01-10.pdf b/text/presentation_01/bits_goa/bits_goa-2024-01-10.pdf new file mode 100644 index 0000000..8c0eafc Binary files /dev/null and b/text/presentation_01/bits_goa/bits_goa-2024-01-10.pdf differ diff --git a/text/presentation_01/bits_goa/bits_goa-2024-01-10.tex b/text/presentation_01/bits_goa/bits_goa-2024-01-10.tex new file mode 100644 index 0000000..e5d86ce --- /dev/null +++ b/text/presentation_01/bits_goa/bits_goa-2024-01-10.tex @@ -0,0 +1,1210 @@ +% !TeX program=xelatex +\documentclass[bigger]{beamer} + +\useinnertheme{circles} +\usefonttheme[onlymath]{serif} +\usefonttheme{structurebold} +\setbeamertemplate{navigation symbols}{} +\setbeamertemplate{footline}[frame number] +\usepackage{xcolor} +\setbeamercolor{highlight block}{bg=gray} + +\usepackage{tikz} +\usetikzlibrary{ +automata,% +positioning,% +calc,% +patterns, +decorations,% +decorations.pathmorphing,% +arrows.meta, +shapes.misc +} +\tikzset{ + vert/.style={draw, circle, inner sep=2pt}, + event/.style={}, + smodel/.style={fill=gray!25}, + tchoice/.style={draw, circle}, + indep/.style={},%{draw, dashed}, + proptc/.style = {-latex, dashed}, + propsm/.style = {-latex, thick}, + doubt/.style = {gray} +} + +\usepackage{tkz-graph} +\newcommand{\qlr}[2]{\ensuremath{\begin{matrix}#1\cr\begin{aligned}\hline #2\end{aligned}\end{matrix}}} + +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[normalem]{ulem} % To strikeout +\usepackage{commath} +\usepackage{amssymb} +\usepackage[normalem]{ulem} +\usepackage[nice]{nicefrac} +\usepackage{stmaryrd} +\usepackage{acronym} +\usepackage{multicol} +\usepackage{cleveref} + +\usepackage{listings} + +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} +\newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}} +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} +\newcommand{\condsymb}[2]{\ensuremath{p_{#1|#2}}} +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} +\newcommand{\deft}[1]{\textbf{#1}} +\newcommand{\emptyevent}{\ensuremath{\vartriangle}} +\newcommand{\ent}{\ensuremath{\lhd}} +\newcommand{\err}[1]{\ensuremath{\mathrm{err}\at{#1}}} +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} +\newcommand{\given}{\ensuremath{~\middle|~}} +\newcommand{\inconsistent}{\bot} +\newcommand{\indepclass}{\ensuremath{\Diamond}} +\newcommand{\isep}{~,~} +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} +\newcommand{\larr}{\ensuremath{\leftarrow}} +\newcommand{\lclass}[1]{\ensuremath{\intoc{#1}}} +\newcommand{\lfrac}[2]{\ensuremath{{#1}/{#2}}} +\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}} +\newcommand{\lset}[1]{\ensuremath{#1_{\ast}}} +\newcommand{\naf}{\ensuremath{\sim\!}} +\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}} +\newcommand{\probfact}[2]{\ensuremath{#2:#1}} +\newcommand{\probrule}[3]{\probfact{#2}{#1} \leftarrow #3} +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} +\newcommand{\pw}[1]{\ensuremath{\mu\at{#1}}} +\newcommand{\pwc}[1]{\ensuremath{\pwcfname\at{#1}}} +\newcommand{\pwcfname}{\ensuremath{\mu_{\textrm{TC}}}} +\newcommand{\q}[1]{\mathbf{#1}} +\newcommand{\rep}[2]{\ensuremath{\left\langle #1 \middle| #2 \right\rangle}} +\newcommand{\smclass}[1]{\ensuremath{\intcc{#1}}} +\newcommand{\stablecore}[1]{\ensuremath{\left\llbracket #1 \right\rrbracket}} +\newcommand{\tcgen}[1]{\ensuremath{\left<#1\right>}} +\newcommand{\uclass}[1]{\ensuremath{\intco{#1}}} +\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}} +\newcommand{\uset}[1]{\ensuremath{#1^{\ast}}} +\newcommand{\yset}[1]{\ensuremath{\left\langle #1 \right\rangle}} +%\newcommand{\tcgen}[1]{\ensuremath{\widehat{#1}}} + +\title{Stochastic Answer Set Programming} +\subtitle{A Research Program} +\author{Francisco Coelho} +\date{January 9, 2024} +\institute[\texttt{fc@uevora.pt}]{ +NOVA LINCS\\ +High Performance Computing Chair\\ +Departamento de Informática, Universidade de Évora +} +% +%=============================================================== +% +\begin{document} +% +\lstset{language=Prolog} +% +%=============================================================== +% +\begin{frame}[plain] + \titlepage + + \begin{center} + \footnotesize This is a join work with \textbf{Salvador Abreu}@DInf and \textbf{Bruno Dinis}@DMat. + \end{center} +\end{frame} +% +%=============================================================== +% +\section*{Motivation} +% +%=============================================================== +% +\begin{frame} + \frametitle{In Short} + + + \begin{itemize} + \item About \textbf{Machine Learning}: + \begin{itemize} + \item Vector or matrix based models lack ``structure''. + \item Large models don't \emph{explain} data. + \end{itemize} + \item About \textbf{Logic Programs}: + \begin{itemize} + \item Logic programs formalize knowledge. + \item Logic doesn't \emph{capture} uncertainty and is \emph{fragile} to noise. + \end{itemize} + \item \textbf{Probabilistic Logic Programs} extend formal knowledge with probabilities. + \begin{itemize} + \item How to propagate probabilities through rules? + \end{itemize} + \end{itemize} + \vfill + \begin{center} + \alert{\bf Goal:} Combine Logic and Statistics. + \end{center} +\end{frame} +% +%=============================================================== +% \begin{frame} +% %------------------------------------------------------------- +% %[fragile] +% %------------------------------------------------------------- +% \frametitle{Statistics and Machine Learning} +% %------------------------------------------------------------- +% \vfill +% \begin{itemize} +% \item \textbf{Data Analysis:} understand and summarize. +% \item \textbf{Model Building:} tools and techniques. +% \item \textbf{Model Evaluation:} performance. +% \item [] +% \item To calculate the accuracy/precision/recall of a model. +% \item To choose the right model for a problem. +% \item To tune the hyperparameters of a model. +% \end{itemize} +% \vfill +% \begin{center}\footnotesize +% Highlights of \texttt{Bard}'s (Google's LLM) reply to ``Explain what is the role of statistics in machine learning''. +% \end{center} +% \end{frame} +% +%=============================================================== +% +\section{Machine Learning} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\bf Machine Learning} + \end{center} + \vfill + \begin{itemize} + \item Standard Example --- Iris Classification + \item Assumptions of Machine Learning + \item Where Machine Learning Fails + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{The Standard Example --- Iris Classification} + %------------------------------------------------------------- + \small + \begin{center} + Learning Functions: \href{https://en.wikipedia.org/wiki/Iris_flower_data_set}{The famous Iris database} + \end{center} + \begin{columns} + \column{0.4\textwidth} + \begin{itemize}\setlength{\itemsep}{-0.5em} + \item[$x_1$] sepal length. + \item[$x_2$] sepal width. + \item[$x_3$] petal length. + \item[$x_4$] petal width. + \item[$y$] species (one of \emph{setosa}, \emph{versicolor}, \emph{virginica}). + \end{itemize} + \column{0.6\textwidth} + \begin{center} + \includegraphics[width=\textwidth]{iris_plot.pdf} + \end{center} + \end{columns} + \vfill + \begin{itemize} + \item A \emph{setosa} model: $ - 0.40 -0.65x_1 + 1.00x_2 > 0.00$. + \item A general \textbf{model template}: + $$ + f_\theta(\vec{x}) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_3 + \theta_4 x_4> 0 + $$ + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Assumptions of Machine Learning} + %------------------------------------------------------------- + \begin{itemize} + \item Each instance is described in a \textbf{single row} by a \textbf{fixed set of features} + $$ + \begin{array}{cccc|c} + \mathbf{x_1} & \mathbf{x_2} & \ldots & \mathbf{x_n} & \mathbf{y} \\ + \hline + x_{11} & x_{21} & \ldots & x_{n1} & y_1 \\ + & & \vdots \\ + x_{1m} & x_{2m} & \ldots & x_{nm} & y_m \\ + \end{array} + .$$ + \item Instances are \textbf{independent} of one another, \textbf{given the model} + $$ + y = f_\theta(\vec{x}). + $$ + \item Parameters \textbf{minimize estimation error} \emph{e.g.} + $$ + \hat{\theta} = \arg\min_\theta \sum_i \norm{y_i - f_\theta(\vec{x_{i}})}. + $$ + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Failing Assumptions\hfill~1/2} + %------------------------------------------------------------- + \vfill + \begin{columns}[c] + \column{0.5\textwidth} + $$ + \begin{array}{ccc} + \text{student} & \text{course} & \text{grade} \\ + \hline + s_1 & c_1 & a \\ + s_2 & c_1 & c \\ + s_1 & c_2 & b \\ + s_2 & c_3 & b \\ + s_3 & c_2 & b \\ + s_4 & c_3 & b \\ + s_3 & c_4 & X \\ + s_4 & c_4 & Y \\ + \end{array} + $$ + \column{0.5\textwidth} + \emph{Infer $X \succ_g Y$.} + + \bigskip + ``\emph{Proof.}'' + + \bigskip + $s_1 \succ_s s_2$ (rows 1 \& 2)\\ so $c_2 \succ_c c_3$ (3 \& 4). + + \bigskip + Then $s_3 \succ_s s_4$ (5 \& 6)\\ therefore $X \succ_g Y\qed$. + \end{columns} + \vfill + \begin{itemize} + \item Instances are not independent. + \item Small dataset, so \emph{no statistics}. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Failing Assumptions\hfill~2/2} + %------------------------------------------------------------- + \footnotesize + \begin{center} + \textbf{Prompt:} Given \texttt{f(machine) = e}, \texttt{f(learning) = g} and \texttt{f(algorithm) = m}, what is the value of \texttt{f(ksadbf)}? + \end{center} + + \vfill + \begin{columns}[t] + \scriptsize + \column{0.35\textwidth} + $$ + \begin{array}{r|l} + \textbf{input} & \textbf{output} \\ + \hline + machine & e \\ + learning & g \\ + algorithm & m \\ + ksadbf & ? + \end{array} + $$ + + \bigskip + \textbf{Bard:} \tt f(ksadbf) = 11. + + We can see that the value of f(x) is simply the first letter of x minus 96. + + \column{0.65\textwidth} + + \textbf{ChatGPT:} \tt The information provided seems to define a function $f$ where certain inputs are associated with specific outputs. Based on the given information: + \begin{itemize} + \item $f(\text{machine}) = e$ + \item $f(\text{learning}) = g$ + \item $f(\text{algorithm}) = m$ + \end{itemize} + However, there is no information given about \( f(\text{ksadbf}) \). Without additional information or a specific rule that determines the output for \( f(\text{ksadbf}) \), \emph{it is not possible to determine its value}. + % + If there is a pattern or rule for how the outputs are assigned based on the inputs, you would need to provide that information for me to help determine the value of \( f(\text{ksadbf}) \). + \end{columns} + + \vfill + \begin{center} + No help from state-of-the-art Natural Language Processing or LLMs. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Where Machine Learning Fails} + %------------------------------------------------------------- + \begin{itemize} + \item No \emph{interpretability}: large models ($\dim \theta \sim \text{thousands of millions}$) are \alert{not understandable by humans}. + \item Need for \emph{large number of training} examples. + \item Unfit for \emph{multiple relations}, \emph{intersample dependencies} and \emph{variable features}: The \textit{Students, Courses and Results} and \emph{Last Letter} examples. + \item Poor \emph{generalization}: A model trained in the iris dataset is useless in any other domain. + \item \alert{No use of background knowledge}. + \end{itemize} +\end{frame} +% +%=============================================================== +% +\section{Logic Programming} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\bf Logic Programming} + \end{center} + \vfill + \begin{itemize} + \item An Example of Logic Programming. + \item Inductive Logic Programming. + \item Where ILP Fails. + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + [fragile] + %------------------------------------------------------------- + \frametitle{An Example of Logic Programming} + %------------------------------------------------------------- + \begin{columns}[c] + \column{0.4\textwidth} + \begin{tikzpicture}[>=Latex] + \node[vert] (v6) {6}; + \node[vert, above left = of v6] (v3) {3}; + \node[vert, above right = of v6] (v5) {5}; + \node[vert, below left = of v6] (v1) {1}; + \node[vert, below right = of v6] (v2) {2}; + \node[vert, below right = of v1] (v4) {4}; + \draw[->] (v6) to (v3); + \draw[<->] (v6) to (v5); + \draw[<->] (v6) to (v2); + \draw[<->] (v3) to (v5); + \draw[<->] (v3) to (v1); + \draw[->] (v3) to[bend right, out=225, in =180,relative=false] (v4); + \draw[->] (v1) to (v2); + \draw[<->] (v1) to (v4); + \draw[->] (v2) to (v5); + \draw[<->] (v2) to (v4); + \draw[->] (v5) to[bend left, out=-45, in=0,relative=false] (v4); + \end{tikzpicture} + \column{0.6\textwidth} + \scriptsize + \begin{lstlisting} +node(1..6). + +edge(1,2). edge(2,4). edge(3,1). +edge(4,1). edge(5,3). edge(6,2). +edge(1,3). edge(2,5). edge(3,4). +edge(4,2). edge(5,4). edge(6,3). +edge(1,4). edge(2,6). edge(3,5). +edge(5,6). edge(6,5). + +col(r). col(b). col(g). + +1 { color(X,C) : col(C) } 1 :- node(X). +:- edge(X,Y), color(X,C), color(Y,C). + +#show color/2. + \end{lstlisting} + \end{columns} + \vfill + \scriptsize + \begin{lstlisting} +color(2,b) color(1,g) color(4,r) color(3,b) color(5,g) color(6,r) +color(1,r) color(2,b) color(4,g) color(3,b) color(5,r) color(6,g) +color(1,r) color(2,g) color(4,b) color(3,g) color(5,r) color(6,b) +color(1,b) color(2,g) color(4,r) color(3,g) color(5,b) color(6,r) +color(2,r) color(1,g) color(4,b) color(3,r) color(5,g) color(6,b) +color(2,r) color(1,b) color(4,g) color(3,r) color(5,b) color(6,g) + \end{lstlisting} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Inductive Logic Programming} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + Learning Logic Programs from Examples. + \end{center} + + \vfill + \alert{Generate rules} that\ldots + \begin{itemize} + \item use \textbf{background knowledge} + $$ + \begin{aligned} + & parent(john,mary), & & parent(david,steve), \\ + & parent(kathy,mary), & & female(kathy), \\ + & male(john), & & male(david). + \end{aligned} + $$ + \item to entail all the \textbf{positive examples}, $father(john,mary), father(david,steve)$, + \item but none of the \textbf{negative examples}. $father(kathy,mary), father(john,steve)$, + + \end{itemize} + + \vfill + A \textbf{solution} is $$father(X,Y) \leftarrow parent(X,Y) \wedge male(X).$$ + +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Where Logic Programming Fails} + %------------------------------------------------------------- + \begin{center} + Meanwhile, in the \textbf{real world}, samples are \emph{incomplete} and come with \emph{noise}. + \end{center} + \vfill + \textbf{Logic inference is \alert{fragile}}: a mistake in the transcription of a fact is dramatic to the consequences: + \begin{itemize} + \item $parent(david,mary)$. + \item $parent(jonh,mary)$. + \end{itemize} + \vfill + \begin{center} + The statistic essence of machine learning provides \alert{robustness}. + \end{center} +\end{frame} +% +%=============================================================== +% +\section{Probabilistic Logic Programming} +% +%=============================================================== +% +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \vfill + \begin{center} + {\huge\bf Probabilistic Logic Programming} + \end{center} + \vfill + \begin{itemize} + \item Define distributions from logic programs. + \item Stochastic ASP: Specifying distributions. + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilistic Logic Programs (PLPs)} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + Logic programs \textbf{annotated} with probabilities. + \end{center} + + \vfill + \begin{columns}[c] + + \column{0.4\textwidth} + \begin{tikzpicture}[>=Latex] + \node[draw, rounded rectangle] (A) {$Alarm$}; + \node[draw, rounded rectangle, below = of A] (J) {$Johncalls$}; + \draw[->] (A) to (J); + \end{tikzpicture} + + \column{0.6\textwidth} + $ + \begin{aligned} + alarm:0.00251, & \\ + johncalls:0.9 & \leftarrow alarm, \\ + johncalls:0.05 & \leftarrow \neg alarm + \end{aligned} + $ + \end{columns} + + \vfill + \begin{itemize} + \item \alert{$alarm:0.00251$} is $alarm \vee \neg alarm$ plus $P(Alarm = true) = 0.00251$. + \item \alert{$johncalls:0.9 \leftarrow alarm$} is + $$P\del{Johncalls = true \middle| Alarm = true} = 0.9$$ + \end{itemize} + + \vfill + \begin{center} + Any bayesian network can be represented by a PLP. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Distributions from Logic Programs} + %------------------------------------------------------------- + \vfill + The program + $$ + \begin{aligned} + alarm:0.00251, & \\ + johncalls:0.9 & \leftarrow alarm, \\ + johncalls:0.05 & \leftarrow \neg alarm + \end{aligned} + $$ + entails four possible models (or worlds): + $$ + \begin{array}{r|r} + \text{model} & \text{probability} \\ + \hline + alarm, johncalls & 0.002259 \\ + alarm, \neg johncalls & 0.000251 \\ + \neg alarm, johncalls & 0.049874 \\ + \neg alarm, \neg johncalls & 0.947616 \\ + \end{array} + $$ + \begin{itemize} + \item \alert{\textbf{Models}} are special sets of \emph{literals} \textbf{entailed} from the program. + \item Probabilities \emph{propagate} from facts, through rules. + \end{itemize} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{There's a Problem\ldots} + %------------------------------------------------------------- + \vfill + The program + $$ + \begin{aligned} + alarm:0.00251, & \\ + johncalls \vee marycalls & \leftarrow alarm + \end{aligned} + $$ + entails three \alert{stable} (\emph{i.e.