From 241956f9bb8224fb6e8b759f89a14ad8b2b48b68 Mon Sep 17 00:00:00 2001 From: Francisco Coelho Date: Wed, 9 Nov 2022 10:13:37 +0000 Subject: [PATCH] continuing 00_PASP --- text/00_PASP.aux | 163 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++----------------------------------------------------------------------------- text/00_PASP.fdb_latexmk | 18 +++++++++--------- text/00_PASP.fls | 2 +- text/00_PASP.log | 188 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++------------------------------------------------------------------------------------------- text/00_PASP.nav | 96 +++++++++++++++++++++++++++++++++++++++++++++++++++--------------------------------------------- text/00_PASP.out | 4 ++-- text/00_PASP.pdf | Bin 87748 -> 0 bytes text/00_PASP.snm | 24 ++++++++++++------------ text/00_PASP.synctex.gz | Bin 76592 -> 0 bytes text/00_PASP.tex | 105 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[2][]{Outline0.6}{\376\377\000A\000S\000P\000\040\000\046\000\040\000r\000e\000l\000a\000t\000e\000d\000\040\000d\000e\000f\000i\000n\000i\000t\000i\000o\000n\000s}{}% 8 diff --git a/text/00_PASP.pdf b/text/00_PASP.pdf index c5512a5..b172662 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 14041bc..5eb076d 100644 --- a/text/00_PASP.snm +++ b/text/00_PASP.snm @@ -1,14 +1,14 @@ \beamer@slide {eq:prob.tc}{5} \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} +\beamer@slide {prop:lucases}{14} +\beamer@slide {prop:lucases.a}{14} +\beamer@slide {prop:lucases.b}{14} +\beamer@slide {prop:lucases.c}{14} +\beamer@slide {prop:lucases.d}{14} +\beamer@slide {def:w.inconsistent}{16} +\beamer@slide {eq:prob.sm}{16} +\beamer@slide {def:w.disj}{16} +\beamer@slide {def:w.conj}{16} +\beamer@slide {def:w.empty}{16} +\beamer@slide {eq:def.prob}{16} +\beamer@slide {eq:def.prob.event}{16} diff --git a/text/00_PASP.synctex.gz b/text/00_PASP.synctex.gz index 636ba77..56d8ecd 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 7ffc136..7f256f5 100644 --- a/text/00_PASP.tex +++ b/text/00_PASP.tex @@ -49,6 +49,7 @@ \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 % @@ -95,10 +96,10 @@ \item \textbf{Atoms} $\fml{A}$, $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$, % -------------------------------- - \item \textbf{Observations} $\fml{Z}$: + \item \textbf{Inputs} $\fml{Z}$: $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$ % -------------------------------- - \item \textbf{Interpretations} or \textit{consistent observations} $\fml{I}$ : + \item \textbf{Interpretations} or \textit{consistent inputs} $\fml{I}$ : $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$ % -------------------------------- \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where @@ -111,7 +112,7 @@ % -------------------------------- \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 $P$. + \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{inputs} and \textit{interpretations} of $P$. \end{itemize} % -------------------------------- \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$. @@ -124,7 +125,7 @@ % -------------------------------- \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 \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are $\theta = \cset{t_c}{c \in C}$ where $c=\alpha::a$ 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 $\fml{S}_\theta = \cset{s \in \fml{S}}{\theta \subset s}$. @@ -133,30 +134,20 @@ % -------------------------------- \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} + \pr{\theta} = \prod_{a \in \theta}\alpha \prod_{\neg a \in \theta}\co{\alpha}.\label{eq:prob.tc} \end{equation} % -------------------------------- \end{itemize} % -------------------------------- - \begin{quote} - This is the \emph{Distribution Semantics} as set by Sato. - \end{quote} + This is the \emph{distribution semantic} as set by Sato. \end{frame} % ================================================================ \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} + How to \textit{extend} probability from total choices to stable models, interpretations and inputs? % -------------------------------- \begin{quotation} - \textbf{But} there is a problem extending probability from total choices to stable models. + There's a problem right at extending to stable models. \end{quotation} % -------------------------------- \end{frame} @@ -200,32 +191,96 @@ % -------------------------------- \end{frame} % ================================================================ +\begin{frame}{Lack of Information \& Parametrization} + % -------------------------------- + \begin{itemize} + % -------------------------------- + \item The specification \textit{lacks information} to set $\alpha\in\intcc{0,1}$ in + $$ + \begin{aligned} + \pr{\set{a, b}} &= 0.3 \alpha, \cr + \pr{\set{a, c}} &= 0.3 \co{\alpha}. + \end{aligned} + $$ + \item A \textit{random variable} captures this: + $$ + \begin{aligned} + \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr + \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr + \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}. + \end{aligned} + $$ + \item Other uncertainties lead to further parameters: + $$ + \pr{s \given\ A_1 = \alpha_1, \ldots, A_n = \alpha_n }. + $$ + % -------------------------------- + \end{itemize} + Reducing \textbf{specification uncertainty}, \textit{e.g.