diff --git a/README.md b/README.md index f9a8b2f..86c88ba 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,7 @@ # Probabilistic ILP +**Check** Conformal prediction. + > Fonte: [Turning 30: New Ideas in Inductive Logic Programming](https://arxiv.org/abs/2002.11002) ## Introduction @@ -62,4 +64,4 @@ Recursion; Predicate Invention; Higher order, ASP Hypotheses; Optimality; Prolog ## Applications -### ELearning \ No newline at end of file +### ELearning diff --git a/SCASP-best-practices.pdf b/SCASP-best-practices.pdf new file mode 100644 index 0000000..ea0feef Binary files /dev/null and b/SCASP-best-practices.pdf differ diff --git a/msc/Empty File b/msc/Empty File new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/msc/Empty File diff --git a/task_01/proposal.md b/task_01/proposal.md index 09cc04b..6c04164 100644 --- a/task_01/proposal.md +++ b/task_01/proposal.md @@ -6,7 +6,7 @@ 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 $1 − p$. +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 labeled 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. @@ -36,7 +36,7 @@ A team of two **(or three?)** researchers and a graduate student, working over s - 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, _etc._ + - Software tools: [`s(casp)`](https://ciao-lang.org/playground/scasp.html), [Potassco suit](https://potassco.org/), _etc._ ## References @@ -44,4 +44,4 @@ A team of two **(or three?)** researchers and a graduate student, working over s 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) \ No newline at end of file +6. 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LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold' -(Font) T1/lmss/m/it --> T1/lmss/b/it on input line 45. +(Font) T1/lmss/m/it --> T1/lmss/b/it on input line 64. LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' -(Font) T1/lmtt/m/n --> T1/lmtt/b/n on input line 45. +(Font) T1/lmtt/m/n --> T1/lmtt/b/n on input line 64. (/usr/share/texmf-dist/tex/latex/translator/translator-basic-dictionary-English.dict Dictionary: translator-basic-dictionary, Language: English ) (/usr/share/texmf-dist/tex/latex/translator/translator-bibliography-dictionary-English.dict @@ -753,91 +753,112 @@ Dictionary: translator-numbers-dictionary, Language: English Dictionary: translator-theorem-dictionary, Language: English ) (./00_PASP.nav) [1.1 +] (./00_PASP.toc) [2.2 + ] -LaTeX Font Info: Trying to load font information for U+msa on input line 65. +LaTeX Font Info: Trying to load font information for U+msa on input line 89. 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a/text/00_PASP.pdf b/text/00_PASP.pdf index 2072180..c5512a5 100644 Binary files a/text/00_PASP.pdf and b/text/00_PASP.pdf differ diff --git a/text/00_PASP.snm b/text/00_PASP.snm index f07ad09..14041bc 100644 --- a/text/00_PASP.snm +++ b/text/00_PASP.snm @@ -1,12 +1,14 @@ -\beamer@slide {prop:lucases.a}{4} -\beamer@slide {prop:lucases.b}{4} -\beamer@slide {prop:lucases.c}{4} -\beamer@slide {prop:lucases.d}{4} \beamer@slide {eq:prob.tc}{5} -\beamer@slide {def:w.inconsistent}{9} -\beamer@slide {eq:prob.sm}{9} -\beamer@slide {def:w.disj}{9} -\beamer@slide {def:w.conj}{9} -\beamer@slide {def:w.empty}{9} -\beamer@slide {eq:def.prob}{9} -\beamer@slide {eq:def.prob.event}{9} +\beamer@slide {prop:unique.ext.tcsm}{7} +\beamer@slide {prop:lucases}{11} +\beamer@slide {prop:lucases.a}{11} +\beamer@slide {prop:lucases.b}{11} +\beamer@slide {prop:lucases.c}{11} +\beamer@slide {prop:lucases.d}{11} +\beamer@slide {def:w.inconsistent}{13} +\beamer@slide {eq:prob.sm}{13} +\beamer@slide {def:w.disj}{13} +\beamer@slide {def:w.conj}{13} +\beamer@slide {def:w.empty}{13} +\beamer@slide {eq:def.prob}{13} +\beamer@slide {eq:def.prob.event}{13} diff --git a/text/00_PASP.synctex.gz b/text/00_PASP.synctex.gz index 590c8d0..636ba77 100644 Binary files a/text/00_PASP.synctex.gz and b/text/00_PASP.synctex.gz differ diff --git a/text/00_PASP.tex b/text/00_PASP.tex index ca3e700..7ffc136 100644 --- a/text/00_PASP.tex +++ b/text/00_PASP.tex @@ -1,9 +1,25 @@ \documentclass{beamer} - +%------------------------------------------ +\usecolortheme{rose} +%------------------------------------------ +\useinnertheme{circles} +%------------------------------------------ \setbeamertemplate{navigation symbols}{} -\setbeamertemplate{itemize items}[circle] - - +%------------------------------------------ +\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} @@ -28,142 +44,225 @@ \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}}} % % Identificação deste documento % \title{Zugzwang} -\subtitle{Stochastic Adventures in Inductive Logic Specifications} +\subtitle{Stochastic Adventures in Inductive Logic} \author{Francisco Coelho} \institute[\texttt{fc@uevora.pt}]{ - Departamento de Informática\\ - Universidade de Évora + Departamento de Informática, Universidade de Évora\\ + High Performance Computing Chair\\ + NOVA-LINCS } \begin{document} % \begin{frame}[plain] \titlepage -\note{The goal of this text is to **explore how ASP specifications with probabilistic facts** can lead to characterizations of the **joint distributions** of the specification's atoms.} \end{frame} \section{Introduction} \begin{frame}{Notation and Assumptions} - \note{We start with **common notations and assumptions**.} + % -------------------------------- \begin{itemize} - \item The \textbf{complement} of $x$ is $\co{x} = 1 - x$. - \item A \textbf{probabilistic atomic choice} $\alpha:a$ defines the disjunction $a \lor \neg a$ and assigns probabilities $P\at{a} = \alpha, P\at{\neg a} = \co{\alpha}$. - \item $\delta a$ denotes the \textbf{disjunction} $a \lor \neg a$ associated to the probabilistic choice $\alpha : a$ and $\delta\! \set{\alpha: a, a \in A} = \set{\delta a, a \in A}$ for any set of atoms $A$. - \item We adopt the the \textbf{closed world assumption}, where $\naf x \models \neg x$. - \item And also assume that \textbf{probabilistic choices} and \textbf{subgoals} are iid. + % -------------------------------- + \item $\co{x} = 1 - x$. + % -------------------------------- + \item \textbf{Probabilistic Atomic Choice (PAC):} $\alpha :: a$ defines $a \lor \neg a$ and probabilities $\pr{a} = \alpha, \pr{\neg a} = \co{\alpha}$. + % -------------------------------- + \item $\delta a$ denotes $a \lor \neg a$ and $\delta\! \set{\alpha :: a, a \in \fml{A}} = \set{\delta a, a \in \fml{A}}$ for a set of atoms $\fml{A}$. + % -------------------------------- + \item \textbf{Closed World Assumption:} $\naf x \models \neg x$. + % -------------------------------- + % \item Probabilistic choices and sub-goals are independent. + % -------------------------------- \end{itemize} - \note{**Subgoals are IID** means that ???} + % -------------------------------- \end{frame} - +% ================================================================ \begin{frame}{General Setting} - \note{Next, we consider the following **general setting**} - Let $\fml{A}$ be a set of \textbf{atoms}, $\overline{\fml{A}} = \set{\neg a \middle| a \in \fml{A}}$ and $\fml{Z}$ the respective set of \textbf{observations}, - $$\fml{Z} = \set{z = \alpha \cup \beta \middle| \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$ and $\fml{I}$ the set of consistent observations or \textbf{interpretations}, $$\fml{I} = \set{z \in \fml{Z} \middle| \forall a \in \fml{A}~\abs{ \set{a, \neg a} \cap z} \leq 1}.$$ + % -------------------------------- \begin{itemize} - \item A \textbf{PASP program} is $P = C \land F \land R$ where - - \begin{itemize} - \item $C = C_P = \set{\alpha_i : a_i \middle| i = 1:n}$ is a set of probabilistic atomic choices, - - \item $F = F_P$ is a set of (common) facts and - - \item $R = R_P$ is a set of (common) rules. - \end{itemize} - - and the sets of atoms, observations and interpretations of program $P$ are denoted $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$ + % -------------------------------- + \item \textbf{Atoms} $\fml{A}$, + $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$, + % -------------------------------- + \item \textbf{Observations} $\fml{Z}$: + $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$ + % -------------------------------- + \item \textbf{Interpretations} or \textit{consistent observations} $\fml{I}$ : + $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$ + % -------------------------------- + \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item $C = C_P = \cset{\alpha_i :: a_i }{ i \in 1:n \land a_i \in \fml{A}}$ \textit{pacs}. + % -------------------------------- + \item $F = F_P$ \textit{facts}. + % -------------------------------- + \item $R = R_P$ \textit{rules}. + % -------------------------------- + \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{observations} and \textit{interpretations} 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 $\theta = \set{c_1, \ldots, c_n}$ where $c_i$ is $a_i$ or $\neg a_i$. + % -------------------------------- + %\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 The \textbf{stable models} of $P$ are the stable models of $\delta P = \delta C + F + R$ and the respective set is denoted $\fml{S} = \fml{S}_P$. + % -------------------------------- + \item \textbf{Total Choice Probability:} + \begin{equation} + \pr{\theta} = \prod_{a_i \in \theta}\alpha_i \prod_{\neg a_i \in \theta}\co{\alpha_i}.\label{eq:prob.tc} + \end{equation} + % -------------------------------- \end{itemize} + % -------------------------------- + \begin{quote} + This is the \emph{Distribution Semantics} as set by Sato. + \end{quote} \end{frame} - -\begin{frame} - \note{A model x has lower and upper "bounds".} +% ================================================================ +\begin{frame}{Problem Statement} + % -------------------------------- + \begin{quotation} + How to extend probability from the total choices to interpretations and observations? + \end{quotation} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item \textbf{Todo:} Extend probability to \textit{stable models}, \textit{interpretations} and \textit{observations}. + % -------------------------------- + \end{itemize} + % -------------------------------- + \begin{quotation} + \textbf{But} there is a problem extending probability from total choices 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{unbiased} choice for $\alpha\in\intcc{0,1}$ in + $$ + \begin{aligned} + \pr{\set{a, b}} &= 0.3 \alpha, \cr + \pr{\set{a, c}} &= 0.3 \co{\alpha}. + \end{aligned} + $$ + % -------------------------------- + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ +\section{Motivation} +% ================================================================ +\begin{frame}{Specification, Data \& Evaluation} + % -------------------------------- + Given some procedure to assign probabilities to observations from specifications and: + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item $P$, a specification. + % -------------------------------- + \item $p$, the distribution of observations from above. + % -------------------------------- + \item $Z$, a dataset of observations. + % -------------------------------- + \item $e$, the respective empirical distribution. + % -------------------------------- + \item $D$, some probability divergence, \textit{e.g.} Kullback-Leibler. + % -------------------------------- + \end{itemize} + % -------------------------------- + Given a dataset $Z$, $D\at{P} = D\at{e, p}$ is a \textit{performance} measure of $P$ and can be used, \textit{e.g.} fitness, by algorithms searching for optimal specifications of a dataset. + % -------------------------------- +\end{frame} +% ================================================================ +\section{Resolution} +% ================================================================ +\begin{frame}{Bounds of Interpretations} + % -------------------------------- \begin{itemize} - \item \textbf{Proposition.