}\ minimal) models + $$ + \begin{array}{r|l} + \text{model} & \text{probability} \\ + \hline + alarm, johncalls & x \\ + alarm, marycalls & y \\ + \neg alarm & 0.99749 + \end{array} + $$ + but \textbf{no single way to set $x,y$}. + \vfill + \begin{center} + Some \emph{Probabilistic Logic Programs} define more than one joint distribution. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{\ldots and an Oportunity} + %------------------------------------------------------------- + \begin{center} + Some \emph{PLP}s define more than one joint distribution. + \end{center} + \vfill + \begin{itemize} + \item There is \textbf{no single probability assignment} from the facts stable models: $x,y \in \intcc{0,1}$. + \item But any assignment is bound by Kolmogorov's axioms, and \textbf{forms equations} such as: + $$x + y = P\del{alarm}.$$ + \item Existing \textbf{data can be used to estimate the unknowns} in those equations. + \end{itemize} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Stable Models, Events and Probabilities} + %------------------------------------------------------------- + \vfill + \begin{center} + What are we talking about? + \end{center} + \vfill + \begin{itemize} + \item A logic program has \textbf{atoms} (and \textbf{literals}) and \textbf{rules}: + $$ + \begin{aligned} + &male(john), \neg parent(kathy,mary), \\ + &father(X, Y) \leftarrow parent(X, Y) \wedge male(X). + \end{aligned} + $$ + \item A \alert{\textbf{stable model}} is a \textbf{minimal} model that contains: + \begin{itemize} + \item program's \emph{facts}: $parent(john,mary),~male(john)$. + \item consequences, by the \emph{rules}: $father(john,mary)$. + \end{itemize} + \item Some programs have more than one model: + \begin{tabular}{c|c} + \textbf{Logic Program} & \textbf{Stable Models} \\ + \hline + $a \vee \neg a, b \vee c \leftarrow a$ + & + $\set{\neg a}, \set{a, b}, \set{a, c}$ + \end{tabular} + \end{itemize} + \vfill + \begin{center} + How to propagate probability from annotated facts to other \emph{events}? + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Logic Programs and Probabilities} + %------------------------------------------------------------- + % \vfill + % \begin{center} + % The \textbf{space of events}, $\Omega$, is the set of all sets of literals. + % \end{center} + \vfill + \begin{itemize} + \item Consider the literals of a logic program $$L = \set{a_1, \ldots a_n, \neg a_1, \ldots \neg a_n}.$$ + \item Any model of that program is a (consistent) subset of $L$. + \item Let $\Omega = \mathbf{P}\del{L}$, \emph{i.e.} an \alert{event} $e$ is a subset of $L$, $e \subseteq L$. + \begin{itemize} + \item Setting a probability for some events seems straightforward: $P\del{\neg alarm} = 0.997483558$. + \item For others, not so much: + \begin{itemize} + \item $P\del{alarm, johncalls}$, $P\del{johncalls, marycalls, alarm}$, $P\del{marycalls}$? + \item $P\del{alarm, \neg alarm}$, $P\del{\neg marycalls}$? + \end{itemize} + \end{itemize} + \end{itemize} + \vfill + \begin{center} + How to \alert{propagate} probability from \emph{facts} to \emph{consequences} and other \emph{events}? + \end{center} +\end{frame} +% +\newcommand{\diagram}{ + \resizebox{!}{24ex}{ + \begin{tikzpicture}%[scale=0.6, every node/.style={scale=0.6}] + \node[event] (E) {$\emptyevent$}; + \node[tchoice, above left = of E, pin={[red!50!black]175:$0.3$}] (a) {$a$}; + \node[smodel, above left = of a, pin={[red!50!black]175:$0.3\theta$}] (ab) {$ab$}; + \node[smodel, above right = of a, pin={[red!50!black]175:$0.3\co{\theta}$}] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[event, above left = of ab] (abC) {$\co{c}ab$}; + \node[event, above right = of ac] (aBc) {$\co{b}ac$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc, pin={[red!50!black]175:$0.7$}] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \path[draw, rounded corners, fill=cyan, opacity=0.1] + (ab.west) -- + (ab.north west) -- + % + (abC.south west) -- + (abC.north west) -- + (abC.north) -- + % + (abc.north east) -- + (abc.east) -- + (abc.south east) -- + % + (ab.north east) -- + (ab.east) -- + (ab.south east) -- + % + (a.north east) -- + % + (E.north east) -- + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (b.south west) -- + % + (ab.west) + ; + % ---- + \path[draw, rounded corners, fill=yellow, opacity=0.1] + (ac.south west) -- + (ac.west) -- + (ac.north west) -- + % + (abc.south west) -- + (abc.west) -- + (abc.north west) -- + % + (aBc.north east) -- + (aBc.east) -- + (aBc.south east) -- + % + (ac.north east) -- + % + (c.east) -- + % + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (a.south west) -- + (a.west) -- + (a.north west) -- + (a.north) -- + % + (ac.south west) + ; + % ---- + \path[draw, rounded corners, fill=magenta, opacity=0.1] + % (A.north west) -- + % + (Ac.north west) -- + % + (Abc.north west) -- + (Abc.north) -- + (Abc.north east) -- + (Abc.south east) -- + % + % (Ac.north east) -- + % (Ac.east) -- + % + % (A.east) -- + (A.south east) -- + % + (E.south east) -- + (E.south) -- + (E.south west) -- + (E.west) -- + (E.north west) -- + % + (Ac.north west) + ; + \end{tikzpicture} + } +} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Classes of Events} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + \diagram + \end{center} + + \vfill + \begin{center} + \begin{tabular}{lr} + $\begin{aligned} + \probfact{0.3}{a} \\ + b \vee c & \leftarrow a + \end{aligned}$ + & + $\co{a} = \set{\neg a}, ab = \set{a, b}, ac = \set{a, c}$ + \end{tabular} + \end{center} + + \vfill + \begin{itemize} + \item Define \alert{equivalence classes} for all events, based on $\subseteq, \supseteq$ relations with the \textbf{stable models}. + \item This example shows $6$ out of $2^3 + 1$ classes. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilities for all Events} + %------------------------------------------------------------- + \vfill + \begin{center} + \diagram + \end{center} + \vfill + \footnotesize + \begin{enumerate} + \item Set \alert{weights} in the stable models (shaded nodes), using parameters when needed: $\mu\del{\co{a}} = 0.7; \mu\del{ab} = 0.3\theta; \mu\del{ac}=0.3\del{1 - \theta}$ + \item Assume that the stable models are \alert{disjoint events}. + \item Define \alert{weight of an event} as the sum of the weights of the related stable models. + \item Normalize weights to get a (probability) \alert{distribution}. + \end{enumerate} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilities for all Events} + %------------------------------------------------------------- + \vfill + % \begin{center} + % \diagram + % \end{center} + % \vfill + % \scriptsize + + \begin{equation*} + \begin{array}{clr|cc|cc} + & \stablecore{e} + & \# \class{e} + & \pw{\class{e}} + & \pw{e} + & \pr{E = e} + & \pr{E \in \class{e}} + \\ + \hline + % + & \inconsistent + & 37 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + \square + & \indepclass + & 9 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + {\color{magenta!20}\blacksquare } + & \co{a} + & 9 + & \frac{7}{10} + & \frac{7}{90} + & \frac{7}{207} + & \frac{7}{23} + \\[4pt] + % + {\color{cyan!20}\blacksquare } + & ab + & 3 + & \frac{3}{10}\theta + & \frac{1}{10}\theta + & \frac{1}{23}\alert{\theta} + & \frac{3}{23}\theta + \\[4pt] + % + {\color{yellow!20}\blacksquare } + & ac + & 3 + & \frac{3}{10}\co{\theta} + & \frac{1}{10}\co{\theta} + & \frac{1}{23}\alert{\co{\theta}} + & \frac{3}{23}\co{\theta} + \\[4pt] + % + & \co{a}, ab + & 0 + & \frac{7 + 3\theta}{10} + & 0 + & 0 + & 0 + \\[4pt] + % + & \co{a}, ac + & 0 + & \frac{7 + 3\co{\theta}}{10} + & 0 + & 0 + & 0 + % + \\[4pt] + % + {\color{green!20}\blacksquare } + & ab, ac + & 2 + & \frac{3}{10} + & \frac{3}{20} + & \frac{3}{46} + & \frac{3}{23} + \\[4pt] + % + {\color{gray!20}\blacksquare } + & \co{a}, ab, ac + & 1 + & 1 + & 1 + & \frac{10}{23} + & \frac{10}{23} + \\[4pt] + % + \hline + & + & 64 + & + & Z = \frac{23}{10} + & + %& \Sigma = 1 + \end{array} + \end{equation*} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Estimating the Parameters} + %------------------------------------------------------------- + A \alert{sample} can be used to estimate the parameters $\theta$, by minimizing + \begin{equation*} + \err{\theta} := \sum_{e\in\fml{E}} \del{\pr{E = e\given \Theta = \theta} - \pr{S = e}}^2.\label{eq:err.e.s} + \end{equation*} + where + \begin{itemize} + \item $\fml{E}$ is the set of all events, + \item $\pr{E\given \Theta}$ the \textbf{model+parameters} based distribution, + \item $\pr{S}$ is the \textbf{empiric} distribution from the given sample. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Behind Parameter Estimation} + %------------------------------------------------------------- + \vfill + So, we can derive a distribution $\pr{E\given \Theta = \hat{\theta}}$ from a program $P$ and a sample $S$. + \begin{itemize} + \item The sample defines an empiric distribution $\pr{S}$\ldots + \item \ldots that is used to estimate $\theta$ in $\pr{E\given \Theta}$\ldots + \item \ldots and \alert{score the program} $P$ w.r.t.\ that sample using, \emph{e.g.} the $\err{}$ function. + \end{itemize} + \vfill + +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Back to Inductive Logic Programming} + %------------------------------------------------------------- + \vfill + Recall the \emph{Learning Logic Programs from Examples} setting: + \begin{itemize} + \item Given \textbf{positive} and \textbf{negative} examples, and \textbf{background knowledge}\ldots + \item find a \textbf{program}\ldots + \begin{itemize} + \item \ldots using the facts and relations from the \textbf{BK}\ldots + \item \ldots such that \textbf{all the PE} and \textbf{none the NE} examples are entailed. + \end{itemize} + \end{itemize} + \vfill + \begin{quotation} + Given a sample of events, and a set of programs, \alert{the score} of those programs (w.r.t. the sample) \alert{can be used in evolutionary algorithms} while searching for better solutions. + \end{quotation} +\end{frame} +% +%=============================================================== +% +\section{In Conclusion} +% +%=============================================================== +% +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \vfill + \begin{center} + {\huge\bf In Conclusion} + \end{center} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + \begin{itemize} + \item \textbf{Machine Learning} has limitations. + \item As does \textbf{Inductive Logic Programming}. + \item But, distributions can be defined by \textbf{Stochastic Logic Programs}. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + Distributions can be defined by \textbf{Stochastic Logic Programs}. + + \vfill + Here we: + \begin{enumerate} + \item Look at the program's \textbf{stable models} and + \item Use them to partition the \textbf{events} and then + \item Using annotated probabilities, define: + \begin{enumerate} + \item a finite \textbf{measure}\ldots + \item that, normalized, is a \textbf{distribution} on all events. + \end{enumerate} + \end{enumerate} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + Distributions can be defined by \textbf{Stochastic Logic Programs}. + + \vfill + \begin{itemize} + \item These distributions might have some \textbf{parameters}, due to indeterminism in the program. + \item A \textbf{sample} can be used to estimate those parameters\ldots + \item \ldots and \alert{score} programs concurring to describe it. + \item This score a key ingredient in \textbf{evolutionary algorithms}. + \end{itemize} + + \vfill + \begin{quotation} + \ldots and a step towards the \alert{induction of stochastic logic programs} using \textbf{data} and \textbf{background knowledge}. + \end{quotation} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Future Work} + %------------------------------------------------------------- + \vfill + \begin{center} + Induction of Stochastic Logic (ASP) Programs. + \end{center} + \vfill + \begin{enumerate} + \item \textbf{Meta-programming:} formal rules for rule generation. + \item \textbf{Generation}, \textbf{Combination} and \textbf{Mutation} operators. + \item \textbf{Complexity.} + \item \textbf{Applications.} + \item \textbf{Profit.} + \end{enumerate} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\alert{\bf Thank You!}} + \vfill + Questions? + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{References} + \begin{itemize} + \item \href{https://arxiv.org/abs/1801.00631}{Gary Marcus, \emph{Deep Learning: A Critical Appraisal}, 2018}. + \item \href{https://arxiv.org/abs/1911.01547}{François Chollet, \emph{On the Measure of Intelligence}, 2019}. + \item \href{https://arxiv.org/abs/1801.00631}{Bengio \emph{et al.}, \emph{A Meta-Transfer Objective for Learning to Disentangle Causal Mechanisms}, 2019}. + \item \href{https://arxiv.org/abs/1801.00631}{Cropper \emph{et al.}, \emph{Turning 30: New Ideas in Inductive Logic Programming}, 2020}. + \item \href{https://doi.org/10.1201/9781003427421}{Fabrizio Riguzzi, \emph{Foundations of Probabilistic Logic Programming }, 2023}. + \end{itemize} + %------------------------------------------------------------- +\end{frame} +% +\end{document} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{TITLE} + %------------------------------------------------------------- +\end{frame} +% + + diff --git a/text/presentation_01/bits_goa/iris_plot.pdf b/text/presentation_01/bits_goa/iris_plot.pdf new file mode 100644 index 0000000..ec92c9b Binary files /dev/null and b/text/presentation_01/bits_goa/iris_plot.pdf differ diff --git a/text/presentation_01/cima_pdoutmat_2023-12-11/cima_pdoutmat_2023-12-11.pdf b/text/presentation_01/cima_pdoutmat_2023-12-11/cima_pdoutmat_2023-12-11.pdf new file mode 100644 index 0000000..80998e4 Binary files /dev/null and b/text/presentation_01/cima_pdoutmat_2023-12-11/cima_pdoutmat_2023-12-11.pdf differ diff --git a/text/presentation_01/cima_pdoutmat_2023-12-11/cima_pdoutmat_2023-12-11.tex b/text/presentation_01/cima_pdoutmat_2023-12-11/cima_pdoutmat_2023-12-11.tex new file mode 100644 index 0000000..4eccd7e --- /dev/null +++ b/text/presentation_01/cima_pdoutmat_2023-12-11/cima_pdoutmat_2023-12-11.tex @@ -0,0 +1,1203 @@ +% !TeX program=xelatex +\documentclass[bigger]{beamer} + +\useinnertheme{circles} +\usefonttheme[onlymath]{serif} +\usefonttheme{structurebold} +\setbeamertemplate{navigation symbols}{} +\setbeamertemplate{footline}[frame number] +\usepackage{xcolor} +\setbeamercolor{highlight block}{bg=gray} + +\usepackage{tikz} +\usetikzlibrary{ +automata,% +positioning,% +calc,% +patterns, +decorations,% +decorations.pathmorphing,% +arrows.meta, +shapes.misc +} +\tikzset{ + vert/.style={draw, circle, inner sep=2pt}, + event/.style={}, + smodel/.style={fill=gray!25}, + tchoice/.style={draw, circle}, + indep/.style={},%{draw, dashed}, + proptc/.style = {-latex, dashed}, + propsm/.style = {-latex, thick}, + doubt/.style = {gray} +} + +\usepackage{tkz-graph} +\newcommand{\qlr}[2]{\ensuremath{\begin{matrix}#1\cr\begin{aligned}\hline #2\end{aligned}\end{matrix}}} + +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[normalem]{ulem} % To strikeout +\usepackage{commath} +\usepackage{amssymb} +\usepackage[normalem]{ulem} +\usepackage[nice]{nicefrac} +\usepackage{stmaryrd} +\usepackage{acronym} +\usepackage{multicol} +\usepackage{cleveref} + +\usepackage{listings} + +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} +\newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}} +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} +\newcommand{\condsymb}[2]{\ensuremath{p_{#1|#2}}} +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} +\newcommand{\deft}[1]{\textbf{#1}} +\newcommand{\emptyevent}{\ensuremath{\vartriangle}} +\newcommand{\ent}{\ensuremath{\lhd}} +\newcommand{\err}[1]{\ensuremath{\mathrm{err}\at{#1}}} +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} +\newcommand{\given}{\ensuremath{~\middle|~}} +\newcommand{\inconsistent}{\bot} +\newcommand{\indepclass}{\ensuremath{\Diamond}} +\newcommand{\isep}{~,~} +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} +\newcommand{\larr}{\ensuremath{\leftarrow}} +\newcommand{\lclass}[1]{\ensuremath{\intoc{#1}}} +\newcommand{\lfrac}[2]{\ensuremath{{#1}/{#2}}} +\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}} +\newcommand{\lset}[1]{\ensuremath{#1_{\ast}}} +\newcommand{\naf}{\ensuremath{\sim\!