} setting $A = 0.21$, must result from \textbf{observations}. + % -------------------------------- +\end{frame} +% ================================================================ +\begin{frame}{Main Research Question} + % -------------------------------- + A \textit{random variable} captures this: + $$ + \begin{aligned} + \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr + \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr + \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}. + \end{aligned} + $$ + % -------------------------------- + \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, consider the possible uncertainty scenarios. + % -------------------------------- + \item The disjunction example illustrates one such step. + % -------------------------------- + \end{itemize} + % -------------------------------- +\end{frame} +% ================================================================ \section{Motivation} % ================================================================ \begin{frame}{Specification, Data \& Evaluation} % -------------------------------- - Given some procedure to assign probabilities to observations from specifications and: + Given some procedure to extend probabilities to stable models, interpretations and inputs, and given: % -------------------------------- \begin{itemize} % -------------------------------- \item $P$, a specification. % -------------------------------- - \item $p$, the distribution of observations from above. + \item $p$, the distribution of inputs from above. % -------------------------------- - \item $Z$, a dataset of observations. + \item $Z$, a dataset of inputs. % -------------------------------- \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. + % -------------------------------- + \begin{quotation} + For a dataset $Z$, $D\at{P} = D\at{e, p}$ is a \textbf{performance} measure of $P$ and can be used, \textit{e.g.} as fitness, by algorithms searching for \textbf{optimal specifications of a dataset}. + \end{quotation} % -------------------------------- \end{frame} % ================================================================ \section{Resolution} % ================================================================ +\begin{frame}{Resolution Path} + \begin{enumerate} + \item \textit{Parametrize} lack of knowledge \emph{e.g.} $\alpha$ in the disjunction example. + \item This extends probability from total choices to \textit{standard models}. + \item Extend probability from standard models to \textit{interpretations}. \textbf{How?} Later. + \item Set $\pr{z} = 0$ for $z \in \fml{Z} \setminus \fml{I}$. + \end{enumerate} +\end{frame} \begin{frame}{Bounds of Interpretations} % -------------------------------- \begin{itemize} @@ -294,9 +349,9 @@ \end{equation} \end{enumerate} % - \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 The last point defines a ``weight'' function on the inputs 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 $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$. + \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} input $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$. % \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} diff --git a/text/00_PASP.toc b/text/00_PASP.toc index 97397a7..4f507b5 100644 --- a/text/00_PASP.toc +++ b/text/00_PASP.toc @@ -1,7 +1,7 @@ \beamer@sectionintoc {1}{Introduction}{2}{0}{1} -\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} +\beamer@sectionintoc {2}{Motivation}{10}{0}{2} +\beamer@sectionintoc {3}{Resolution}{12}{0}{3} +\beamer@sectionintoc {4}{Cases \& Examples}{17}{0}{4} +\beamer@subsectionintoc {4}{1}{Programs with disjunctive heads}{18}{0}{4} +\beamer@subsectionintoc {4}{2}{Non-stratified programs}{23}{0}{4} +\beamer@sectionintoc {5}{Conclusions}{28}{0}{5} diff --git a/text/00_PASP.xdv b/text/00_PASP.xdv index a286a18..f0966dd 100644 Binary files a/text/00_PASP.xdv and b/text/00_PASP.xdv differ -- libgit2 0.21.2