} Let $x\in\fml{I}$ be an interpretation. + % -------------------------------- + \item For $x\in\fml{I}$: + % -------------------------------- \begin{itemize} - \item[Lower Models] $\lset{x} = \set{s\in \fml{S} \middle| s \subseteq x}$. - - \item[Upper Models] $\uset{x} = \set{s\in \fml{S} \middle| x \subseteq s}$. + % -------------------------------- + \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} - \note{If $a$ is a lower model and $b$ an upper model, since stable models are minimal, must be $a = b = x$.} - - \item Exactly one of the following cases takes place: + % -------------------------------- + \item\label{prop:lucases} \textbf{Proposition.} Stable models are \textit{minimal} so \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. - % + % -------------------------------- \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$. - % + % -------------------------------- \item\label{prop:lucases.c} $\lset{x} = \emptyset \land \uset{x} \neq \emptyset$. - % + % -------------------------------- \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. + % -------------------------------- \end{enumerate} + % -------------------------------- \end{itemize} + % -------------------------------- \end{frame} - -\begin{frame} - \note{Total choice are key to define probability of a clause.} - \begin{itemize} - \item The probabilistic facts $C$ define a set $\Theta = \Theta_C$ of \textbf{total choices}, with $2^n$ elements, each one a set $\theta = \set{c_1, \ldots, c_n}$ where $c_i$ is either $a_i$ or $\neg a_i$. - - \item For each stable model $s\in\fml{S}$ let $\theta_s$ be the unique \textbf{total choice} contained in $s$ and $\fml{S}_\theta \subseteq \fml{S}$ the set of stable models that contains $\theta$. - - \item Define - \begin{equation} - p\at{\theta} = \prod_{a_i \in \theta}\alpha_i \prod_{\neg a_i \in \theta}\co{\alpha_i}.\label{eq:prob.tc} - \end{equation} - \end{itemize} -\end{frame} - - -\begin{frame} - \note{Relate stable models with Sato's probabilistic semantics} - \begin{quotation} - The problem we address is how to \textbf{assign probabilities to observations} given that a total choice might entail zero or many stable models \emph{i.e.} How to assign probabilities to the stable models of $\fml{S}_\theta$ when $\envert{\fml{S}_\theta} \not= 1$? - \end{quotation} -\end{frame} - -\begin{frame} - \note{There are some problems} - - As it turns out, it is quite easy to come out with a program from which result no single probability distribution. For example - $$ - \begin{aligned} - 0.3:a,& \cr - b \lor c \larr& a. - \end{aligned} - $$ - has three stable models - $$ - \begin{aligned} - s_1 &= \set{\neg a} \cr - s_2 &= \set{a, b} \cr - s_3 &= \set{a, c} - \end{aligned} - $$ - and while $p\at{\set{\neg a}} = 0.7$ is quite natural, we have no further information to support the choice of a singular $\alpha\in\intcc{0,1}$ in the assignment - $$ - \begin{aligned} - p\at{\set{a, b}} &= 0.3 \alpha \cr - p\at{\set{a, c}} &= 0.3 \co{\alpha} - \end{aligned} - $$ -\end{frame} - +% ================================================================ \begin{frame} - Next we try to formalize the possible configurations of this scenario. Consider the ASP program $P = C \land F \land R$ with total choices $\Theta $ and stable models $\fml{S}$. Let $d : \fml{S} \to \intcc{0,1}$ such that $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$. + Next we try to formalize the possible configurations of this scenario. Consider the ASP program $P = C \land F \land R$ with total choices $\Theta $ and stable models $\fml{S}$. Let $d :: \fml{S} \to \intcc{0,1}$ such that $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$. \end{frame} - +% ================================================================ \begin{frame} \begin{enumerate} @@ -176,7 +275,7 @@ % \item $z$ is an interpretation and $\lset{z} = \set{z} = \uset{x}$. Then $z = s$ is a stable model and \textbf{define} \begin{equation} - w_d\at{z} = w\at{s} = d\at{s} p\at{\theta_s}.\label{eq:prob.sm} + w_d\at{z} = w\at{s} = d\at{s} \pr{\theta_s}.