}} +\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}} +\newcommand{\probfact}[2]{\ensuremath{#2:#1}} +\newcommand{\probrule}[3]{\probfact{#2}{#1} \leftarrow #3} +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} +\newcommand{\pw}[1]{\ensuremath{\mu\at{#1}}} +\newcommand{\pwc}[1]{\ensuremath{\pwcfname\at{#1}}} +\newcommand{\pwcfname}{\ensuremath{\mu_{\textrm{TC}}}} +\newcommand{\q}[1]{\mathbf{#1}} +\newcommand{\rep}[2]{\ensuremath{\left\langle #1 \middle| #2 \right\rangle}} +\newcommand{\smclass}[1]{\ensuremath{\intcc{#1}}} +\newcommand{\stablecore}[1]{\ensuremath{\left\llbracket #1 \right\rrbracket}} +\newcommand{\tcgen}[1]{\ensuremath{\left<#1\right>}} +\newcommand{\uclass}[1]{\ensuremath{\intco{#1}}} +\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}} +\newcommand{\uset}[1]{\ensuremath{#1^{\ast}}} +\newcommand{\yset}[1]{\ensuremath{\left\langle #1 \right\rangle}} +%\newcommand{\tcgen}[1]{\ensuremath{\widehat{#1}}} + +\title{Stochastic Answer Set Programming} +\subtitle{A Research Program} +\author{Francisco Coelho} +\date{December 11, 2023} +\institute[\texttt{fc@uevora.pt}]{ +NOVA LINCS\\ +High Performance Computing Chair\\ +Departamento de Informática, Universidade de Évora +} +% +%=============================================================== +% +\begin{document} +% +\lstset{language=Prolog} +% +%=============================================================== +% +\begin{frame}[plain] + \titlepage + + \begin{center} + \footnotesize This is a join work with \textbf{Salvador Abreu}@DInf and \textbf{Bruno Dinis}@DMat. + \end{center} +\end{frame} +% +%=============================================================== +% +\section*{Motivation} +% +%=============================================================== +% +\begin{frame} + \frametitle{In Short} + + + \begin{itemize} + \item About \textbf{Machine Learning}: + \begin{itemize} + \item Vector or matrix based models lack ``structure''. + \item Large models don't \emph{explain} data. + \end{itemize} + \item About \textbf{Logic Programs}: + \begin{itemize} + \item Logic programs formalize knowledge. + \item Logic doesn't \emph{capture} uncertainty and is \emph{fragile} to noise. + \end{itemize} + \item \textbf{Probabilistic Logic Programs} extend formal knowledge with probabilities. + \begin{itemize} + \item How to propagate probabilities through rules? + \end{itemize} + \end{itemize} + \vfill + \begin{center} + \alert{\bf Goal:} Combine Logic and Statistics. + \end{center} +\end{frame} +% +%=============================================================== +% \begin{frame} +% %------------------------------------------------------------- +% %[fragile] +% %------------------------------------------------------------- +% \frametitle{Statistics and Machine Learning} +% %------------------------------------------------------------- +% \vfill +% \begin{itemize} +% \item \textbf{Data Analysis:} understand and summarize. +% \item \textbf{Model Building:} tools and techniques. +% \item \textbf{Model Evaluation:} performance. +% \item [] +% \item To calculate the accuracy/precision/recall of a model. +% \item To choose the right model for a problem. +% \item To tune the hyperparameters of a model. +% \end{itemize} +% \vfill +% \begin{center}\footnotesize +% Highlights of \texttt{Bard}'s (Google's LLM) reply to ``Explain what is the role of statistics in machine learning''. +% \end{center} +% \end{frame} +% +%=============================================================== +% +\section{Machine Learning} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\bf Machine Learning} + \end{center} + \vfill + \begin{itemize} + \item Standard Example --- Iris Classification + \item Assumptions of Machine Learning + \item Where Machine Learning Fails + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{The Standard Example --- Iris Classification} + %------------------------------------------------------------- + \small + \begin{center} + Learning Functions: \href{https://en.wikipedia.org/wiki/Iris_flower_data_set}{The famous Iris database} + \end{center} + \begin{columns} + \column{0.4\textwidth} + \begin{itemize}\setlength{\itemsep}{-0.5em} + \item[$x_1$] sepal length. + \item[$x_2$] sepal width. + \item[$x_3$] petal length. + \item[$x_4$] petal width. + \item[$y$] species (one of \emph{setosa}, \emph{versicolor}, \emph{virginica}). + \end{itemize} + \column{0.6\textwidth} + \begin{center} + \includegraphics[width=\textwidth]{iris_plot.pdf} + \end{center} + \end{columns} + \vfill + \begin{itemize} + \item A \emph{setosa} model: $ - 0.40 -0.65x_1 + 1.00x_2 > 0.00$. + \item A general \textbf{model template}: + $$ + f_\theta(\vec{x}) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_3 + \theta_4 x_4> 0 + $$ + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Assumptions of Machine Learning} + %------------------------------------------------------------- + \begin{itemize} + \item Each instance is described in a \textbf{single row} by a \textbf{fixed set of features} + $$ + \begin{array}{cccc|c} + \mathbf{x_1} & \mathbf{x_2} & \ldots & \mathbf{x_n} & \mathbf{y} \\ + \hline + x_{11} & x_{21} & \ldots & x_{n1} & y_1 \\ + & & \vdots \\ + x_{1m} & x_{2m} & \ldots & x_{nm} & y_m \\ + \end{array} + .$$ + \item Instances are \textbf{independent} of one another, \textbf{given the model} + $$ + y = f_\theta(\vec{x}). + $$ + \item Parameters \textbf{minimize estimation error} \emph{e.g.} + $$ + \hat{\theta} = \arg\min_\theta \sum_i \norm{y_i - f_\theta(\vec{x_{i}})}. + $$ + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Failing Assumptions\hfill~1/2} + %------------------------------------------------------------- + \vfill + \begin{columns}[c] + \column{0.5\textwidth} + $$ + \begin{array}{ccc} + \text{student} & \text{course} & \text{grade} \\ + \hline + s_1 & c_1 & a \\ + s_2 & c_1 & c \\ + s_1 & c_2 & b \\ + s_2 & c_3 & b \\ + s_3 & c_2 & b \\ + s_4 & c_3 & b \\ + s_3 & c_4 & X \\ + s_4 & c_4 & Y \\ + \end{array} + $$ + \column{0.5\textwidth} + \emph{Infer $X \succ_g Y$.} + + \bigskip + ``\emph{Proof.}'' + + \bigskip + $s_1 \succ_s s_2$ (rows 1 \& 2)\\ so $c_2 \succ_c c_3$ (3 \& 4). + + \bigskip + Then $s_3 \succ_s s_4$ (5 \& 6)\\ therefore $X \succ_g Y\qed$. + \end{columns} + \vfill + \begin{itemize} + \item Instances are not independent. + \item Small dataset, so \emph{no statistics}. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Failing Assumptions\hfill~2/2} + %------------------------------------------------------------- + \footnotesize + \begin{center} + \textbf{Prompt:} Given \texttt{f(machine) = e}, \texttt{f(learning) = g} and \texttt{f(algorithm) = m}, what is the value of \texttt{f(ksadbf)}? + \end{center} + + \vfill + \begin{columns}[t] + \scriptsize + \column{0.35\textwidth} + $$ + \begin{array}{r|l} + \textbf{input} & \textbf{output} \\ + \hline + machine & e \\ + learning & g \\ + algorithm & m \\ + ksadbf & ? + \end{array} + $$ + + \bigskip + \textbf{Bard:} \tt f(ksadbf) = 11. + + We can see that the value of f(x) is simply the first letter of x minus 96. + + \column{0.65\textwidth} + + \textbf{ChatGPT:} \tt The information provided seems to define a function $f$ where certain inputs are associated with specific outputs. Based on the given information: + \begin{itemize} + \item $f(\text{machine}) = e$ + \item $f(\text{learning}) = g$ + \item $f(\text{algorithm}) = m$ + \end{itemize} + However, there is no information given about \( f(\text{ksadbf}) \). Without additional information or a specific rule that determines the output for \( f(\text{ksadbf}) \), \emph{it is not possible to determine its value}. + % + If there is a pattern or rule for how the outputs are assigned based on the inputs, you would need to provide that information for me to help determine the value of \( f(\text{ksadbf}) \). + \end{columns} + + \vfill + \begin{center} + No help from state-of-the-art Natural Language Processing or LLMs. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Where Machine Learning Fails} + %------------------------------------------------------------- + \begin{itemize} + \item No \emph{interpretability}: large models ($\dim \theta \sim \text{millions}$) are \alert{not understandable by humans}. + \item Need for \emph{large number of training} examples. + \item Unfit for \emph{multiple relations}, \emph{intersample dependencies} and \emph{variable features}: The \textit{Students, Courses and Results} and \emph{Last Letter} examples. + \item Poor \emph{generalization}: A model trained in the iris dataset is useless in any other domain. + \item \alert{No use of background knowledge}. + \end{itemize} +\end{frame} +% +%=============================================================== +% +\section{Logic Programming} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\bf Logic Programming} + \end{center} + \vfill + \begin{itemize} + \item An Example of Logic Programming. + \item Inductive Logic Programming. + \item Where ILP Fails. + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + [fragile] + %------------------------------------------------------------- + \frametitle{An Example of Logic Programming} + %------------------------------------------------------------- + \begin{columns}[c] + \column{0.4\textwidth} + \begin{tikzpicture}[>=Latex] + \node[vert] (v6) {6}; + \node[vert, above left = of v6] (v3) {3}; + \node[vert, above right = of v6] (v5) {5}; + \node[vert, below left = of v6] (v1) {1}; + \node[vert, below right = of v6] (v2) {2}; + \node[vert, below right = of v1] (v4) {4}; + \draw[->] (v6) to (v3); + \draw[<->] (v6) to (v5); + \draw[<->] (v6) to (v2); + \draw[<->] (v3) to (v5); + \draw[<->] (v3) to (v1); + \draw[->] (v3) to[bend right, out=225, in =180,relative=false] (v4); + \draw[->] (v1) to (v2); + \draw[<->] (v1) to (v4); + \draw[->] (v2) to (v5); + \draw[<->] (v2) to (v4); + \draw[->] (v5) to[bend left, out=-45, in=0,relative=false] (v4); + \end{tikzpicture} + \column{0.6\textwidth} + \scriptsize + \begin{lstlisting} +node(1..6). + +edge(1,2). edge(2,4). edge(3,1). +edge(4,1). edge(5,3). edge(6,2). +edge(1,3). edge(2,5). edge(3,4). +edge(4,2). edge(5,4). edge(6,3). +edge(1,4). edge(2,6). edge(3,5). +edge(5,6). edge(6,5). + +col(r). col(b). col(g). + +1 { color(X,C) : col(C) } 1 :- node(X). +:- edge(X,Y), color(X,C), color(Y,C). + +#show color/2. + \end{lstlisting} + \end{columns} + \vfill + \scriptsize + \begin{lstlisting} +color(2,b) color(1,g) color(4,r) color(3,b) color(5,g) color(6,r) +color(1,r) color(2,b) color(4,g) color(3,b) color(5,r) color(6,g) +color(1,r) color(2,g) color(4,b) color(3,g) color(5,r) color(6,b) +color(1,b) color(2,g) color(4,r) color(3,g) color(5,b) color(6,r) +color(2,r) color(1,g) color(4,b) color(3,r) color(5,g) color(6,b) +color(2,r) color(1,b) color(4,g) color(3,r) color(5,b) color(6,g) + \end{lstlisting} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Inductive Logic Programming} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + Learning Logic Programs from Examples. + \end{center} + + \vfill + \alert{Generate rules} that\ldots + \begin{itemize} + \item use \textbf{background knowledge} + $$ + \begin{aligned} + & parent(john,mary), & & parent(david,steve), \\ + & parent(kathy,mary), & & female(kathy), \\ + & male(john), & & male(david). + \end{aligned} + $$ + \item to entail all the \textbf{positive examples}, $father(john,mary), father(david,steve)$, + \item but none of the \textbf{negative examples}. $father(kathy,mary), father(john,steve)$, + + \end{itemize} + + \vfill + A \textbf{solution} is $$father(X,Y) \leftarrow parent(X,Y) \wedge male(X).$$ + +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Where Logic Programming Fails} + %------------------------------------------------------------- + \begin{center} + Meanwhile, in the \textbf{real world}, samples are \emph{incomplete} and come with \emph{noise}. + \end{center} + \vfill + \textbf{Logic inference is \alert{fragile}}: a mistake in the transcription of a fact is dramatic to the consequences: + \begin{itemize} + \item $parent(david,mary)$. + \item $parent(jonh,mary)$. + \end{itemize} + \vfill + \begin{center} + The statistic essence of machine learning provides \alert{robustness}. + \end{center} +\end{frame} +% +%=============================================================== +% +\section{Probabilistic Logic Programming} +% +%=============================================================== +% +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \vfill + \begin{center} + {\huge\bf Probabilistic Logic Programming} + \end{center} + \vfill + \begin{itemize} + \item Define distributions from logic programs. + \item Stochastic ASP: Specifying distributions. + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilistic Logic Programs (PLPs)} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + Logic programs \textbf{annotated} with probabilities. + \end{center} + + \vfill + \begin{columns}[c] + + \column{0.4\textwidth} + \begin{tikzpicture}[>=Latex] + \node[draw, rounded rectangle] (A) {$Alarm$}; + \node[draw, rounded rectangle, below = of A] (J) {$Johncalls$}; + \draw[->] (A) to (J); + \end{tikzpicture} + + \column{0.6\textwidth} + $ + \begin{aligned} + alarm:0.00251, & \\ + johncalls:0.9 & \leftarrow alarm, \\ + johncalls:0.05 & \leftarrow \neg alarm + \end{aligned} + $ + \end{columns} + + \vfill + \begin{itemize} + \item \alert{$alarm:0.00251$} is $alarm \vee \neg alarm$ plus $P(Alarm = true) = 0.00251$. + \item \alert{$johncalls:0.9 \leftarrow alarm$} is + $$P\del{Johncalls = true \middle| Alarm = true} = 0.9$$ + \end{itemize} + + \vfill + \begin{center} + Any bayesian network can be represented by a PLP. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Distributions from Logic Programs} + %------------------------------------------------------------- + \vfill + The program + $$ + \begin{aligned} + alarm:0.00251, & \\ + johncalls:0.9 & \leftarrow alarm, \\ + johncalls:0.05 & \leftarrow \neg alarm + \end{aligned} + $$ + entails four possible models (or worlds): + $$ + \begin{array}{r|r} + \text{model} & \text{probability} \\ + \hline + alarm, johncalls & 0.002259 \\ + alarm, \neg johncalls & 0.000251 \\ + \neg alarm, johncalls & 0.049874 \\ + \neg alarm, \neg johncalls & 0.947616 \\ + \end{array} + $$ + \begin{itemize} + \item \alert{\textbf{Models}} are special sets of \emph{literals} \textbf{entailed} from the program. + \item Probabilities \emph{propagate} from facts, through rules. + \end{itemize} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{There's a Problem\ldots} + %------------------------------------------------------------- + \vfill + The program + $$ + \begin{aligned} + alarm:0.00251, & \\ + johncalls \vee marycalls & \leftarrow alarm + \end{aligned} + $$ + entails three \alert{stable} (\emph{i.e.}\ minimal) models + $$ + \begin{array}{r|l} + \text{model} & \text{probability} \\ + \hline + alarm, johncalls & x \\ + alarm, marycalls & y \\ + \neg alarm & 0.99749 + \end{array} + $$ + but \textbf{no single way to set $x,y$}. + \vfill + \begin{center} + Some \emph{Probabilistic Logic Programs} define more than one joint distribution. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{\ldots and an Oportunity} + %------------------------------------------------------------- + \begin{center} + Some \emph{PLP}s define more than one joint distribution. + \end{center} + \vfill + \begin{itemize} + \item There is \textbf{no single probability assignment} from the facts stable models: $x,y \in \intcc{0,1}$. + \item But any assignment is bound by Kolmogorov's axioms, and \textbf{forms equations} such as: + $$x + y = P\del{alarm}.$$ + \item Existing \textbf{data can be used to estimate the unknowns} in those equations. + \end{itemize} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Stable Models, Events and Probabilities} + %------------------------------------------------------------- + \vfill + \begin{center} + What are we talking about? + \end{center} + \vfill + \begin{itemize} + \item A logic program has \textbf{atoms} (and \textbf{literals}) and \textbf{rules}: + $$ + \begin{aligned} + &male(john), \neg parent(kathy,mary), \\ + &father(X, Y) \leftarrow parent(X, Y) \wedge male(X). + \end{aligned} + $$ + \item A \alert{\textbf{stable model}} is a \textbf{minimal} model that contains: + \begin{itemize} + \item program's \emph{facts}: $parent(john,mary),~male(john)$. + \item consequences, by the \emph{rules}: $father(john,mary)$. + \end{itemize} + \item Some programs have more than one model: + \begin{tabular}{c|c} + \textbf{Logic Program} & \textbf{Stable Models} \\ + \hline + $a \vee \neg a, b \vee c \leftarrow a$ + & + $\set{\neg a}, \set{a, b}, \set{a, c}$ + \end{tabular} + \end{itemize} + \vfill + \begin{center} + How to propagate probability from annotated facts to other \emph{events}? + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Logic Programs and Probabilities} + %------------------------------------------------------------- + % \vfill + % \begin{center} + % The \textbf{space of events}, $\Omega$, is the set of all sets of literals. + % \end{center} + \vfill + \begin{itemize} + \item Consider the literals of a logic program $$L = \set{a_1, \ldots a_n, \neg a_1, \ldots \neg a_n}.$$ + \item Any model of that program is a (consistent) subset of $L$. + \item Let $\Omega = \mathbf{P}\del{L}$, \emph{i.e.} an \alert{event} $e$ is a subset of $L$, $e \subseteq L$. + \begin{itemize} + \item Setting a probability for some events seems straightforward: $P\del{\neg alarm} = 0.997483558$. + \item For others, not so much: + \begin{itemize} + \item $P\del{alarm, johncalls}$, $P\del{johncalls, marycalls, alarm}$, $P\del{marycalls}$? + \item $P\del{alarm, \neg alarm}$, $P\del{\neg marycalls}$? + \end{itemize} + \end{itemize} + \end{itemize} + \vfill + \begin{center} + How to \alert{propagate} probability from \emph{facts} to \emph{consequences} and other \emph{events}? + \end{center} +\end{frame} +% +\newcommand{\diagram}{ + \resizebox{!