From 237d62bda61e206c29f2a7e73a9a81960869bef8 Mon Sep 17 00:00:00 2001 From: Francisco Coelho Date: Wed, 9 Nov 2022 16:49:40 +0000 Subject: [PATCH] Further rewriting of 00_PASP --- text/00_PASP.aux | 122 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++--------------------------------------------------------------- text/00_PASP.fdb_latexmk | 19 +++++++++---------- text/00_PASP.fls | 4 ---- text/00_PASP.log | 169 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-------------------------------------------------------------------------------------- text/00_PASP.nav | 73 +++++++++++++++++++++++++++++++++++-------------------------------------- text/00_PASP.out | 13 ++++++------- text/00_PASP.pdf | Bin 92937 -> 0 bytes text/00_PASP.snm | 14 +++++++------- text/00_PASP.synctex.gz | Bin 83177 -> 0 bytes text/00_PASP.tex | 290 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"/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" 1667935656 32020 e7e67df128d9c518e87afbb7c6491ef3 "" +["xelatex"] 1668012209 "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" "00_PASP.xdv" "00_PASP" 1668012216 + "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" 1668012208 32880 d2ff92ff8b2ef9e5a1bd016ddcdcb92a "" "/usr/share/texmf-dist/fonts/map/fontname/texfonts.map" 1650183167 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/share/texmf-dist/fonts/tfm/adobe/zapfding/pzdr.tfm" 1650183167 1528 f853c4d1b4e0550255e02831fdc8496f "" "/usr/share/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1650183167 1004 54797486969f23fa377b128694d548df "" @@ -32,7 +32,6 @@ "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss10.tfm" 1650183167 11176 53ebf7a171df1f9447b387b178768bb5 "" "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss12.tfm" 1650183167 11232 955a7245396175d9219648eadc654ac9 "" "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss8.tfm" 1650183167 11180 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ff89349dbd8e3874420c8d8c57eb7504 "xelatex" - "00_PASP.tex" 1667935656 32020 e7e67df128d9c518e87afbb7c6491ef3 "" - "00_PASP.toc" 1667935663 384 d39373a67753c9668ace5a0f122dc338 "xelatex" + "00_PASP.aux" 1668012215 10175 631506c17974104545ec9f7326871ea2 "xelatex" + "00_PASP.nav" 1668012215 4415 70ff7441078bcca01fe957123d46aa96 "xelatex" + "00_PASP.out" 1668012215 1123 b2bcf7d8938de98bdfd00e76ac45d7c1 "xelatex" + "00_PASP.tex" 1668012208 32880 d2ff92ff8b2ef9e5a1bd016ddcdcb92a "" + "00_PASP.toc" 1668012215 359 90550d08e709a3bd1267d8fa48d5648b "xelatex" "Pabc_alpha03.pdf" 1663331875 12659 e50124c327299028f63ecb501d23eb0d "" (generated) "00_PASP.aux" diff --git a/text/00_PASP.fls b/text/00_PASP.fls index 7592f11..99123d0 100644 --- a/text/00_PASP.fls +++ b/text/00_PASP.fls @@ -1471,10 +1471,6 @@ INPUT /usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss12.tfm INPUT ./00_PASP.toc INPUT 00_PASP.toc INPUT 00_PASP.toc -INPUT /usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmssbo10.tfm -INPUT 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[4.4 ] [5.5 @@ -794,52 +794,49 @@ LaTeX Font Info: Font shape `T1/lmss/m/it' in size <5> not available ] [9.9 -] (./00_PASP.toc) [10.10 +] [10.10 -] -LaTeX Font Info: Font shape `T1/lmss/bx/it' in size <10.95> not available -(Font) Font shape `T1/lmss/bx/sl' tried instead on input line 272. - [11.11 +] (./00_PASP.toc) [11.11 -] (./00_PASP.toc) [12.12 +] [12.12 ] [13.13 -] [14.14 +] +Overfull \hbox (81.56876pt too wide) in paragraph at lines 398--398 + [] + [] + +[14.14 ] [15.15 +] [16.16 + ] -Overfull \hbox (18.42326pt too wide) detected at line 390 +Overfull \hbox (18.42326pt too wide) detected at line 481 [][] [] -Overfull \vbox (448.34027pt too high) detected at line 390 +Overfull \vbox (227.00671pt too high) detected at line 481 [] -[16.16 - -] (./00_PASP.toc) [17.17 +[17.17 ] (./00_PASP.toc) [18.18 -] [19.19 +] (./00_PASP.toc) [19.19 ] [20.20 -] -Overfull \hbox (81.56876pt too wide) in paragraph at lines 494--494 - [] - [] - -[21.21 +] [21.21 ] File: Pabc_alpha03.pdf Graphic file (type pdf) -Overfull \vbox (479.67523pt too high) detected at line 553 +Overfull \vbox (479.67523pt too high) detected at line 581 [] [22.22 @@ -861,7 +858,7 @@ Overfull \vbox (479.67523pt too high) detected at line 553 ] (./00_PASP.toc) [30.30 ] -Overfull \vbox (411.79784pt too high) detected at line 688 +Overfull \vbox (411.79784pt too high) detected at line 716 [] [31.31 @@ -882,15 +879,15 @@ File: se-pdfdoc.def 2019/11/29 v1.12 stringenc: PDFDocEncoding File: se-utf32be.def 2019/11/29 v1.12 stringenc: UTF-32BE ) (./00_PASP.aux) Package rerunfilecheck Info: File `00_PASP.out' has not changed. -(rerunfilecheck) Checksum: FF89349DBD8E3874420C8D8C57EB7504;1097. +(rerunfilecheck) Checksum: B2BCF7D8938DE98BDFD00E76AC45D7C1;1123. ) Here is how much of TeX's memory you used: - 26344 strings out of 476156 - 520379 string characters out of 5814963 - 832946 words of memory out of 5000000 - 46552 multiletter control sequences out of 15000+600000 - 544952 words of font info for 84 fonts, out of 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[2][]{Outline0.5}{\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}{}% 7 diff --git a/text/00_PASP.pdf b/text/00_PASP.pdf index b172662..f900ad3 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 5eb076d..b35d472 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}{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 {prop:lucases}{13} +\beamer@slide {prop:lucases.a}{13} +\beamer@slide {prop:lucases.b}{13} +\beamer@slide {prop:lucases.c}{13} +\beamer@slide {prop:lucases.d}{13} \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} +\beamer@slide {eq:def.prob}{17} +\beamer@slide {eq:def.prob.event}{17} diff --git a/text/00_PASP.synctex.gz b/text/00_PASP.synctex.gz index 56d8ecd..4b0a202 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 7f256f5..8852d0e 100644 --- a/text/00_PASP.tex +++ b/text/00_PASP.tex @@ -23,6 +23,18 @@ \usepackage[overridenote]{pdfpc} \usepackage{tikz} +\tikzset{ +rrect/.style={ + rectangle, + rounded corners, + fill=gray!10, + % minimum width=1em, + % text width=2em, + % draw=blue!75, + % very thick, + align=center, + } +} \usepackage{commath} \usepackage{amssymb} @@ -96,10 +108,10 @@ \item \textbf{Atoms} $\fml{A}$, $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$, % -------------------------------- - \item \textbf{Inputs} $\fml{Z}$: + \item \textbf{Samples} $\fml{Z}$: $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$ % -------------------------------- - \item \textbf{Interpretations} or \textit{consistent inputs} $\fml{I}$ : + \item \textbf{Interpretations} or \textit{consistent samples} $\fml{I}$ : $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$ % -------------------------------- \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where @@ -112,7 +124,7 @@ % -------------------------------- \item $R = R_P$ \textit{rules}. % -------------------------------- - \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{inputs} and \textit{interpretations} of $P$. + \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{samples} and \textit{interpretations} of $P$. \end{itemize} % -------------------------------- \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$. @@ -142,9 +154,11 @@ This is the \emph{distribution semantic} as set by Sato. \end{frame} % ================================================================ -\begin{frame}{Problem Statement} - % -------------------------------- - How to \textit{extend} probability from total choices to stable models, interpretations and inputs? +\begin{frame} + % -------------------------------- + \begin{block}{Problem Statement} + How to \textit{extend} probability from total choices to stable models, interpretations and samples? + \end{block} % -------------------------------- \begin{quotation} There's a problem right at extending to stable models. @@ -179,7 +193,7 @@ % -------------------------------- \item $\pr{\set{\neg a}} = 0.7$ is straightforward. % -------------------------------- - \item But, no \textit{unbiased} choice for $\alpha\in\intcc{0,1}$ in + \item But, no \textit{informed} choice for $\alpha\in\intcc{0,1}$ in $$ \begin{aligned} \pr{\set{a, b}} &= 0.3 \alpha, \cr @@ -202,17 +216,17 @@ \pr{\set{a, c}} &= 0.3 \co{\alpha}. \end{aligned} $$ - \item A \textit{random variable} captures this: + \item A \textit{random variable} captures this uncertainty: $$ \begin{aligned} - \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr - \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr - \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}. + \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr + \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr + \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}. \end{aligned} $$ \item Other uncertainties lead to further parameters: $$ - \pr{s \given\ A_1 = \alpha_1, \ldots, A_n = \alpha_n }. + \pr{s \given A_1 = \alpha_1, \ldots, A_n = \alpha_n }. $$ % -------------------------------- \end{itemize} @@ -220,14 +234,14 @@ % -------------------------------- \end{frame} % ================================================================ -\begin{frame}{Main Research Question} +\begin{frame}%{Main Research Question} % -------------------------------- - A \textit{random variable} captures this: + A \textit{random variable} captures this uncertainty: $$ \begin{aligned} - \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr - \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr - \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}. + \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr + \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr + \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}. \end{aligned} $$ % -------------------------------- @@ -237,49 +251,51 @@ % -------------------------------- \begin{itemize} % -------------------------------- - \item Follow ASP syntax; for each case, consider the possible uncertainty scenarios. + \item Follow ASP syntax; for each case, what are the uncertainty scenarios? + % -------------------------------- + \item The disjunction example illustrates one such scenario. % -------------------------------- - \item The disjunction example illustrates one such step. + \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} % ================================================================ -\section{Motivation} -% ================================================================ -\begin{frame}{Specification, Data \& Evaluation} +\begin{frame}%{Specification, Data \& Evaluation} % -------------------------------- - Given some procedure to extend probabilities to stable models, interpretations and inputs, and given: + Given a method that produces a distribution of samples, $p$, from a specification, $P$ and: % -------------------------------- \begin{itemize} % -------------------------------- - \item $P$, a specification. - % -------------------------------- - \item $p$, the distribution of inputs from above. - % -------------------------------- - \item $Z$, a dataset of inputs. + \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{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} + % -------------------------------- + \begin{block}{Motivation} + Then $D\at{P} = D\at{e, p}$ 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{block} % -------------------------------- \end{frame} % ================================================================ -\section{Resolution} +\section{Extending Probability to Samples} % ================================================================ \begin{frame}{Resolution Path} + Prior to \textit{conciliation} with data: \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 $z \in \fml{Z} \setminus \fml{I}$. + \item Use \textit{conditional parameters} to represent lack of knowledge \emph{e.g.} $\alpha$ in the disjunction example. + \item This extends probability from total choices to \textit{standard models} --- \alert{hopefully}. + \item Assume probability set on standard models; Extend it to \textit{interpretations}. \textbf{How?} Later. + \item Set $\pr{z} = 0$ for $z \in \fml{Z} \setminus \fml{I}$ (inconsistent samples). \end{enumerate} + Assuming a conditional probability set on stable models, \textit{How to extend it to interpretations?} \end{frame} \begin{frame}{Bounds of Interpretations} % -------------------------------- @@ -295,7 +311,7 @@ % -------------------------------- \end{itemize} % -------------------------------- - \item\label{prop:lucases} \textbf{Proposition.} Stable models are \textit{minimal} so \textit{one} of the following cases takes place: + \item\label{prop:lucases} \textbf{Proposition.} Exactly \textit{one} of the following cases takes place: % -------------------------------- \begin{enumerate} % -------------------------------- @@ -308,50 +324,125 @@ \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. % -------------------------------- \end{enumerate} + because stable models are \textit{minimal}. % -------------------------------- \end{itemize} % -------------------------------- \end{frame} % ================================================================ +\begin{frame}{Disjunction Example | The Interpretation's Lattice} + \begin{center} + \begin{tikzpicture} + % + \draw [help lines, color=gray!20] grid (11,7); + % + \node [fill=gray!50] (sm) at (1, 7) {sm}; + \node [draw,rrect] (sub) at (2, 7) {sub}; + \node [draw, fill=gray!10] (sup) at (3, 7) {sup}; + \node (ind) at (4, 7) {ind}; + % + \node [draw, rrect] (E) at (5.5,0) {$\emptyset$}; + % + \node [draw, rrect] (a) at (1.5,1.5) {$a$}; + \node [draw, rrect] (b) at (0,1.5) {$b$}; + \node [draw, rrect] (c) at (4.5,1.5) {$c$}; + \node [fill=gray!50] (A) at (8.5,1.5) {$\co{a}$}; + \node (B) at (9.5,1.5) {$\co{b}$}; + \node (C) at (10.5,1.5) {$\co{c}$}; + % + \node [fill=gray!50] (ab) at (0,4) {$ab$}; + \node [fill=gray!50] (ac) at (3,4) {$ac$}; + \node (aB) at (1,4) {$a\co{b}$}; + \node (aC) at (2,4) {$a\co{c}$}; + \node [draw, fill=gray!10] (Ab) at (4,4) {$\co{a}b$}; + \node [draw, fill=gray!10] (Ac) at (5,4) {$\co{a}c$}; + \node [draw, fill=gray!10] (AB) at (6,4) {$\co{a}\co{b}$}; + \node [draw, fill=gray!10] (AC) at (7,4) {$\co{a}\co{c}$}; + \node [fill=white] (bc) at (10,4) {$bc$}; + \node [fill=white] (bC) at (11,4) {$b\co{c}$}; + \node [fill=white] (Bc) at (12,4) {$\co{b}c$}; + \node [fill=white] (BC) at (13,4) {$\co{b}\co{c}$}; + % + \node [draw, fill=gray!10] (abc) at (1.5,6) {$abc$}; + \node (abC) at (3,6) {$ab\co{c}$}; + \node (aBc) at (4,6) {$a\co{b}c$}; + \node (aBC) at (5,6) {$a\co{b}\co{c}$}; + \node [draw, fill=gray!10] (Abc) at (7,6) {$\co{a}bc$}; + \node [draw, fill=gray!10] (AbC) at (8,6) {$\co{a}b\co{c}$}; + \node [draw, fill=gray!10] (ABc) at (9,6) {$\co{a}\co{b}c$}; + \node [draw, fill=gray!10] (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 [->] (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 [->] (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} - 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 $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$. + + \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 For each $z\in\fml{Z}$ only one of the following cases takes place + \item $z$ is inconsistent. Then \textbf{define} + \begin{equation} + w_d\at{x} = 0.\label{def:w.inconsistent} + \end{equation} + % + \item $z$ is an interpretation 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 interpretation 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 interpretation 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 interpretation 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 $z$ is inconsistent. Then \textbf{define} - \begin{equation} - w_d\at{x} = 0.\label{def:w.inconsistent} - \end{equation} - % - \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} \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} - \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 interpretation 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 interpretation 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} % - \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 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 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 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} 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} @@ -432,69 +523,6 @@ % ================================================================ \begin{frame} - Now consider all the interpretations for this program: - \begin{center} - \begin{tikzpicture} - %\draw [help lines] grid (11,3); - % - \node [draw, circle] (E) at (5.5,0) {$\emptyset$}; - % - \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 (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) {$ab$}; - \node [fill=gray!50] (ac) at (1,4) {$ac$}; - \node (aB) at (2,4) {$a\co{b}$}; - \node (aC) at (3,4) {$a\co{c}$}; - \node [draw] (Ab) at (4,4) {$\co{a}b$}; - \node [draw] (Ac) at (5,4) {$\co{a}c$}; - \node [draw] (AB) at (6,4) {$\co{a}\co{b}$}; - \node [draw] (AC) at (7,4) {$\co{a}\co{c}$}; - \node [fill=white] (bc) at (10,4) {$bc$}; - \node [fill=white] (bC) at (11,4) {$b\co{c}$}; - \node [fill=white] (Bc) at (12,4) {$\co{b}c$}; - \node [fill=white] (BC) at (13,4) {$\co{b}\co{c}$}; - % - \node [draw] (abc) at (0.5,6) {$abc$}; - \node (abC) at (3,6) {$ab\co{c}$}; - \node (aBc) at (4,6) {$a\co{b}c$}; - \node (aBC) at (5,6) {$a\co{b}\co{c}$}; - \node [draw] (Abc) at (7,6) {$\co{a}bc$}; - \node [draw] (AbC) at (8,6) {$\co{a}b\co{c}$}; - \node [draw] (ABc) at (9,6) {$\co{a}\co{b}c$}; - \node [draw] (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 [->] (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 [->] (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} - 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] {$ab$}}. diff --git a/text/00_PASP.toc b/text/00_PASP.toc index 4f507b5..3d8ca93 100644 --- a/text/00_PASP.toc +++ b/text/00_PASP.toc @@ -1,7 +1,6 @@ \beamer@sectionintoc {1}{Introduction}{2}{0}{1} -\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} +\beamer@sectionintoc {2}{Extending Probability to Samples}{11}{0}{2} +\beamer@sectionintoc {3}{Cases \& Examples}{18}{0}{3} +\beamer@subsectionintoc {3}{1}{Programs with disjunctive heads}{19}{0}{3} +\beamer@subsectionintoc {3}{2}{Non-stratified programs}{23}{0}{3} +\beamer@sectionintoc {4}{Conclusions}{28}{0}{4} diff --git a/text/00_PASP.xdv b/text/00_PASP.xdv index f0966dd..8249b8a 100644 Binary files a/text/00_PASP.xdv and b/text/00_PASP.xdv differ -- libgit2 0.21.2