\label{eq:prob.sm} \end{equation} % \item $z$ is an interpretation and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define} @@ -197,36 +296,36 @@ % \item The last point defines a ``weight'' function on the observations 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 interpretation, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} observation $\top = \fml{A} \cup \set{\neg a \middle| a \in \fml{A}}$. + \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 interpretation, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} observation $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$. % \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 $P\at{E} \in \intcc{0, 1}$ for any $E \in F$. - \item $P\at{\Omega} = 1$. - \item if $E_1 \cap E_2 = \emptyset$ then $P\at{E_1 \cup E_2} = P\at{E_1} + P\at{E_2}$. + \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} - P\at{z} = \frac{w\at{z}}{Z}, z \in \Omega \label{eq:def.prob} + \pr{z} = \frac{w\at{z}}{Z}, z \in \Omega \label{eq:def.prob} \end{equation} and \begin{equation} - P\at{E} = \sum_{x\in E} P\at{x}, E \subseteq \Omega. \label{eq:def.prob.event} + \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 $P\at{\Omega} = 1$ also results directly from the definitions. + \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 $$ \begin{aligned} - P\at{A \cup B} &= \sum_{x \in A \cup B} P\at{x} \cr - &= \sum_{x \in A} P\at{x} + \sum_{x \in B} P\at{x} - \sum_{x \in A \cap B} P\at{x} \cr - &= \sum_{x \in A} P\at{x} + \sum_{x \in B} P\at{x} &\text{because}~A\cap B = \emptyset \cr - &= P\at{A} + P\at{B}. + \pr{A \cup B} &= \sum_{x \in A \cup B} \pr{x} \cr + &= \sum_{x \in A} \pr{x} + \sum_{x \in B} \pr{x} - \sum_{x \in A \cap B} \pr{x} \cr + &= \sum_{x \in A} \pr{x} + \sum_{x \in B} \pr{x} &\text{because}~A\cap B = \emptyset \cr + &= \pr{A} + \pr{B}. \end{aligned} $$ \item So $\del{\Omega = \fml{Z}, F = \pset{\Omega}, P}$ is a probability space. {$\blacksquare$} @@ -234,10 +333,11 @@ \end{enumerate} \end{frame} - +% ================================================================ \section{Cases \& Examples} +% ================================================================ \subsection{Programs with disjunctive heads} - +% ================================================================ \begin{frame} Consider the program: @@ -263,10 +363,9 @@ \end{aligned} $$ \end{frame} - - +% ================================================================ \begin{frame} - Suppose that we add an annotation $\alpha:a$, which entails $\co{\alpha}:\neg a$. This is enough to get $w\at{s_1} = \co{\alpha}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $\alpha$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get: + Suppose that we add an annotation $\alpha :: a$, which entails $\co{\alpha} :: \neg a$. This is enough to get $w\at{s_1} = \co{\alpha}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $\alpha$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get: $$ \begin{cases} w\at{s_1 } = &\co{\alpha}\cr @@ -275,7 +374,7 @@ \end{cases} $$ \end{frame} - +% ================================================================ \begin{frame} Now consider all the interpretations for this program: @@ -285,11 +384,11 @@ % \node [draw, circle] (E) at (5.5,0) {$\emptyset$}; % - \node [draw, circle] (a) at (2,2) {$a$}; - \node [draw, circle] (b) at (3,2) {$b$}; + \node [draw, circle] (a) at (3,2) {$a$}; + \node [draw, circle] (b) at (2,2) {$b$}; \node [draw, circle] (c) at (4,2) {$c$}; - \node [fill=gray!50] (A) at (9,2) {$\co{a}$}; - \node (B) at (8,2) {$\co{b}$}; + \node [fill=gray!50] (A) at (8,2) {$\co{a}$}; + \node (B) at (9,2) {$\co{b}$}; \node (C) at (7,2) {$\co{c}$}; % \node [fill=gray!50] (ab) at (0,4) {$ab$}; @@ -338,7 +437,7 @@ \end{tikzpicture} \end{center} \end{frame} - +% ================================================================ \begin{frame} In this diagram: @@ -378,7 +477,7 @@ $$ \item Now some statistics are possible. For example we get $$ - P\at{abc \mid \alpha = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9} + \pr{abc \mid \alpha = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9} $$. \item This expression can be plotted for $d\in\intcc{0,1}$ @@ -386,10 +485,10 @@ \includegraphics[height=15em]{Pabc_alpha03.pdf} \end{center} - \item If a data set $E$ entails \emph{e.g.