}{24ex}{ + \begin{tikzpicture}%[scale=0.6, every node/.style={scale=0.6}] + \node[event] (E) {$\emptyevent$}; + \node[tchoice, above left = of E, pin={[red!50!black]175:$0.3$}] (a) {$a$}; + \node[smodel, above left = of a, pin={[red!50!black]175:$0.3\theta$}] (ab) {$ab$}; + \node[smodel, above right = of a, pin={[red!50!black]175:$0.3\co{\theta}$}] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[event, above left = of ab] (abC) {$\co{c}ab$}; + \node[event, above right = of ac] (aBc) {$\co{b}ac$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc, pin={[red!50!black]175:$0.7$}] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \path[draw, rounded corners, fill=cyan, opacity=0.1] + (ab.west) -- + (ab.north west) -- + % + (abC.south west) -- + (abC.north west) -- + (abC.north) -- + % + (abc.north east) -- + (abc.east) -- + (abc.south east) -- + % + (ab.north east) -- + (ab.east) -- + (ab.south east) -- + % + (a.north east) -- + % + (E.north east) -- + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (b.south west) -- + % + (ab.west) + ; + % ---- + \path[draw, rounded corners, fill=yellow, opacity=0.1] + (ac.south west) -- + (ac.west) -- + (ac.north west) -- + % + (abc.south west) -- + (abc.west) -- + (abc.north west) -- + % + (aBc.north east) -- + (aBc.east) -- + (aBc.south east) -- + % + (ac.north east) -- + % + (c.east) -- + % + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (a.south west) -- + (a.west) -- + (a.north west) -- + (a.north) -- + % + (ac.south west) + ; + % ---- + \path[draw, rounded corners, fill=magenta, opacity=0.1] + % (A.north west) -- + % + (Ac.north west) -- + % + (Abc.north west) -- + (Abc.north) -- + (Abc.north east) -- + (Abc.south east) -- + % + % (Ac.north east) -- + % (Ac.east) -- + % + % (A.east) -- + (A.south east) -- + % + (E.south east) -- + (E.south) -- + (E.south west) -- + (E.west) -- + (E.north west) -- + % + (Ac.north west) + ; + \end{tikzpicture} + } +} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Classes of Events} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + \diagram + \end{center} + + \vfill + \begin{center} + \begin{tabular}{lr} + $\begin{aligned} + \probfact{0.3}{a} \\ + b \vee c & \leftarrow a + \end{aligned}$ + & + $\co{a} = \set{\neg a}, ab = \set{a, b}, ac = \set{a, c}$ + \end{tabular} + \end{center} + + \vfill + \begin{itemize} + \item Define \alert{equivalence classes} for all events, based on $\subseteq, \supseteq$ relations with the \textbf{stable models}. + \item This example shows $6$ out of $2^3 + 1$ classes. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilities for all Events} + %------------------------------------------------------------- + \vfill + \begin{center} + \diagram + \end{center} + \vfill + \footnotesize + \begin{enumerate} + \item Set \alert{weights} in the stable models (shaded nodes), using parameters when needed: $\mu\del{\co{a}} = 0.7; \mu\del{ab} = 0.3\theta; \mu\del{ac}=0.3\del{1 - \theta}$ + \item Assume that the stable models are \alert{disjoint events}. + \item Define \alert{weight of an event} as the sum of the weights of the related stable models. + \item Normalize weights to get a (probability) \alert{distribution}. + \end{enumerate} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilities for all Events} + %------------------------------------------------------------- + \vfill + % \begin{center} + % \diagram + % \end{center} + % \vfill + % \scriptsize + + \begin{equation*} + \begin{array}{clr|cc|cc} + & \stablecore{e} + & \# \class{e} + & \pw{\class{e}} + & \pw{e} + & \pr{E = e} + & \pr{E \in \class{e}} + \\ + \hline + % + & \inconsistent + & 37 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + \square + & \indepclass + & 9 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + {\color{magenta!20}\blacksquare } + & \co{a} + & 9 + & \frac{7}{10} + & \frac{7}{90} + & \frac{7}{207} + & \frac{7}{23} + \\[4pt] + % + {\color{cyan!20}\blacksquare } + & ab + & 3 + & \frac{3}{10}\theta + & \frac{1}{10}\theta + & \frac{1}{23}\alert{\theta} + & \frac{3}{23}\theta + \\[4pt] + % + {\color{yellow!20}\blacksquare } + & ac + & 3 + & \frac{3}{10}\co{\theta} + & \frac{1}{10}\co{\theta} + & \frac{1}{23}\alert{\co{\theta}} + & \frac{3}{23}\co{\theta} + \\[4pt] + % + & \co{a}, ab + & 0 + & \frac{7 + 3\theta}{10} + & 0 + & 0 + & 0 + \\[4pt] + % + & \co{a}, ac + & 0 + & \frac{7 + 3\co{\theta}}{10} + & 0 + & 0 + & 0 + % + \\[4pt] + % + {\color{green!20}\blacksquare } + & ab, ac + & 2 + & \frac{3}{10} + & \frac{3}{20} + & \frac{3}{46} + & \frac{3}{23} + \\[4pt] + % + {\color{gray!20}\blacksquare } + & \co{a}, ab, ac + & 1 + & 1 + & 1 + & \frac{10}{23} + & \frac{10}{23} + \\[4pt] + % + \hline + & + & 64 + & + & Z = \frac{23}{10} + & + %& \Sigma = 1 + \end{array} + \end{equation*} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Estimating the Parameters} + %------------------------------------------------------------- + A \alert{sample} can be used to estimate the parameters $\theta$, by minimizing + \begin{equation*} + \err{\theta} := \sum_{e\in\fml{E}} \del{\pr{E = e\given \Theta = \theta} - \pr{S = e}}^2.\label{eq:err.e.s} + \end{equation*} + where + \begin{itemize} + \item $\fml{E}$ is the set of all events, + \item $\pr{E\given \Theta}$ the \textbf{model+parameters} based distribution, + \item $\pr{S}$ is the \textbf{empiric} distribution from the given sample. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Behind Parameter Estimation} + %------------------------------------------------------------- + \vfill + So, we can derive a distribution $\pr{E\given \Theta = \hat{\theta}}$ from a program $P$ and a sample $S$. + \begin{itemize} + \item The sample defines an empiric distribution $\pr{S}$\ldots + \item \ldots that is used to estimate $\theta$ in $\pr{E\given \Theta}$\ldots + \item \ldots and \alert{score the program} $P$ w.r.t.\ that sample using, \emph{e.g.} the $\err{}$ function. + \end{itemize} + \vfill + +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Back to Inductive Logic Programming} + %------------------------------------------------------------- + \vfill + Recall the \emph{Learning Logic Programs from Examples} setting: + \begin{itemize} + \item Given \textbf{positive} and \textbf{negative} examples, and \textbf{background knowledge}\ldots + \item find a \textbf{program}\ldots + \begin{itemize} + \item \ldots using the facts and relations from the \textbf{BK}\ldots + \item \ldots such that \textbf{all the PE} and \textbf{none the NE} examples are entailed. + \end{itemize} + \end{itemize} + \vfill + \begin{quotation} + Given a sample of events, and a set of programs, \alert{the score} of those programs (w.r.t. the sample) \alert{can be used in evolutionary algorithms} while searching for better solutions. + \end{quotation} +\end{frame} +% +%=============================================================== +% +\section{In Conclusion} +% +%=============================================================== +% +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \vfill + \begin{center} + {\huge\bf In Conclusion} + \end{center} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + \begin{itemize} + \item \textbf{Machine Learning} has limitations. + \item As does \textbf{Inductive Logic Programming}. + \item But, distributions can be defined by \textbf{Stochastic Logic Programs}. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + Distributions can be defined by \textbf{Stochastic Logic Programs}. Here we: + \begin{enumerate} + \item Look at the program's \textbf{stable models} and + \item Use them to partition the \textbf{events} and then + \item Using annotated probabilities, define: + \begin{enumerate} + \item a finite \textbf{measure}\ldots + \item that, normalized, is a \textbf{distribution} on all events. + \end{enumerate} + \end{enumerate} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + Distributions can be defined by \textbf{Stochastic Logic Programs}. + \begin{itemize} + \item These distributions might have some \textbf{parameters}, due to indeterminism in the program. + \item A \textbf{sample} can be used to estimate those parameters\ldots + \item \ldots and \textbf{score} programs concurring to describe it. + \item This score a key ingredient in \textbf{evolutionary algorithms}. + \end{itemize} + \begin{quotation} + \ldots and a step towards the \alert{induction of stochastic logic programs} using \textbf{data} and \textbf{background knowledge}. + \end{quotation} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Future Work} + %------------------------------------------------------------- + \vfill + \begin{center} + Induction of Stochastic Logic (ASP) Programs. + \end{center} + \vfill + \begin{enumerate} + \item \textbf{Meta-programming:} formal rules for rule generation. + \item \textbf{Generation}, \textbf{Combination} and \textbf{Mutation} operators. + \item \textbf{Complexity.} + \item \textbf{Applications.} + \item \textbf{Profit.} + \end{enumerate} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\alert{\bf Thank You!}} + \vfill + Questions? + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{References} + \begin{itemize} + \item \href{https://arxiv.org/abs/1801.00631}{Gary Marcus, \emph{Deep Learning: A Critical Appraisal}, 2018}. + \item \href{https://arxiv.org/abs/1911.01547}{François Chollet, \emph{On the Measure of Intelligence}, 2019}. + \item \href{https://arxiv.org/abs/1801.00631}{Bengio \emph{et al.}, \emph{A Meta-Transfer Objective for Learning to Disentangle Causal Mechanisms}, 2019}. + \item \href{https://arxiv.org/abs/1801.00631}{Cropper \emph{et al.}, \emph{Turning 30: New Ideas in Inductive Logic Programming}, 2020}. + \item \href{https://doi.org/10.1201/9781003427421}{Fabrizio Riguzzi, \emph{Foundations of Probabilistic Logic Programming }, 2023}. + \end{itemize} + %------------------------------------------------------------- +\end{frame} +% +\end{document} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{TITLE} + %------------------------------------------------------------- +\end{frame} +% + + diff --git a/text/presentation_01/cima_pdoutmat_2023-12-11/iris_plot.pdf b/text/presentation_01/cima_pdoutmat_2023-12-11/iris_plot.pdf new file mode 100644 index 0000000..ec92c9b Binary files 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+\usepackage[normalem]{ulem} +\usepackage[nice]{nicefrac} +\usepackage{stmaryrd} +\usepackage{acronym} +\usepackage{multicol} +\usepackage{cleveref} + +\usepackage{listings} + +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}} +\newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}} +\newcommand{\co}[1]{\ensuremath{\overline{#1}}} +\newcommand{\condsymb}[2]{\ensuremath{p_{#1|#2}}} +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}} +\newcommand{\deft}[1]{\textbf{#1}} +\newcommand{\emptyevent}{\ensuremath{\vartriangle}} +\newcommand{\ent}{\ensuremath{\lhd}} +\newcommand{\err}[1]{\ensuremath{\mathrm{err}\at{#1}}} +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}} +\newcommand{\given}{\ensuremath{~\middle|~}} +\newcommand{\inconsistent}{\bot} +\newcommand{\indepclass}{\ensuremath{\Diamond}} +\newcommand{\isep}{~,~} +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}} +\newcommand{\larr}{\ensuremath{\leftarrow}} +\newcommand{\lclass}[1]{\ensuremath{\intoc{#1}}} +\newcommand{\lfrac}[2]{\ensuremath{{#1}/{#2}}} +\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}} +\newcommand{\lset}[1]{\ensuremath{#1_{\ast}}} +\newcommand{\naf}{\ensuremath{\sim\!}} +\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}} +\newcommand{\probfact}[2]{\ensuremath{#2:#1}} +\newcommand{\probrule}[3]{\probfact{#2}{#1} \leftarrow #3} +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}} +\newcommand{\pw}[1]{\ensuremath{\mu\at{#1}}} +\newcommand{\pwc}[1]{\ensuremath{\pwcfname\at{#1}}} +\newcommand{\pwcfname}{\ensuremath{\mu_{\textrm{TC}}}} +\newcommand{\q}[1]{\mathbf{#1}} +\newcommand{\rep}[2]{\ensuremath{\left\langle #1 \middle| #2 \right\rangle}} +\newcommand{\smclass}[1]{\ensuremath{\intcc{#1}}} +\newcommand{\stablecore}[1]{\ensuremath{\left\llbracket #1 \right\rrbracket}} +\newcommand{\tcgen}[1]{\ensuremath{\left<#1\right>}} +\newcommand{\uclass}[1]{\ensuremath{\intco{#1}}} +\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}} +\newcommand{\uset}[1]{\ensuremath{#1^{\ast}}} +\newcommand{\yset}[1]{\ensuremath{\left\langle #1 \right\rangle}} +%\newcommand{\tcgen}[1]{\ensuremath{\widehat{#1}}} + +\title{Stochastic Answer Set Programming} +\subtitle{A Research Program} +\author{Francisco Coelho} +\date{November 15, 2023} +\institute[\texttt{fc@uevora.pt}]{ +NOVA LINCS\\ +High Performance Computing Chair\\ +Departamento de Informática, Universidade de Évora +} +% +%=============================================================== +% +\begin{document} +% +\lstset{language=Prolog} +% +%=============================================================== +% +\begin{frame}[plain] + \titlepage + + \begin{center} + \footnotesize This is a join work with \textbf{Salvador Abreu}@DInf and \textbf{Bruno Dinis}@DMat. + \end{center} +\end{frame} +% +%=============================================================== +% +\section*{Motivation} +% +%=============================================================== +% +\begin{frame} + \frametitle{In Short} + + + \begin{itemize} + \item About \textbf{Machine Learning}: + \begin{itemize} + \item Vector or matrix based models lack ``structure''. + \item Large models don't \emph{explain} data. + \end{itemize} + \item About \textbf{Logic Programs}: + \begin{itemize} + \item Logic programs formalize knowledge. + \item Logic doesn't \emph{capture} uncertainty and is \emph{fragile} to noise. + \end{itemize} + \item \textbf{Probabilistic Logic Programs} extend formal knowledge with probabilities. + \begin{itemize} + \item How to propagate probabilities through rules? + \end{itemize} + \end{itemize} + \vfill + \begin{center} + \alert{\bf Goal:} Combine Logic and Statistics. + \end{center} +\end{frame} +% +%=============================================================== +% \begin{frame} +% %------------------------------------------------------------- +% %[fragile] +% %------------------------------------------------------------- +% \frametitle{Statistics and Machine Learning} +% %------------------------------------------------------------- +% \vfill +% \begin{itemize} +% \item \textbf{Data Analysis:} understand and summarize. +% \item \textbf{Model Building:} tools and techniques. +% \item \textbf{Model Evaluation:} performance. +% \item [] +% \item To calculate the accuracy/precision/recall of a model. +% \item To choose the right model for a problem. +% \item To tune the hyperparameters of a model. +% \end{itemize} +% \vfill +% \begin{center}\footnotesize +% Highlights of \texttt{Bard}'s (Google's LLM) reply to ``Explain what is the role of statistics in machine learning''. +% \end{center} +% \end{frame} +% +%=============================================================== +% +\section{Machine Learning} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\bf Machine Learning} + \end{center} + \vfill + \begin{itemize} + \item Standard Example --- Iris Classification + \item Assumptions of Machine Learning + \item Where Machine Learning Fails + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{The Standard Example --- Iris Classification} + %------------------------------------------------------------- + \small + \begin{center} + Learning Functions: \href{https://en.wikipedia.org/wiki/Iris_flower_data_set}{The famous Iris database} + \end{center} + \begin{columns} + \column{0.4\textwidth} + \begin{itemize}\setlength{\itemsep}{-0.5em} + \item[$x_1$] sepal length. + \item[$x_2$] sepal width. + \item[$x_3$] petal length. + \item[$x_4$] petal width. + \item[$y$] species (one of \emph{setosa}, \emph{versicolor}, \emph{virginica}). + \end{itemize} + \column{0.6\textwidth} + \begin{center} + \includegraphics[width=\textwidth]{iris_plot.pdf} + \end{center} + \end{columns} + \vfill + \begin{itemize} + \item A \emph{setosa} model: $ - 0.40 -0.65x_1 + 1.00x_2 > 0.00$. + \item A general \textbf{model template}: + $$ + f_\theta(\vec{x}) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \theta_3 x_3 + \theta_4 x_4> 0 + $$ + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Assumptions of Machine Learning} + %------------------------------------------------------------- + \begin{itemize} + \item Each instance is described in a \textbf{single row} by a \textbf{fixed set of features} + $$ + \begin{array}{cccc|c} + \mathbf{x_1} & \mathbf{x_2} & \ldots & \mathbf{x_n} & \mathbf{y} \\ + \hline + x_{11} & x_{21} & \ldots & x_{n1} & y_1 \\ + & & \vdots \\ + x_{1m} & x_{2m} & \ldots & x_{nm} & y_m \\ + \end{array} + .$$ + \item Instances are \textbf{independent} of one another, \textbf{given the model} + $$ + y = f_\theta(\vec{x}). + $$ + \item Parameters \textbf{minimize estimation error} \emph{e.g.} + $$ + \hat{\theta} = \arg\min_\theta \sum_i \norm{y_i - f_\theta(\vec{x_{i}})}. + $$ + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Failing Assumptions\hfill~1/2} + %------------------------------------------------------------- + \vfill + \begin{columns}[c] + \column{0.5\textwidth} + $$ + \begin{array}{ccc} + \text{student} & \text{course} & \text{grade} \\ + \hline + s_1 & c_1 & a \\ + s_2 & c_1 & c \\ + s_1 & c_2 & b \\ + s_2 & c_3 & b \\ + s_3 & c_2 & b \\ + s_4 & c_3 & b \\ + s_3 & c_4 & X \\ + s_4 & c_4 & Y \\ + \end{array} + $$ + \column{0.5\textwidth} + \emph{Infer $X \succ_g Y$.} + + \bigskip + ``\emph{Proof.}'' + + \bigskip + $s_1 \succ_s s_2$ (rows 1 \& 2)\\ so $c_2 \succ_c c_3$ (3 \& 4). + + \bigskip + Then $s_3 \succ_s s_4$ (5 \& 6)\\ therefore $X \succ_g Y\qed$. + \end{columns} + \vfill + \begin{itemize} + \item Instances are not independent. + \item Small dataset, so \emph{no statistics}. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Failing Assumptions\hfill~2/2} + %------------------------------------------------------------- + \footnotesize + \begin{center} + \textbf{Prompt:} Given \texttt{f(machine) = e}, \texttt{f(learning) = g} and \texttt{f(algorithm) = m}, what is the value of \texttt{f(ksadbf)}? + \end{center} + + \vfill + \begin{columns}[t] + \scriptsize + \column{0.35\textwidth} + $$ + \begin{array}{r|l} + \textbf{input} & \textbf{output} \\ + \hline + machine & e \\ + learning & g \\ + algorithm & m \\ + ksadbf & ? + \end{array} + $$ + + \bigskip + \textbf{Bard:} \tt f(ksadbf) = 11. + + We can see that the value of f(x) is simply the first letter of x minus 96. + + \column{0.65\textwidth} + + \textbf{ChatGPT:} \tt The information provided seems to define a function $f$ where certain inputs are associated with specific outputs. Based on the given information: + \begin{itemize} + \item $f(\text{machine}) = e$ + \item $f(\text{learning}) = g$ + \item $f(\text{algorithm}) = m$ + \end{itemize} + However, there is no information given about \( f(\text{ksadbf}) \). Without additional information or a specific rule that determines the output for \( f(\text{ksadbf}) \), \emph{it is not possible to determine its value}. + % + If there is a pattern or rule for how the outputs are assigned based on the inputs, you would need to provide that information for me to help determine the value of \( f(\text{ksadbf}) \). + \end{columns} + + \vfill + \begin{center} + No help from state-of-the-art Natural Language Processing or LLMs. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Where Machine Learning Fails} + %------------------------------------------------------------- + \begin{itemize} + \item No \emph{interpretability}: large models ($\dim \theta \sim \text{millions}$) are \alert{not understandable by humans}. + \item Need for \emph{large number of training} examples. + \item Unfit for \emph{multiple relations}, \emph{intersample dependencies} and \emph{variable features}: The \textit{Students, Courses and Results} and \emph{Last Letter} examples. + \item Poor \emph{generalization}: A model trained in the iris dataset is useless in any other domain. + \item \alert{No use of background knowledge}. + \end{itemize} +\end{frame} +% +%=============================================================== +% +\section{Logic Programming} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\bf Logic Programming} + \end{center} + \vfill + \begin{itemize} + \item An Example of Logic Programming. + \item Inductive Logic Programming. + \item Where ILP Fails. + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + [fragile] + %------------------------------------------------------------- + \frametitle{An Example of Logic Programming} + %------------------------------------------------------------- + \begin{columns}[c] + \column{0.4\textwidth} + \begin{tikzpicture}[>=Latex] + \node[vert] (v6) {6}; + \node[vert, above left = of v6] (v3) {3}; + \node[vert, above right = of v6] (v5) {5}; + \node[vert, below left = of v6] (v1) {1}; + \node[vert, below right = of v6] (v2) {2}; + \node[vert, below right = of v1] (v4) {4}; + \draw[->] (v6) to (v3); + \draw[<->] (v6) to (v5); + \draw[<->] (v6) to (v2); + \draw[<->] (v3) to (v5); + \draw[<->] (v3) to (v1); + \draw[->] (v3) to[bend right, out=225, in =180,relative=false] (v4); + \draw[->] (v1) to (v2); + \draw[<->] (v1) to (v4); + \draw[->] (v2) to (v5); + \draw[<->] (v2) to (v4); + \draw[->] (v5) to[bend left, out=-45, in=0,relative=false] (v4); + \end{tikzpicture} + \column{0.6\textwidth} + \scriptsize + \begin{lstlisting} +node(1..6). + +edge(1,2). edge(2,4). edge(3,1). +edge(4,1). edge(5,3). edge(6,2). +edge(1,3). edge(2,5). edge(3,4). +edge(4,2). edge(5,4). edge(6,3). +edge(1,4). edge(2,6). edge(3,5). +edge(5,6). edge(6,5). + +col(r). col(b). col(g). + +1 { color(X,C) : col(C) } 1 :- node(X). +:- edge(X,Y), color(X,C), color(Y,C). + +#show color/2. + \end{lstlisting} + \end{columns} + \vfill + \scriptsize + \begin{lstlisting} +color(2,b) color(1,g) color(4,r) color(3,b) color(5,g) color(6,r) +color(1,r) color(2,b) color(4,g) color(3,b) color(5,r) color(6,g) +color(1,r) color(2,g) color(4,b) color(3,g) color(5,r) color(6,b) +color(1,b) color(2,g) color(4,r) color(3,g) color(5,b) color(6,r) +color(2,r) color(1,g) color(4,b) color(3,r) color(5,g) color(6,b) +color(2,r) color(1,b) color(4,g) color(3,r) color(5,b) color(6,g) + \end{lstlisting} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Inductive Logic Programming} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + Learning Logic Programs from Examples. + \end{center} + + \vfill + \alert{Generate rules} that\ldots + \begin{itemize} + \item use \textbf{background knowledge} + $$ + \begin{aligned} + & parent(john,mary), & & parent(david,steve), \\ + & parent(kathy,mary), & & female(kathy), \\ + & male(john), & & male(david). + \end{aligned} + $$ + \item to entail all the \textbf{positive examples}, $father(john,mary), father(david,steve)$, + \item but none of the \textbf{negative examples}. $father(kathy,mary), father(john,steve)$, + + \end{itemize} + + \vfill + A \textbf{solution} is $$father(X,Y) \leftarrow parent(X,Y) \wedge male(X).$$ + +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Where Logic Programming Fails} + %------------------------------------------------------------- + \begin{center} + Meanwhile, in the \textbf{real world}, samples are \emph{incomplete} and come with \emph{noise}. + \end{center} + \vfill + \textbf{Logic inference is \alert{fragile}}: a mistake in the transcription of a fact is dramatic to the consequences: + \begin{itemize} + \item $parent(david,mary)$. + \item $parent(jonh,mary)$. + \end{itemize} + \vfill + \begin{center} + The statistic essence of machine learning provides \alert{robustness}. + \end{center} +\end{frame} +% +%=============================================================== +% +\section{Probabilistic Logic Programming} +% +%=============================================================== +% +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \vfill + \begin{center} + {\huge\bf Probabilistic Logic Programming} + \end{center} + \vfill + \begin{itemize} + \item Define distributions from logic programs. + \item Stochastic ASP: Specifying distributions. + \end{itemize} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilistic Logic Programs (PLPs)} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + Logic programs \textbf{annotated} with probabilities. + \end{center} + + \vfill + \begin{columns}[c] + + \column{0.4\textwidth} + \begin{tikzpicture}[>=Latex] + \node[draw, rounded rectangle] (A) {$Alarm$}; + \node[draw, rounded rectangle, below = of A] (J) {$Johncalls$}; + \draw[->] (A) to (J); + \end{tikzpicture} + + \column{0.6\textwidth} + $ + \begin{aligned} + alarm:0.00251, & \\ + johncalls:0.9 & \leftarrow alarm, \\ + johncalls:0.05 & \leftarrow \neg alarm + \end{aligned} + $ + \end{columns} + + \vfill + \begin{itemize} + \item \alert{$alarm:0.00251$} is $alarm \vee \neg alarm$ plus $P(Alarm = true) = 0.00251$. + \item \alert{$johncalls:0.9 \leftarrow alarm$} is + $$P\del{Johncalls = true \middle| Alarm = true} = 0.9$$ + \end{itemize} + + \vfill + \begin{center} + Any bayesian network can be represented by a PLP. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Distributions from Logic Programs} + %------------------------------------------------------------- + \vfill + The program + $$ + \begin{aligned} + alarm:0.00251, & \\ + johncalls:0.9 & \leftarrow alarm, \\ + johncalls:0.05 & \leftarrow \neg alarm + \end{aligned} + $$ + entails four possible models (or worlds): + $$ + \begin{array}{r|r} + \text{model} & \text{probability} \\ + \hline + alarm, johncalls & 0.002259 \\ + alarm, \neg johncalls & 0.000251 \\ + \neg alarm, johncalls & 0.049874 \\ + \neg alarm, \neg johncalls & 0.947616 \\ + \end{array} + $$ + \begin{itemize} + \item \alert{\textbf{Models}} are special sets of \emph{literals} \textbf{entailed} from the program. + \item Probabilities \emph{propagate} from facts, through rules. + \end{itemize} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{There's a Problem\ldots} + %------------------------------------------------------------- + \vfill + The program + $$ + \begin{aligned} + alarm:0.00251, & \\ + johncalls \vee marycalls & \leftarrow alarm + \end{aligned} + $$ + entails three \alert{stable} (\emph{i.e.}\ minimal) models + $$ + \begin{array}{r|l} + \text{model} & \text{probability} \\ + \hline + alarm, johncalls & x \\ + alarm, marycalls & y \\ + \neg alarm & 0.99749 + \end{array} + $$ + but \textbf{no single way to set $x,y$}. + \vfill + \begin{center} + Some \emph{Probabilistic Logic Programs} define more than one joint distribution. + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{\ldots and an Oportunity} + %------------------------------------------------------------- + \begin{center} + Some \emph{PLP}s define more than one joint distribution. + \end{center} + \vfill + \begin{itemize} + \item There is \textbf{no single probability assignment} from the facts stable models: $x,y \in \intcc{0,1}$. + \item But any assignment is bound by Kolmogorov's axioms, and \textbf{forms equations} such as: + $$x + y = P\del{alarm}.$$ + \item Existing \textbf{data can be used to estimate the unknowns} in those equations. + \end{itemize} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Stable Models, Events and Probabilities} + %------------------------------------------------------------- + \vfill + \begin{center} + What are we talking about? + \end{center} + \vfill + \begin{itemize} + \item A logic program has \textbf{atoms} (and \textbf{literals}) and \textbf{rules}: + $$ + \begin{aligned} + &male(john), \neg parent(kathy,mary), \\ + &father(X, Y) \leftarrow parent(X, Y) \wedge male(X). + \end{aligned} + $$ + \item A \alert{\textbf{stable model}} is a \textbf{minimal} model that contains: + \begin{itemize} + \item program's \emph{facts}: $parent(john,mary),~male(john)$. + \item consequences, by the \emph{rules}: $father(john,mary)$. + \end{itemize} + \item Some programs have more than one model: + \begin{tabular}{c|c} + \textbf{Logic Program} & \textbf{Stable Models} \\ + \hline + $a \vee \neg a, b \vee c \leftarrow a$ + & + $\set{\neg a}, \set{a, b}, \set{a, c}$ + \end{tabular} + \end{itemize} + \vfill + \begin{center} + How to propagate probability from annotated facts to other \emph{events}? + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Logic Programs and Probabilities} + %------------------------------------------------------------- + % \vfill + % \begin{center} + % The \textbf{space of events}, $\Omega$, is the set of all sets of literals. + % \end{center} + \vfill + \begin{itemize} + \item Consider the literals of a logic program $$L = \set{a_1, \ldots a_n, \neg a_1, \ldots \neg a_n}.$$ + \item Any model of that program is a (consistent) subset of $L$. + \item Let $\Omega = \mathbf{P}\del{L}$, \emph{i.e.} an \alert{event} $e$ is a subset of $L$, $e \subseteq L$. + \begin{itemize} + \item Setting a probability for some events seems straightforward: $P\del{\neg alarm} = 0.997483558$. + \item For others, not so much: + \begin{itemize} + \item $P\del{alarm, johncalls}$, $P\del{johncalls, marycalls, alarm}$, $P\del{marycalls}$? + \item $P\del{alarm, \neg alarm}$, $P\del{\neg marycalls}$? + \end{itemize} + \end{itemize} + \end{itemize} + \vfill + \begin{center} + How to \alert{propagate} probability from \emph{facts} to \emph{consequences} or other \emph{events}? + \end{center} +\end{frame} +% +\newcommand{\diagram}{ + \resizebox{!}{24ex}{ + \begin{tikzpicture}%[scale=0.6, every node/.style={scale=0.6}] + \node[event] (E) {$\emptyevent$}; + \node[tchoice, above left = of E, pin={[red!50!black]175:$0.3$}] (a) {$a$}; + \node[smodel, above left = of a, pin={[red!50!black]175:$0.3\theta$}] (ab) {$ab$}; + \node[smodel, above right = of a, pin={[red!50!black]175:$0.3\co{\theta}$}] (ac) {$ac$}; + \node[event, below = of ab] (b) {$b$}; + \node[event, below = of ac] (c) {$c$}; + \node[event, above right = of ab] (abc) {$abc$}; + \node[event, above left = of ab] (abC) {$\co{c}ab$}; + \node[event, above right = of ac] (aBc) {$\co{b}ac$}; + \node[indep, right = of ac] (bc) {$bc$}; + \node[tchoice, smodel, below right = of bc, pin={[red!50!black]175:$0.7$}] (A) {$\co{a}$}; + \node[event, above = of A] (Ac) {$\co{a}c$}; + \node[event, above right = of Ac] (Abc) {$\co{a}bc$}; + % ---- + \path[draw, rounded corners, fill=cyan, opacity=0.1] + (ab.west) -- + (ab.north west) -- + % + (abC.south west) -- + (abC.north west) -- + (abC.north) -- + % + (abc.north east) -- + (abc.east) -- + (abc.south east) -- + % + (ab.north east) -- + (ab.east) -- + (ab.south east) -- + % + (a.north east) -- + % + (E.north east) -- + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (b.south west) -- + % + (ab.west) + ; + % ---- + \path[draw, rounded corners, fill=yellow, opacity=0.1] + (ac.south west) -- + (ac.west) -- + (ac.north west) -- + % + (abc.south west) -- + (abc.west) -- + (abc.north west) -- + % + (aBc.north east) -- + (aBc.east) -- + (aBc.south east) -- + % + (ac.north east) -- + % + (c.east) -- + % + (E.east) -- + (E.south east) -- + (E.south) -- + (E.south west) -- + % + (a.south west) -- + (a.west) -- + (a.north west) -- + (a.north) -- + % + (ac.south west) + ; + % ---- + \path[draw, rounded corners, fill=magenta, opacity=0.1] + % (A.north west) -- + % + (Ac.north west) -- + % + (Abc.north west) -- + (Abc.north) -- + (Abc.north east) -- + (Abc.south east) -- + % + % (Ac.north east) -- + % (Ac.east) -- + % + % (A.east) -- + (A.south east) -- + % + (E.south east) -- + (E.south) -- + (E.south west) -- + (E.west) -- + (E.north west) -- + % + (Ac.north west) + ; + \end{tikzpicture} + } +} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Classes of Events} + %------------------------------------------------------------- + \small + + \vfill + \begin{center} + \diagram + \end{center} + + \vfill + \begin{center} + \begin{tabular}{lr} + $\begin{aligned} + \probfact{0.3}{a} \\ + b \vee c & \leftarrow a + \end{aligned}$ + & + $\co{a} = \set{\neg a}, ab = \set{a, b}, ac = \set{a, c}$ + \end{tabular} + \end{center} + + \vfill + \begin{itemize} + \item Define \alert{equivalence classes} for all events, based on $\subseteq, \supseteq$ relations with the \textbf{stable models}. + \item This example shows $6$ out of $2^3 + 1$ classes. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilities for all Events} + %------------------------------------------------------------- + \vfill + \begin{center} + \diagram + \end{center} + \vfill + \footnotesize + \begin{enumerate} + \item Set \alert{weights} in the stable models (shaded nodes), using parameters when needed: $\mu\del{\co{a}} = 0.7; \mu\del{ab} = 0.3\theta; \mu\del{ac}=0.3\del{1 - \theta}$ + \item Assume that the stable models are \alert{disjoint events}. + \item Define \alert{weight of an event} as the sum of the weights of the related stable models. + \item Normalize weights to get a (probability) \alert{distribution}. + \end{enumerate} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Probabilities for all Events} + %------------------------------------------------------------- + \vfill + % \begin{center} + % \diagram + % \end{center} + % \vfill + % \scriptsize + + \begin{equation*} + \begin{array}{clr|cc|cc} + & \stablecore{e} + & \# \class{e} + & \pw{\class{e}} + & \pw{e} + & \pr{E = e} + & \pr{E \in \class{e}} + \\ + \hline + % + & \inconsistent + & 37 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + \square + & \indepclass + & 9 + & 0 + & 0 + & 0 + & 0 + \\[4pt] + % + {\color{magenta!20}\blacksquare } + & \co{a} + & 9 + & \frac{7}{10} + & \frac{7}{90} + & \frac{7}{207} + & \frac{7}{23} + \\[4pt] + % + {\color{cyan!20}\blacksquare } + & ab + & 3 + & \frac{3}{10}\theta + & \frac{1}{10}\theta + & \frac{1}{23}\alert{\theta} + & \frac{3}{23}\theta + \\[4pt] + % + {\color{yellow!20}\blacksquare } + & ac + & 3 + & \frac{3}{10}\co{\theta} + & \frac{1}{10}\co{\theta} + & \frac{1}{23}\alert{\co{\theta}} + & \frac{3}{23}\co{\theta} + \\[4pt] + % + & \co{a}, ab + & 0 + & \frac{7 + 3\theta}{10} + & 0 + & 0 + & 0 + \\[4pt] + % + & \co{a}, ac + & 0 + & \frac{7 + 3\co{\theta}}{10} + & 0 + & 0 + & 0 + % + \\[4pt] + % + {\color{green!20}\blacksquare } + & ab, ac + & 2 + & \frac{3}{10} + & \frac{3}{20} + & \frac{3}{46} + & \frac{3}{23} + \\[4pt] + % + {\color{gray!20}\blacksquare } + & \co{a}, ab, ac + & 1 + & 1 + & 1 + & \frac{10}{23} + & \frac{10}{23} + \\[4pt] + % + \hline + & + & 64 + & + & Z = \frac{23}{10} + & + %& \Sigma = 1 + \end{array} + \end{equation*} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Estimating the Parameters} + %------------------------------------------------------------- + A \alert{sample} can be used to estimate the parameters $\theta$, by minimizing + \begin{equation*} + \err{\theta} := \sum_{e\in\fml{E}} \del{\pr{E = e\given \Theta = \theta} - \pr{S = e}}^2.\label{eq:err.e.s} + \end{equation*} + where + \begin{itemize} + \item $\fml{E}$ is the set of all events, + \item $\pr{E\given \Theta}$ the \textbf{model+parameters} based distribution, + \item $\pr{S}$ is the \textbf{empiric} distribution from the given sample. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Behind Parameter Estimation} + %------------------------------------------------------------- + \vfill + So, we can derive a distribution $\pr{E\given \Theta = \hat{\theta}}$ from a program $P$ and a sample $S$. + \begin{itemize} + \item The sample defines an empiric distribution $\pr{S}$\ldots + \item \ldots that is used to estimate $\theta$ in $\pr{E\given \Theta}$\ldots + \item \ldots and \alert{score the program} $P$ w.r.t.\ that sample using, \emph{e.g.} the $\err{}$ function. + \end{itemize} + \vfill + +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Back to Inductive Logic Programming} + %------------------------------------------------------------- + \vfill + Recall the \emph{Learning Logic Programs from Examples} setting: + \begin{itemize} + \item Given \textbf{positive} and \textbf{negative} examples, and \textbf{background knowledge}\ldots + \item find a \textbf{program}\ldots + \begin{itemize} + \item \ldots using the facts and relations from the \textbf{BK}\ldots + \item \ldots such that \textbf{all the PE} and \textbf{none the NE} examples are entailed. + \end{itemize} + \end{itemize} + \vfill + \begin{quotation} + Given a sample of events, and a set of programs, \alert{the score} of those programs (w.r.t. the sample) \alert{can be used in evolutionary algorithms} while searching for better solutions. + \end{quotation} +\end{frame} +% +%=============================================================== +% +\section{In Conclusion} +% +%=============================================================== +% +\begin{frame} + %------------------------------------------------------------- %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \vfill + \begin{center} + {\huge\bf In Conclusion} + \end{center} + \vfill + %------------------------------------------------------------- +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + \begin{itemize} + \item \textbf{Machine Learning} has limitations. + \item As does \textbf{Inductive Logic Programming}. + \item But, distributions can be defined by \textbf{Stochastic Logic Programs}. + \end{itemize} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + Distributions can be defined by \textbf{Stochastic Logic Programs}. Here we: + \begin{enumerate} + \item Look at the program's \textbf{stable models} and + \item Use them to partition the \textbf{events} and then + \item Using annotated probabilities, define: + \begin{enumerate} + \item a finite \textbf{measure}\ldots + \item that, normalized, is a \textbf{distribution} on all events. + \end{enumerate} + \end{enumerate} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + % \frametitle{In Conclusion} + %------------------------------------------------------------- + Distributions can be defined by \textbf{Stochastic Logic Programs}. + \begin{itemize} + \item These distributions might have some \textbf{parameters}, due to indeterminism in the program. + \item A \textbf{sample} can be used to estimate those parameters\ldots + \item \ldots and \textbf{score} programs concurring to describe it. + \item This score a key ingredient in \textbf{evolutionary algorithms}. + \end{itemize} + \begin{quotation} + \ldots and a step towards the \alert{induction of stochastic logic programs} using \textbf{data} and \textbf{background knowledge}. + \end{quotation} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{Future Work} + %------------------------------------------------------------- + \vfill + \begin{center} + Induction of Stochastic Logic (ASP) Programs. + \end{center} + \vfill + \begin{enumerate} + \item \textbf{Meta-programming:} formal rules for rule generation. + \item \textbf{Generation}, \textbf{Combination} and \textbf{Mutation} operators. + \item \textbf{Complexity.} + \item \textbf{Applications.} + \item \textbf{Profit.} + \end{enumerate} + \vfill +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{} + %------------------------------------------------------------- + \begin{center} + \vfill + {\huge\alert{\bf Thank You!}} + \vfill + Questions? + \end{center} +\end{frame} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{References} + \begin{itemize} + \item \href{https://arxiv.org/abs/1801.00631}{Gary Marcus, \emph{Deep Learning: A Critical Appraisal}, 2018}. + \item \href{https://arxiv.org/abs/1911.01547}{François Chollet, \emph{On the Measure of Intelligence}, 2019}. + \item \href{https://arxiv.org/abs/1801.00631}{Bengio \emph{et al.}, \emph{A Meta-Transfer Objective for Learning to Disentangle Causal Mechanisms}, 2019}. + \item \href{https://arxiv.org/abs/1801.00631}{Cropper \emph{et al.}, \emph{Turning 30: New Ideas in Inductive Logic Programming}, 2020}. + \item \href{https://doi.org/10.1201/9781003427421}{Fabrizio Riguzzi, \emph{Foundations of Probabilistic Logic Programming }, 2023}. + \end{itemize} + %------------------------------------------------------------- +\end{frame} +% +\end{document} +% +%=============================================================== +\begin{frame} + %------------------------------------------------------------- + %[fragile] + %------------------------------------------------------------- + \frametitle{TITLE} + %------------------------------------------------------------- +\end{frame} +% + + diff --git a/text/summaries/00_ASP.md b/text/summaries/00_ASP.md new file mode 100644 index 0000000..029c631 --- /dev/null +++ b/text/summaries/00_ASP.md @@ -0,0 +1,239 @@ +# Answer Set Programming + +> **Answer set programming (ASP) is a form of declarative programming oriented towards difficult (primarily NP-hard) search problems.** +> +> It is **based on the stable model (answer set) semantics** of logic programming. In ASP, search problems are reduced to computing stable models, and answer set solvers ---programs for generating stable models--- are used to perform search. + +--- + +**ASP** "programs" generates "deduction-minimal" models _aka_ **stable models** or **answer sets**. +- Given an ASP program $P$, a model $X$ of $P$ is a set where each element $x \in X$ has a proof using $P$. +- In a "deduction-minimal" model $X$ each element $x \in X$ has a proof using $P$. Non-minimal models have elements without a proof. + +## Key Questions + +1. What is the relation between ASP and Prolog? + 1. **Prolog** performs **top-down query evaluation**. Solutions are extracted from the instantiation of variables of successful queries. + 2. **ASP** proceeds in two steps: first, **grounding** generates a (finite) _propositional representation of the program_; second, **solving** computes the _stable models_ of that representation. +2. What are the roles of **grounding** with `gringo` and **solving** with `clasp`? +3. Can ASP be used to **pLP**? + 1. What are the key probabilistic tasks/questions/problems? + 2. Where does distribution semantics enters? What about **pILP**? +4. Can the probabilistic task control the grounding (`gringo`) or solving (`clasp`) steps in ASP? +5. Can ASP replace kanren? + 1. As much as ASP can replace Prolog. + +## Formal Foundations + +### Common Concepts and Notation + + context | true, false | if | and | or | iff | default negation | classical negation +---------|------------|----|-----|----|-----|------|----- +source | | `:-` | `,` | `|` | | `not` | `-` +logic prog. | | ← | , | ; | | ̃ | ¬ +formula | ⊤, ⊥ | → | ∧ | ∨ | ↔ | ̃ | ¬ + +> - **default negation** or **negation as failure (naf)**, `not a` ($\sim a$), means "_no information about `a`_". +> - **classical negation** or **strong negation**, `-a` ($\neg a$), means "_positive information about `-a`_" ie "_negative information about `a`_". Likewise `a`: "_positive informations about `a`_". +> - The symbol `not` ($\sim$), is a new logical connective; `not a` ($\sim a$) is often read as "_it is not believed that `a` is true_" or "_there is no proof of `a`_". Note that this does not imply that `a` is believed to be false. + +- **Interpretation.** A _boolean_ interpretation is a function from ground atoms to **⊤** and **⊥**. It is represented by the atoms mapped to **⊤**. + - if u, v are two interpretations **u ≤ v** iff u ⊆ v under this representation. + - **partial interpretations** are represented by ( {true atoms}, {false atoms}) leaving the undefined atoms implicit. + - an **ordered boolean assignment** $a$ over $dom(a)$ in represented by a sequence $a = (V_ix_i | i \in 1:n)$ where $V_i$ is either $\top$ or $\bot$ and each $x_i\in dom(a)$. + - $a^\top \subseteq a$ such that $\top x \in a$; $a^\bot \subseteq a$ such that $\bot x \in a$. + - An ordered assignment $(a^\top, a^\bot)$ is a partial boolean interpretation. +- Subsets have a partial order for the $\subset$ relation; remember maximal and minimal elements. +- Directed graphs; Path; **Strongly connected** iff all vertex pairs (a,b) are connected; The **strongly connected components** are the strongly connected subgraphs. + +### Basic ASP syntax and semantics + +- A **definite clause** is, by definition, $a_0 \vee \neg a_1 \vee \cdots \vee \neg a_n$, a disjunction with exactly one positive atom. + - Also denoted $a_0 \leftarrow a_1 \wedge \cdots \wedge a_n$. + - **A set of definite clauses has exactly one smallest model.** +- A **horn clause** has at most one positive atom. + - A horn clause without positive atom is an _integrity_ constraint - _a conjunction that **can't** hold_. + - **A set of horn clauses has one or zero smallest models.** +- If $P$ is a **positive program**: + - A set $X$ is **closed** under $P$ if $head(r) \in X$ if $body^+(r) \subset X$. + - $Cn(P)$ is, by definition, the set of **consequences of $P$**. + - $Cn(P)$ is the smallest set closed under $P$. + - $Cn(P)$ is the $\subseteq$-smallest model of $P$. + - The **stable model** of $P$ is, by definition, $Cn(P)$. + - If $P$ is a positive program, $Cn(P)$ is the smallest model of the definite clauses of $P$. + +#### Example calculation of stable models + +Consider the program P: +```prolog +person(joey). +male(X); female(X) :- person(X). +bachelor(X) :- male(X), not married(X). +``` + +1. Any SM of P must have the **fact** `person(joey)`. +2. Therefore the **grounded rule** `male(joey) ; female(joey) :- person(joey).` entails that the SMs of P either have `male(joey)` or `female(joey)`. +3. Any **SM must contain** either A: `{person(joey), male(joey)}` or B: `{person(joey), female(joey)}`. +4. In **the reduct** of P in A we get the rule `bachelor(joey) :- male(joey).` and therefore `bachelor(joey)` must be in a SM that contains A. Let A1: `{person(joey), male(joey), bachelor(joey)}`. +5. No further conclusions result from P on A1. Therefore A1 is a SM. +6. Also no further conclusions result from P on B; It is also a SM. +7. The SMs of P are: + 1. `{person(joey), male(joey), bachelor(joey)}` + 2. `{person(joey), female(joey)}` + + +```prolog +-a. +not a. +% +% { -a } +% +-a. +a. +% +% UNSAT. +% +not a. +a. +% +% UNSAT +% +%---------------------------------------- +% +a. +%% Answer: 1 +%% a +%% SATISFIABLE +% +% There is (only) one (stable) model: {a} +% +%---------------------------------------- +% +-a. +%% Answer: 1 +%% -a +%% SATISFIABLE +% +% Same as above. +% +%---------------------------------------- +% +--a. +%% *** ERROR: (clingo): parsing failed +% +% WTF? +% +%---------------------------------------- +% +not a. +%% Answer: 1 +%% +%% SATISFIABLE +% +% ie there is (only) one (stable) model: {} +% +% This program states that there is no information. +% In particular, there is no information about a. +% Therefore there are no provable atoms. +% Hence the empty set is a stable model. +% +%---------------------------------------- +% +not not a. +%% UNSATISFIABLE +% +% ie no models. Because +% 1. No model can contain ~p. +% 2. Any model contains all the facts. +% 3. Suppose X is a model. +% 4. Since ~~a is a fact, by 2, ~~a ∈ X. +% 5. But, by 1, ~~a ∉ X. +% 6. Therefore there are no models. +% +%---------------------------------------- +% +not -a. +%% Answer: 1 +%% +%% SATISFIABLE +% +% Same as ~a. +% +%---------------------------------------- +% +b. +a;-a. +not a :- b. +% Answer: 1 +% b -a +% SATISFIABLE +% +% 1. Any model must contain b (fact b). +% 2. Any models entails ~a (rule not a :- b.). +% 3. Any model must contain one of a or ¬a (rule a;-a). +% 4. No model can contain both a and ~a. +% 5. Therefore any model must contain {b, ¬a}, which is stable. +% +% Q: Why ~a does not contradicts -a +% A: Not sure. Maybe because "~a" states that no model can contain a but says nothing about ¬a. +% +%---------------------------------------- +% +b. +a;c. +% Answer: 1 +% b c +% Answer: 2 +% b a +% SATISFIABLE +% +% 1. Any model must have b. +% 2. Any model must have one of a or c. +% 3. No model with both a and c is minimal because either one satisfies a;c +``` + +- Why is the double strong negation, `--a`, a syntax error but the double naf, `not not a` is not? + +#### Definitions and basic propositions +1. Let $\cal{A}$ be a **set of ground atoms**. +2. A **normal rule** $r$ has the form $a \leftarrow b_1, \ldots, b_m, \sim c_1, \ldots, \sim c_n$ with $0 \leq m \leq n$. + - _Intuitively,_ the head $a$ is true if **each one of the $b_i$ has a proof** and **none of the $c_j$ has a proof**. +3. A **program** is a finite set of rules. +4. The **head** of the rule is $\text{head}(r) = a$; The **body** is $\text{body}(r) = \left\lbrace b_1, \ldots, b_m, \sim c_1, \ldots, \sim c_n \right\rbrace$. +5. A **fact** is a rule with empty body and is simply denoted $a$. +6. A **literal** is an atom $a$ or the default negation $\sim a$ of an atom. +7. Let $X$ be a set of literals. $X^+ = X \cap \cal{A}$ and the $X^- = \left\lbrace p\middle| \sim p \in X\right\rbrace$. +9. The set of atoms that occur in program $P$ is denoted $\text{atom}(P)$. Also $\text{body}(P) = \left\lbrace \text{body}(r)~\middle|~r \in P\right\rbrace$. At last, $\text{body}_P(a) = \left\lbrace \text{body}(r)~\middle|~r \in P \wedge \text{head}(r) = a\right\rbrace$. +10. A **model** of the program $P$ is a set of ground atoms $X \subseteq \cal{A}$ such that, for each rule $r \in P$, $$\text{body}^+(r) \subseteq X \wedge \text{body}^-(r) \cap X = \emptyset \to \text{head}(r) \in X.$$ +8. A rule $r$ is **positive** if $\text{body}(r)^- = \emptyset$; A program is positive if all its rules are positive. +11. _A positive program has an unique $\subseteq$-minimal model._ **Is this the link to prolog?** +12. The **reduct** of a formula $f$ relative to $X$ is the formula $f^X$ that results from $f$ replacing each maximal sub-formula _not satisfied by $X$_ by $\bot$. +13. The **reduct** of program $P$ relative to $X$ is $$P^X = \left\lbrace \text{head}(r) \leftarrow \text{body}^+(r) \middle| r \in P \wedge \text{body}^-(r) \cap X = \emptyset \right\rbrace.$$ Thus $P^X$ results from + 1. Remove every rule with a naf literal $\sim a$ where $a \in X$. + 2. Remove the naf literals of the remaining rules. +14. Since $P^X$ is a positive program, it has a unique $\subseteq$-minimal model. +15. $X$ is a **stable model** of $P$ if $X$ is the $\subseteq$-minimal model of $P^X$. +16. **Alternatively,** let ${\cal C}$ be the **consequence operator**, that yields the smallest model of a positive program. A **stable model** $X$ is a solution of $${\cal C}\left(P^X\right) = X.$$ + - _negative literals must only be true, while positive ones must also be provable._ +17. _A stable model is $\subseteq$-minimal but not the converse._ +18. _A positive program has a unique stable model, its smallest model._ +19. _If $X,Y$ are stable models of a normal program then $X \not\subset Y$._ +20. _Also, $X \subseteq {\cal C}(P^X) \subseteq \text{head}(P^X)$._ + +## ASP Programming Strategies + +- **Elimination of unnecessary combinatorics.** The number of grounded instances has an huge impact on performance. Rules can be used as "pre-computation" steps. +- **Boolean Constraint Solving.** This is at the core of the **solving** step, e.g. `clasp`. + +## ASP vs. Prolog + +- The different number of stable models lies precisely at the core difference between Prolog and ASP. **In Prolog, the presence of programs with negation that do not have a unique stable model cause trouble and the SLDNF resolution does not terminate on them [17]**. However, ASP embraces the disparity of stable models and treats the stable models of the programs as solutions to a given search program (from [Prolog and Answer Set Programming: Languages in Logic Programming](https://silviacasacuberta.files.wordpress.com/2020/07/final_paper.pdf) ) +- Prolog programs may not terminate (`p :- \+ p.`); ASP "programs" always terminate (`p :- not p.` has zero solutions). +- ASP doesn't allow function symbols; Prolog does. + + +## References + +1. Martin Gebser, Roland Kaminski, Benjamin Kaufmann, Torsten Schaub - Answer Set Solving in Practice-Morgan & Claypool (2013) +2. [Potassco, clingo and gringo](https://potassco.org/): There are two major approaches to integrating probabilistic reasoning into logical representations: **distribution semantics** and **maximum entropy**. + +> - Is there a **sound interpretation of ASP**, in particular of **stable models**, to any of the two approaches above? +> - Under such interpretation, **what probabilistic problems can be addressed?** MARG? MLE? MAP? Decision? +> - **What is the relation to other logic and uncertainty approaches?** Independent Choice Logic? Abduction? Stochastic Logic Programs? etc. + + +## Maximum Entropy Summary + +> ME approaches annotate uncertainties only at the level of a logical theory. That is, they assume that the predicates in the BK are labelled as either true or false, but the label may be incorrect. + +These approaches are not based on logic programming, but rather on first-order logic. Consequently, the underlying semantics are different: rather than consider proofs, **these approaches consider models or groundings of a theory**. + +This difference primarily changes what uncertainties represent. For instance, Markov Logic Networks (MLN) represent programs as a set of weighted clauses. The weights in MLN do not correspond to probabilities of a formula being true but, intuitively, to a log odds between a possible world (an interpretation) where the clause is true and a world where the clause is false. + +## Distribution Semantics + +> DS approaches explicitly annotate uncertainties in BK. To allow such annotation, they extend Prolog with two primitives for stochastic execution: probabilistic facts and annotated disjunctions. + +Probabilistic facts are the most basic stochastic primitive and they take the form of logical facts labelled with a probability p. **Each probabilistic fact represents a Boolean random variable that is true with probability p and false with probability 1 − p.