} $P\at{abc \mid E} = 0.0015$ we can numerically solve + \item If a data set $E$ entails \emph{e.g.} $\pr{abc \mid E} = 0.0015$ we can numerically solve $$ \begin{aligned} - P\at{abc \mid \alpha = 0.3} &= P\at{abc \mid E} \cr + \pr{abc \mid \alpha = 0.3} &= \pr{abc \mid E} \cr \iff\cr \frac{0.09 d \del{d - 1}}{0.09 d^{2} - 0.69 d - 7.9} &= 0.0015 \end{aligned} @@ -397,10 +496,9 @@ which has two solutions, $d \approx 0.15861$ or $d \approx 0.83138$. \end{itemize} \end{frame} - +% ================================================================ \subsection{Non-stratified programs} - - +% ================================================================ \begin{frame} The following LP is non-stratified, because has a cycle with negated arcs: $$ @@ -429,13 +527,13 @@ }. $$ \end{frame} - +% ================================================================ \begin{frame} - Looking into probabilistic interpretations of the program and/or its models, we define $\alpha = P\at{\Theta = \theta_1}\in\intcc{0, 1}$ and $P\at{\Theta = \theta_2} = \co{\alpha}$. + Looking into probabilistic interpretations of the program and/or its models, we define $\alpha = \pr{\Theta = \theta_1}\in\intcc{0, 1}$ and $\pr{\Theta = \theta_2} = \co{\alpha}$. - Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $P\at{ s_1 } = P\at{\Theta = \theta_1} = \alpha$. However, there is no clear way to assign $P\at{s_2}, P\at{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly, - $$P\at{s_2 \mid \Theta} + P\at{s_3 \mid \Theta} = + Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $\pr{ s_1 } = \pr{\Theta = \theta_1} = \alpha$. 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, + $$\pr{s_2 \mid \Theta} + \pr{s_3 \mid \Theta} = \begin{cases} 0 & \text{if}~\Theta = \theta_1\cr 1 & \text{if}~\Theta = \theta_2 @@ -444,16 +542,15 @@ but further assumptions are not supported \emph{a priori}. So let's \textbf{parameterize} the equation above, $$ \begin{cases} - P\at{s_2 \mid \Theta = \theta_2} = &\beta \in \intcc{0, 1} \cr - P\at{s_3 \mid \Theta = \theta_2} = &\co{\beta}, + \pr{s_2 \mid \Theta = \theta_2} = &\beta \in \intcc{0, 1} \cr + \pr{s_3 \mid \Theta = \theta_2} = &\co{\beta}, \end{cases} $$ in order to explicit our knowledge, or lack of, with numeric values and relations. \end{frame} - - +% ================================================================ \begin{frame} - Now we are able to define the \textbf{joint distribution} of the boolean random variables $A,B,C$, : + Now we are able to define the \textbf{joint distribution} of the boolean random variables $A,B,C$: $$ \begin{array}{cc|l} @@ -467,10 +564,9 @@ $$ where $\alpha, \beta\in\intcc{0,1}$. \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. @@ -479,10 +575,9 @@ \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} @@ -536,4 +631,5 @@ \end{itemize} \end{itemize} \end{frame} +% ================================================================ \end{document} \ No newline at end of file diff --git a/text/00_PASP.toc b/text/00_PASP.toc index 33a80d8..97397a7 100644 --- a/text/00_PASP.toc +++ b/text/00_PASP.toc @@ -1,5 +1,7 @@ \beamer@sectionintoc {1}{Introduction}{2}{0}{1} -\beamer@sectionintoc {2}{Cases \& Examples}{10}{0}{2} -\beamer@subsectionintoc {2}{1}{Programs with disjunctive heads}{10}{0}{2} -\beamer@subsectionintoc {2}{2}{Non-stratified programs}{14}{0}{2} -\beamer@sectionintoc {3}{Conclusions}{18}{0}{3} +\beamer@sectionintoc {2}{Motivation}{8}{0}{2} +\beamer@sectionintoc {3}{Resolution}{10}{0}{3} +\beamer@sectionintoc {4}{Cases \& Examples}{14}{0}{4} +\beamer@subsectionintoc {4}{1}{Programs with disjunctive heads}{15}{0}{4} +\beamer@subsectionintoc {4}{2}{Non-stratified programs}{20}{0}{4} +\beamer@sectionintoc {5}{Conclusions}{25}{0}{5} diff --git a/text/00_PASP.xdv b/text/00_PASP.xdv index 42422e7..a286a18 100644 Binary files a/text/00_PASP.xdv and b/text/00_PASP.xdv differ -- libgit2 0.21.2