** _This is very close to facts in ASP. A "simple" syntax extension would be enough to capture probability annotations. **What about the semantics of such programs?**_ + +Whereas probabilistic facts introduce non-deterministic behaviour on the level of facts, annotated disjunctions introduce non-determinism on the level of clauses. Annotated disjunctions allow for multiple literals in the head, where only one of the head literals can be true at a time. + +### Core Distribution Semantics + +- Let $F$ be a set of **grounded probabilistic facts** and $P:F \to \left[0, 1 \right]$. + +> For example, `F` and `P` result from +> ```prolog +> 0.9::edge(a,c). +> 0.7::edge(c,b). +> 0.6::edge(d,c). +> 0.9::edge(d,b). +> ``` + +- **Facts are assumed marginally independent:** $$\forall a,b \in F, P(a \wedge b) = P(a)P(b).$$ + +- The **probability of $S \subseteq F$** is $$P_F(S) = \prod_{f \in S} P(f) \prod_{f \not\in S} \left(1 - P(f) \right).$$ + +- Let $R$ be a set of **definite clauses** defining further (new) predicates. + +> For example, `R` is +> ```prolog +> path(X,Y) :- edge(X,Y). +> path(X,Y) :- edge(X,Z), path(Z,Y). +> ``` + +- Any combination $S \cup R$ has an **unique least Herbrand model**, $$W = M(S \cup R).$$ + +- **That uniqueness fails for stable models.** Exactly why? - What is the relation of stable models and least Herbrand models? + +- The set of ground facts $S$ is an **explanation** of the world $W = M(S \cup R)$. A world might have multiple explanations. In ASP a explanation can entail 0, 1 or more worlds. + +- The **probability of a possible world** $W$ is +$$P(W) = \sum_{S \subseteq F :~W=M(S\cup R)} P_F(S).$$ + +- The **probability of a ground proposition** $p$ is (defined as) the probability that $p$ has a proof: $$P(p) = \sum_{S :~ S\cup R ~\vdash~ p} P_F(S) = \sum_{W :~ p\in W} P(W).$$ + +- A proposition may have many proofs in a single world $M(S\cup W)$. Without further guarantees, the probabilities of those proofs cannot be summed. The definition above avoids this problem. + + +> For example, a proof of `path(a,b)` employs (only) the facts `edge(a,c)` and `edge(c,b)` _i.e._ these facts are an explanation of `path(a,b)`. Since these facts are (marginally) independent, **the probability of the proof** is $$\begin{aligned}P(\text{path}(a, b)) & = P(\text{edge}(a,c) \wedge\text{edge}(c,b)) \\&= P(\text{edge}(a,c)) \times P(\text{edge}(c,b)) \\ &= 0.9 \times 0.7 \\ &= 0.63. \end{aligned}$$ +> This is the only proof of `path(a,b)` so $P(\text{path}(a,b)) = 0.63$. +> +> On the other hand, since `path(d,b)` has two explanations, `edge(d,b)` and `edge(d,c), edge(c,b)`: $$\begin{aligned} P(\text{path}(d,b)) & = P\left(\text{edge}(d,c) \vee \left(\text{edge}(d,c)\wedge\text{edge}(c,b)\right)\right) \\ &= 0.9 + 0.6 \times 0.7 - 0.9 \times 0.6 \times 0.7 \\ &= 0.942.\end{aligned}$$ + +- With this **semantics of the probability of a possible world**, the probability of an arbitrary proposition is still hard to compute, because of the _disjunct-sum_ problem: **An explanation can have many worlds.** Since the probability is computed via the explanation, if there are many models for a single explanation, **how to assign probability to specific worlds within the same explanation?** + +> Because computing the probability of a fact or goal under the distribution semantics is hard, systems such as Prism [4] and Probabilistic Horn Abduction (PHA) [8] impose additional restrictions that can be used to improve the efficiency of the inference procedure. +> +> **The key assumption is that the explanations for a goal are mutually exclusive, which overcomes the disjoint-sum problem.** If the different explanations of a fact do not overlap, then its probability is simply the sum of the probabilities of its explanations. This directly follows from the inclusion-exclusion formulae as under the exclusive-explanation assumption the conjunctions (or intersections) are empty (_Statistical Relational Learning_, Luc De Raedt and Kristian Kersting, 2010) +> +> **This assumption/restriction is quite _ad-hoc_ and overcoming it requires further inquiry.** + +- Reading Fabio Gagliardi Cozman, Denis Deratani Mauá, _The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference_ (2020) gave a big boost securing my initial intuition. + +- The problem can be illustrated with disjunctive clauses, such as the one in the following example. + +```prolog +a ; -a. % prob(a) = 0.7 +b ; c :- a. +``` + +- More specifically, in the example above, **the explanation `a` entails two possible worlds, `ab` and `ac`. How to assign probability of each one?** + +### Assigning Probabilities on "Multiple Worlds per Explanation" Scenarios + +#### Clause Annotations + +> Assign a probability to each case in the head of the clause. For example, annotate $P(b|a) = 0.8$. + +This case needs further study on the respective consequences, specially concerning the joint probability distribution. + +- In particular, $P(b|a) = 0.8$ entails $P(\neg b | a) = 0.2$. But $\neg b$ is not in any world. +- Also, unless assumed the contrary, the independence of $b$ and $c$ is unknown. + +#### Learn from Observations + +> Leave the probabilities uniformly distributed; update them from observation. + +Under this approach, how do observations affect the assigned probabilities? + +- In particular, how to update the probabilities of the worlds `a b` and `a c` given observations such as `a`, `b`, `ab`, `a-b`, `-ab` or `abc`? + 1. Define a criterium to decide if an observation $z$ is compatible world $w$. For example, $z \subseteq w$. + 2. Define the probability of a world from on the explanation probability and a count of **compatible observations**. + +#### Leave One World Out + +> Define a **compatibility criterium** for observations and worlds, add another world and update its probability on incompatible observations; The probability of this world measures the model+sensors limitations. diff --git a/text/summaries/00_DistSem.pdf b/text/summaries/00_DistSem.pdf new file mode 100644 index 0000000..b504547 Binary files /dev/null and b/text/summaries/00_DistSem.pdf differ diff --git a/text/summaries/00_ILP.md b/text/summaries/00_ILP.md new file mode 100644 index 0000000..0cbcb0a --- /dev/null +++ b/text/summaries/00_ILP.md @@ -0,0 +1,74 @@ +# Inductive Logic Programming + +> Inductive logic programming (ILP) is a form of machine learning (ML). As with +other forms of ML, the goal of ILP is to induce a hypothesis that generalises training examples. However, whereas most forms of ML use vectors/tensors to represent data (examples and hypotheses), ILP uses logic programs (sets of logical rules). Moreover, whereas most forms of ML learn functions, ILP learns relations. + +## Why ILP? + +- **Data efficiency.** Many forms of ML are notorious for their inability to generalise from small numbers of training examples, notably deep learning. By contrast, ILP can induce hypotheses from small numbers of examples, often from a single example. +- **Background knowledge.** ILP learns using BK represented as a logic program. Moreover, because hypotheses are symbolic, hypotheses can be added the to BK, and thus ILP systems naturally support lifelong and transfer learning. +- **Expressivity.** Because of the expressivity of logic programs, ILP can learn complex relational theories. Because of the symbolic nature of logic programs, ILP can reason about hypotheses, which allows it to learn optimal programs. +- **Explainability.** Because of logic’s similarity to natural language, logic programs can be easily read by humans, which is crucial for explainable AI. + +## Recent Advances + +- Search: Meta-level +- Recursion: Yes +- Predicate Invention: Limited +- Hypotheses: Higher-order; ASP +- Optimality: Yes +- Technology: Prolog; ASP; NNs + +### Review + +- **Search.** The fundamental ILP problem is to efficiently search a large hypothesis space. Most older ILP approaches search in either a top-down or bottom-up fashion. A third new search approach has recently emerged called meta-level ILP. + - **Top-down** approaches start with a general hypothesis and then specialise it. + - **Bottom-up** approaches start with the examples and generalise them. + - **Meta-level.** (Most) approaches encode the ILP problem as a program that reasons about programs. +- **Recursion.** Learning recursive programs has long been considered a difficult problem for ILP. The power of recursion is that an infinite number of computations can be described by a finite recursive program. + - Interest in recursion has resurged with the introduction of meta-interpretive learning (MIL) and the MIL system Metagol. The key idea of MIL is to use metarules, or program templates, to restrict the form of inducible programs, and thus the hypothesis space. A metarule is a higher-order clause. Following MIL, many meta-level ILP systems can learn recursive programs. With recursion, ILP systems can now generalise from small numbers of examples, often a single example. Moreover, the ability to learn recursive programs has opened up ILP to new application areas. +- **Predicate invention.** A key characteristic of ILP is the use of BK. BK is similar to features used in most forms of ML. However, whereas features are tables, BK contains facts and rules (extensional and intensional definitions) in the form of a logic program. + - Rather than expecting a user to provide all the necessary BK, the goal of predicate invention (PI) is for an ILP system to automatically invent new auxiliary predicate symbols. Whilst PI has attracted interest since the beginnings of ILP, and has subsequently been repeatedly stated as a major challenge, most ILP systems do not support it. + - Several PI approaches try to address this challenge: Placeholders, Metarules, Pre/post-processing, Lifelong Learning. + - The aforementioned techniques have improved the ability of ILP to invent high-level concepts. However, PI is still difficult and there are many challenges to overcome. The challenges are that (i) many systems struggle to perform PI at all, and (ii) those that do support PI mostly need much user-guidance, metarules to restrict the space of invented symbols or that a user specifies the arity and argument types of invented symbols. +- ILP systems have traditionally induced definite and normal logic programs, typically represented as Prolog programs. A recent development has been to use different **hypothesis representations**. + - **Datalog** is a syntactical subset of Prolog which disallows complex terms as arguments of predicates and imposes restrictions on the use of negation. The general motivation for reducing the expressivity of the representation language from Prolog to Datalog is to allow the problem to be encoded as a satisfiability problem, particularly to leverage recent developments in SAT and SMT. + - **Answer Set Programming** (ASP) is a logic programming paradigm based on the stable model semantics of normal logic programs that can be implemented using the latest advances in SAT solving technology. + - When learning Prolog programs, the procedural aspect of SLD-resolution must be taken into account. By contrast, as ASP is a truly declarative language, no such consideration need be taken into account when learning ASP programs. Compared to Datalog and Prolog, ASP supports additional language constructs, such as disjunction in the head of a clause, choice rules, and hard and weak constraints. + - **A key difference between ASP and Prolog is semantics.** A definite logic program has only one model (the least Herbrand model). By contrast, an ASP program can have one, many, or even no stable models (answer sets). Due to its non-monotonicity, ASP is particularly useful for expressing common-sense reasoning. + - Approaches to learning ASP programs can mostly be divided into two categories: **brave learners**, which aim to learn a program such that at least one answer set covers the examples, and **cautious learners**, which aim to find a program which covers the examples in all answer sets. + - **Higher-order programs** where predicate symbols can be used as terms. + - **Probabilistic logic programs.** A major limitation of logical representations, such as Prolog and its derivatives, is the implicit assumption that the BK is perfect. This assumption is problematic if data is noisy, which is often the case. + - **Integrating probabilistic reasoning into logical representations** is a principled way to handle such uncertainty in data. This integration is the focus of statistical relational artificial intelligence (StarAI). In essence, StarAI hypothesis representations extend BK with probabilities or weights indicating the degree of confidence in the correctness of parts of BK. Generally, StarAI techniques can be divided in two groups: _distribution representations_ and _maximum entropy_ approaches. + - **Distribution semantics** approaches explicitly annotate uncertainties in BK. To allow such annotation, they extend Prolog with two primitives for stochastic execution: probabilistic facts and annotated disjunctions. Probabilistic facts are the most basic stochastic primitive and they take the form of logical facts labelled with a probability p. Each probabilistic fact represents a Boolean random variable that is true with probability p and false with probability 1 − p. Whereas probabilistic facts introduce non-deterministic behaviour on the level of facts, annotated disjunctions introduce non-determinism on the level of clauses. Annotated disjunctions allow for multiple literals in the head, where only one of the head literals can be true at a time. + - **Maximum entropy** approaches annotate uncertainties only at the level of a logical theory. That is, they assume that the predicates in the BK are labelled as either true or false, but the label may be incorrect. These approaches are not based on logic programming, but rather on first-order logic. Consequently, the underlying semantics are different: rather than consider proofs, these approaches consider models or groundings of a theory. This difference primarily changes what uncertainties represent. For instance, Markov Logic Networks (MLN) represent programs as a set of weighted clauses. The weights in MLN do not correspond to probabilities of a formula being true but, intuitively, to a log odds between a possible world (an interpretation) where the clause is true and a world where the clause is false. + - The techniques from learning such probabilistic programs are typically direct extensions of ILP techniques. +- **Optimality.** There are often multiple (sometimes infinitely many) hypotheses that explain the data. Deciding which hypothesis to choose has long been a difficult problem. + - Older ILP systems were not guaranteed to induce optimal programs, where optimal typically means with respect to the size of the induced program or the coverage of examples. A key reason for this limitation was that most search techniques learned a single clause at a time, leading to the construction of sub-programs which were sub-optimal in terms of program size and coverage. + - Newer ILP systems try to address this limitation. As with the ability to learn recursive programs, the main development is to take a global view of the induction task by using meta-level search techniques. In other words, rather than induce a single clause at a time from a single example, the idea is to induce multiple clauses from multiple examples. + - The ability to learn optimal programs opens up ILP to new problems. For instance, learning efficient logic programs has long been considered a difficult problem in ILP, mainly because there is no declarative difference between an efficient program and an inefficient program. +- **Technologies.** Older ILP systems mostly use Prolog for reasoning. Recent work considers using different technologies. + - **Constraint satisfaction and satisfiability.** There have been tremendous recent advances in SAT. + - To leverage these advances, much recent work in ILP uses related techniques, notably ASP. The main motivations for using ASP are to leverage (i) the language benefits of ASP, and (ii) the efficiency and optimisation techniques of modern ASP solvers, which supports conflict propagation and learning. + - With similar motivations, other approaches encode the ILP problem as SAT or SMT problems. + - These approaches have been shown able to **reduce learning times** compared to standard Prolog-based approaches. However, some unresolved issues remain. A key issue is that most approaches **encode an ILP problem as a single (often very large) satisfiability problem**. These approaches therefore often struggle to scale to very large problems, although preliminary work attempts to tackle this issue. + - **Neural Networks.** With the rise of deep learning, several approaches have explored using gradient-based methods to learn logic programs. These approaches all **replace discrete logical reasoning with a relaxed version that yields continuous values** reflecting the confidence of the conclusion. +- **Applications.** + - **Scientific discovery.** Perhaps the most prominent application of ILP is in scientific discovery: identify and predict ligands (substructures responsible for medical activity) and infer missing pathways in protein signalling networks; ecology. + - **Program analysis.** learning SQL queries; programming language semantics, and code search. + - **Robotics.** Robotics applications often require incorporating domain knowledge or imposing certain requirements on the learnt programs. + - **Games.** Inducing game rules has a long history in ILP, where chess has often been the focus + - **Data curation and transformation.** Another successful application of ILP is in data curation and transformation, which is again largely because ILP can learn executable programs. There is much interest in this topic, largely due to success in synthesising programs for end-user problems, such as string transformations. Other transformation tasks include extracting values from semi-structured data (e.g. XML files or medical records), extracting relations from ecological papers, and spreadsheet manipulation. + - **Learning from trajectories.** Learning from interpretation transitions (LFIT) automatically constructs a model of the dynamics of a system from the observation of its state transitions. LFIT has been applied to learn biological models, like Boolean Networks, under several semantics: memory-less deterministic systems, and their multi-valued extensions. The Apperception Engine explain sequential data, such as cellular automata traces, rhythms and simple nursery tunes, image occlusion tasks, game dynamics, and sequence induction intelligence tests. Surprisingly, can achieve human-level performance on the sequence induction intelligence tests in the zero-shot setting (without having been trained on lots of other examples of such tests, and without hand-engineered knowledge of the particular setting). At a high level, these systems take the unique selling point of ILP systems (the ability to strongly generalise from a handful of data), and apply it to the self-supervised setting, producing an explicit human-readable theory that explains the observed state transitions. +- **Limitations and future research.** + - **Better systems.** A problem with ILP is the lack of well engineered tools. They state that whilst over 100 ILP systems have been built, less than a handful of systems can be meaningfully used by ILP researchers. By contrast, driven by industry, other forms of ML now have reliable and well-maintained implementations, which has helped drive research. A frustrating issue with ILP systems is that they use many different language biases or even different syntax for the same biases. _For ILP to be more widely adopted both inside and outside of academia, we must develop more standardised, user-friendly, and better-engineered tools._ + - **Language biases.** One major issue with ILP is choosing an appropriate language bias. Even for ILP experts, determining a suitable language bias is often a frustrating and time-consuming process. We think the need for an almost perfect language bias is severely holding back ILP from being widely adopted. _We think that an important direction for future work in ILP is to develop techniques for automatically identifying suitable language biases._ This area of research is largely under-researched. + - **Better datasets.** Interesting problems, alongside usable systems, drive research and attract interest in a research field. This relationship is most evident in the deep learning community which has, over a decade, grown into the largest AI community. This community growth has been supported by the constant introduction of new problems, datasets, and well-engineered tools. ILP has, unfortunately, failed to deliver on this front: most research is still evaluated on 20-year old datasets. Most new datasets that have been introduced often come from toy domains and are designed to test specific properties of the introduced technique. _We think that the ILP community should learn from the experiences of other AI communities and put significant efforts into developing datasets that identify limitations of existing methods as well as showcase potential applications of ILP._ + - **Relevance.** New methods for predicate invention have improved the abilities of ILP systems to learn large programs. Moreover, these techniques raise the potential for ILP to be used in lifelong learning settings. However, inventing and acquiring new BK could lead to a problem of too much BK, which can overwhelm an ILP system. On this issue, a key under-explored topic is that of relevancy. _Given a new induction problem with large amounts of BK, how does an ILP system decide which BK is relevant?_ One emerging technique is to train a neural network to score how relevant programs are in the BK and to then only use BK with the highest score to learn programs. Without efficient methods of relevance identification, it is unclear how efficient lifelong learning can be achieved. + - **Handling mislabelled and ambiguous data.** A major open question in ILP is how best to handle noisy and ambiguous data. Neural ILP systems are designed from the start to robustly handle mislabelled data. Although there has been work in recent years on designing ILP systems that can handle noisy mislabelled data, there is much less work on the even harder and more fundamental problem of designing ILP systems that can handle raw ambiguous data. ILP systems typically assume that the input has already been preprocessed into symbolic declarative form (typically, a set of ground atoms representing positive and negative examples). But real-world input does not arrive in symbolic form. _For ILP systems to be widely applicable in the real world, they need to be redesigned so they can handle raw ambiguous input from the outset._ + - **Probabilistic ILP.** Real-world data is often noisy and uncertain. Extending ILP to deal with such uncertainty substantially broadens its applicability. While StarAI is receiving growing attention, **learning probabilistic programs from data is still largely under-investigated due to the complexity of joint probabilistic and logical inference.** When working with probabilistic programs, we are interested in the probability that a program covers an example, not only whether the program covers the example. Consequently, probabilistic programs need to compute all possible derivations of an example, not just a single one. Despite added complexity, probabilistic ILP opens many new challenges. Most of the existing work on probabilistic ILP considers the minimal extension of ILP to the probabilistic setting, by assuming that either (i) BK facts are uncertain, or (ii) that learned clauses need to model uncertainty. **These assumptions make it possible to separate structure from uncertainty and simply reuse existing ILP techniques.** Following this minimal extension, the existing work focuses on discriminative learning in which the goal is to learn a program for a single target relation. However, a grand challenge in probabilistic programming is generative learning. That is, learning a program describing a generative process behind the data, not a single target relation. **Learning generative programs is a significantly more challenging problem, which has received very little attention in probabilistic ILP.** + - **Explainability.** Explainability is one of the claimed advantages of a symbolic representation. Recent work evaluates the comprehensibility of ILP hypotheses using Michie’s framework of ultra-strong machine learning, where a learned hypothesis is expected to not only be accurate but to also demonstrably improve the performance of a human being provided with the learned hypothesis. [Some work] empirically demonstrate improved human understanding directly through learned hypotheses. _However, more work is required to better understand the conditions under which this can be achieved, especially given the rise of PI._ + +## Bibliography + +1. Inductive logic programming at 30 \ No newline at end of file diff --git a/text/summaries/00_ILP.pdf b/text/summaries/00_ILP.pdf new file mode 100644 index 0000000..c9295eb Binary files /dev/null and b/text/summaries/00_ILP.pdf differ diff --git a/text/summaries/00_PASP.pdf b/text/summaries/00_PASP.pdf new file mode 100644 index 0000000..ab358a7 Binary files /dev/null and b/text/summaries/00_PASP.pdf differ diff --git a/text/summaries/00_PASP_credal.md b/text/summaries/00_PASP_credal.md new file mode 100644 index 0000000..1e56cbb --- /dev/null +++ b/text/summaries/00_PASP_credal.md @@ -0,0 +1,143 @@ +# Probabilistic Answer Set Programming + +## Non-stratified programs + +> Minimal example of **non-stratified program**. + +The following annotated LP, with clauses $c_1, c_2, c_3$ respectively, is non-stratified (because has a cycle with negated arcs) but no head is disjunctive: +```prolog +0.3::a. % c1 +s :- not w, not a. % c2 +w :- not s. % c3 +``` + +This program has three stable models: +$$ +\begin{aligned} +m_1 &= \set{ a, w } \cr +m_2 &= \set{ \neg a, s } \cr +m_3 &= \set{ \neg a, w } +\end{aligned} +$$ + +The probabilistic clause `0.3::a.` defines a **total choice** +$$ +\Theta = \set{ + \theta_1 = \set{ a }, + \theta_2 = \set{ \neg a } +} +$$ +such that +$$ +\begin{aligned} +P(\Theta = \set{ a }) &= 0.3\cr +P(\Theta = \set{ \neg a }) &= 0.7 \cr +\end{aligned} +$$ + +> While it is natural to extend $P( m_1 ) = 0.3$ from $P(\theta_1) = 0.3$ there is no clear way to assign $P(m_2), P(m_3)$ since both models result from the total choice $\theta_2$. + + + +Under the **CWA**, $\sim\!\!q \models \neg q$, so $c_2, c_3$ induce probabilities: + +$$ +\begin{aligned} +p_a &= P(a | \Theta) \cr +p_s &= P(s | \Theta) &= (1 - p_w)(1 - p_a) \cr +p_w &= P(w | \Theta) &= (1 - p_w) +\end{aligned} +$$ +from which results +$$ +\begin{equation} +p_s = p_s(1 - p_a). +\end{equation} +$$ + +So, if $\Theta = \theta_1 = \set{ a }$ (one stable model): + +- We have $p_a = P(a | \Theta = \set{ a }) = 1$. +- Equation (1) becomes $p_s = 0$. +- From $p_w = 1 - p_s$ we get $P(w | \Theta) = 1$. + +and if $\Theta = \theta_2 = \set{ \neg a }$ (two stable models): + +- We have $p_a = P(a | \Theta = \set{ \neg a }) = 0$. +- Equation (1) becomes $p_s = p_s$; Since we know nothing about $p_s$, we let $p_s = \alpha \in \left[0, 1\right]$. +- We still have the relation $p_w = 1 - p_s$ so $p_w = 1 - \alpha$. + +We can now define the **marginals** for $s, w$: +$$ +\begin{aligned} +P(s) &=\sum_\theta P(s|\theta)P(\theta)= 0.7\alpha \cr +P(w) &=\sum_\theta P(s|\theta)P(\theta)= 0.3 + 0.7(1 - \alpha) \cr +\alpha &\in\left[ 0, 1 \right] +\end{aligned} +$$ + +> The parameter $\alpha$ not only **expresses insufficient information** to sharply define $p_s$ but also **relates** $p_s$ and $p_w$. + +## Disjunctive heads + +> Minimal example of **disjunctive heads** program. + +Consider this LP + +```prolog +0.3::a. +b ; c :- a. +``` + +with three stable models: +$$ +\begin{aligned} +m_1 &= \set{ \neg a } \cr +m_2 &= \set{ a, b } \cr +m_3 &= \set{ a, c } +\end{aligned} +$$ + +Again, $P(m_1) = 0.3$ is quite natural but there are no clear assignments for $P(m_2), P(m_3)$. + +The total choices here are +$$ +\Theta = \set{ + \theta_1 = \set{ a } + \theta_2 = \set{ \neg a } +} +$$ +such that +$$ +\begin{aligned} +P(\Theta = \set{ a }) &= 0.3\cr +P(\Theta = \set{ \neg a }) &= 0.7 \cr +\end{aligned} +$$ +and the LP induces +$$ +P(b \vee c | \Theta) = P(a | \Theta). +$$ + +Since the disjunctive expands as +$$ +\begin{equation} +P(b \vee c | \Theta) = P(b | \Theta) + P( c | \Theta) - P(b \wedge c | \Theta) +\end{equation} +$$ +and we know that $P(b \vee c | \Theta) = P(a | \Theta)$ we need two independent parameters, for example +$$ +\begin{aligned} +P(b | \Theta) &= \beta \cr +P(c | \Theta) &= \gamma \cr +\end{aligned} +$$ +where +$$ +\begin{aligned} + \alpha & \in \left[0, 0.3\right] \cr + \beta & \in \left[0, \alpha\right] +\end{aligned} +$$ + +This example also calls for reconsidering the CWA since it entails that **we should assume that $b$ and $c$ are conditionally independent given $a$.** \ No newline at end of file diff --git a/text/summaries/00_POTASSCO.md b/text/summaries/00_POTASSCO.md new file mode 100644 index 0000000..a53ea17 --- /dev/null +++ b/text/summaries/00_POTASSCO.md @@ -0,0 +1,13 @@ +# Potassco + +> [Potassco](https://potassco.org/), the Potsdam Answer Set Solving Collection, bundles tools for Answer Set Programming developed at the University of Potsdam. + +- [The Potassco Guide](https://github.com/potassco/guide) + +## clingo + +> Current answer set solvers work on variable-free programs. Hence, a grounder is needed that, given an input program with first-order variables, computes an equivalent ground (variable-free) program. gringo is such a grounder. Its output can be processed further with clasp, claspfolio, or clingcon. +> +> [clingo](https://potassco.org/clingo/) combines both gringo and clasp into a monolithic system. This way it offers more control over the grounding and solving process than gringo and clasp can offer individually - e.g., incremental grounding and solving. + +- [Python module list](https://potassco.org/clingo/python-api/current/) \ No newline at end of file diff --git a/text/summaries/00_PROB.md b/text/summaries/00_PROB.md new file mode 100644 index 0000000..c3ac219 --- /dev/null +++ b/text/summaries/00_PROB.md @@ -0,0 +1,41 @@ +# Probability Problems + +>- What are the general tasks we expect to solve with probabilistic programs? +> - The **MAP** task is the one with best applications. It is also the hardest to compute. +> - **MLE** is the limit case of **MAP**. Has simpler computations but overfits the data. + +## Background + +- **Conditional Probability** $$P(A, B) = P(B | A) P(A).$$ +- **Bayes Theorem** $$P(B | A) = \frac{P(A | B) P(B)}{P(A)}.$$ +- **For maximization tasks** $$P(B | A) \propto P(A | B) P(B).$$ +- **Marginal** $$P(A) = \sum_b P(A,b).$$ +- In $P(B | A) \propto P(A | B) P(B)$, if the **posterior** $P(B | A)$ and the **prior** $P(B)$ follow distributions of the same family, $P(B)$ is a **conjugate prior** for the **likelihood** $P(A | B)$. +- **Density Estimation:** Estimate a joint probability distribution from a set of observations; Select a probability distribution function and the parameters that best explains the distributions of the observations. + +## MLE: Maximum Likelihood Estimation + +> Given a probability **distribution** $d$ and a set of **observations** $X$, find the distribution **parameters** $\theta$ that maximize the **likelihood** (_i.e._ the probability of those observations) for that distribution. +> +> **Overfits the data:** high variance of the parameter estimate; sensitive to random variations in the data. Regularization with $P(\theta)$ leads to **MAP**. + +Given $d, X$, find +$$ +\hat{\theta}_{\text{MLE}}(d,X) = \arg_{\theta} \max P_d(X | \theta). +$$ + +## MAP: Maximum A-Priori + +> Given a probability **distribution** $d$ and a set of **observations** $X$, find the distribution **parameters** $\theta$ that best explain those observations. + +Given $d, X$, find +$$ +\hat{\theta}_{\text{MAP}}(d, X) = \arg_{\theta}\max P(\theta | X). +$$ + +Using $P(A | B) \propto P(B | A) P(A)$, +$$\hat{\theta}_{\text{MAP}}(d, X) = \arg_{\theta} \max P_d(X | \theta) P(\theta)$$ + +Variants: +- **Viterbi algorithm:** Find the most likely sequence of hidden states (on HMMs) that results in a sequence of observed events. +- **MPE: Most Probable Explanation** and **Max-sum, Max-product algorithms:** Calculates the marginal distribution for each unobserved node, conditional on any observed nodes; Defines the most likely assignment to all the random variables that is consistent with the given evidence. diff --git a/text/summaries/00_Z3.md b/text/summaries/00_Z3.md new file mode 100644 index 0000000..c755c66 --- /dev/null +++ b/text/summaries/00_Z3.md @@ -0,0 +1,14 @@ +# Z3 - An SMT solver + +> `Z3` is a theorem prover from Microsoft Research. +> **However, `Potassco` seems more to the point.** + +## Introduction + +An Answer Set Program can be solved translating it into a SAP problem. + +## References + +1. [Programming Z3](https://theory.stanford.edu/~nikolaj/programmingz3.html): . +2. [Julia Package](https://www.juliapackages.com/p/z3): . +3. [Repository](https://github.com/Z3Prover): . \ No newline at end of file diff --git a/text/summaries/00_pasp.md b/text/summaries/00_pasp.md new file mode 100644 index 0000000..f64edbc --- /dev/null +++ b/text/summaries/00_pasp.md @@ -0,0 +1,18 @@ +# Probabilistic ASP + +## Weighted Approach + +1. **Total Choices.** $N(C = x) = \prod_{a \in x} w_a \prod_{\neg a \in x} (1 - w_a)$. +2. **Stable Models.** $N(S = x | C = c) = \alpha_{x,c}$, + where the set of parameters $\alpha_{x,c}$ is such that: + $$ + \begin{cases} + \alpha_{x,c} \geq 0, & \forall c, x\cr + \alpha_{x,c} = 0, & \forall x \not\supseteq c \cr + \sum_{x} \alpha_{x,c} = 1, & \forall c. + \end{cases} + $$ +3. **Worlds.** $N(W = x)$ + 1. If $x$ is a _total choice_: $ + N(W = x) = \prod_{a \in x} w_a \prod_{\neg a \in x} (1 - w_a). + $$ \ No newline at end of file diff --git a/zugzwang.code-workspace b/zugzwang.code-workspace new file mode 100644 index 0000000..89e2e1a --- /dev/null +++ b/zugzwang.code-workspace @@ -0,0 +1,29 @@ +{ + "folders": [ + { + "path": "." + }, + { + "path": "../../../projects/contrib/dsd" + }, + { + "path": "../../../prog/contrib/zugzwang" + } + ], + "settings": { + "cSpell.words": [ + "biblatex", + "CREDAL", + "interpretability", + "println", + "stablecore", + "testset", + "uminus" + ], + "cSpell.enableFiletypes": [ + "latex" + ], + "aspLanguage.setConfig": "config.json", + "julia.environmentPath": "/home/fc/sci/projetos/zugzwang" + } +} \ No newline at end of file -- libgit2 0.21.2