Commit 237d62bda61e206c29f2a7e73a9a81960869bef8

Authored by Francisco Coelho
1 parent 241956f9
Exists in master

Further rewriting of 00_PASP

text/00_PASP.aux
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895 892
896 -Output written on 00_PASP.xdv (31 pages, 214680 bytes). 893 +Output written on 00_PASP.xdv (31 pages, 203892 bytes).
text/00_PASP.nav
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text/00_PASP.out
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text/00_PASP.pdf
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text/00_PASP.snm
1 \beamer@slide {eq:prob.tc}{5} 1 \beamer@slide {eq:prob.tc}{5}
2 \beamer@slide {prop:unique.ext.tcsm}{7} 2 \beamer@slide {prop:unique.ext.tcsm}{7}
3 -\beamer@slide {prop:lucases}{14}  
4 -\beamer@slide {prop:lucases.a}{14}  
5 -\beamer@slide {prop:lucases.b}{14}  
6 -\beamer@slide {prop:lucases.c}{14}  
7 -\beamer@slide {prop:lucases.d}{14} 3 +\beamer@slide {prop:lucases}{13}
  4 +\beamer@slide {prop:lucases.a}{13}
  5 +\beamer@slide {prop:lucases.b}{13}
  6 +\beamer@slide {prop:lucases.c}{13}
  7 +\beamer@slide {prop:lucases.d}{13}
8 \beamer@slide {def:w.inconsistent}{16} 8 \beamer@slide {def:w.inconsistent}{16}
9 \beamer@slide {eq:prob.sm}{16} 9 \beamer@slide {eq:prob.sm}{16}
10 \beamer@slide {def:w.disj}{16} 10 \beamer@slide {def:w.disj}{16}
11 \beamer@slide {def:w.conj}{16} 11 \beamer@slide {def:w.conj}{16}
12 \beamer@slide {def:w.empty}{16} 12 \beamer@slide {def:w.empty}{16}
13 -\beamer@slide {eq:def.prob}{16}  
14 -\beamer@slide {eq:def.prob.event}{16} 13 +\beamer@slide {eq:def.prob}{17}
  14 +\beamer@slide {eq:def.prob.event}{17}
text/00_PASP.synctex.gz
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text/00_PASP.tex
@@ -23,6 +23,18 @@ @@ -23,6 +23,18 @@
23 \usepackage[overridenote]{pdfpc} 23 \usepackage[overridenote]{pdfpc}
24 24
25 \usepackage{tikz} 25 \usepackage{tikz}
  26 +\tikzset{
  27 +rrect/.style={
  28 + rectangle,
  29 + rounded corners,
  30 + fill=gray!10,
  31 + % minimum width=1em,
  32 + % text width=2em,
  33 + % draw=blue!75,
  34 + % very thick,
  35 + align=center,
  36 + }
  37 +}
26 38
27 \usepackage{commath} 39 \usepackage{commath}
28 \usepackage{amssymb} 40 \usepackage{amssymb}
@@ -96,10 +108,10 @@ @@ -96,10 +108,10 @@
96 \item \textbf{Atoms} $\fml{A}$, 108 \item \textbf{Atoms} $\fml{A}$,
97 $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$, 109 $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$,
98 % -------------------------------- 110 % --------------------------------
99 - \item \textbf{Inputs} $\fml{Z}$: 111 + \item \textbf{Samples} $\fml{Z}$:
100 $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$ 112 $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$
101 % -------------------------------- 113 % --------------------------------
102 - \item \textbf{Interpretations} or \textit{consistent inputs} $\fml{I}$ : 114 + \item \textbf{Interpretations} or \textit{consistent samples} $\fml{I}$ :
103 $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$ 115 $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$
104 % -------------------------------- 116 % --------------------------------
105 \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where 117 \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where
@@ -112,7 +124,7 @@ @@ -112,7 +124,7 @@
112 % -------------------------------- 124 % --------------------------------
113 \item $R = R_P$ \textit{rules}. 125 \item $R = R_P$ \textit{rules}.
114 % -------------------------------- 126 % --------------------------------
115 - \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{inputs} and \textit{interpretations} of $P$. 127 + \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{samples} and \textit{interpretations} of $P$.
116 \end{itemize} 128 \end{itemize}
117 % -------------------------------- 129 % --------------------------------
118 \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$. 130 \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 @@ @@ -142,9 +154,11 @@
142 This is the \emph{distribution semantic} as set by Sato. 154 This is the \emph{distribution semantic} as set by Sato.
143 \end{frame} 155 \end{frame}
144 % ================================================================ 156 % ================================================================
145 -\begin{frame}{Problem Statement}  
146 - % --------------------------------  
147 - How to \textit{extend} probability from total choices to stable models, interpretations and inputs? 157 +\begin{frame}
  158 + % --------------------------------
  159 + \begin{block}{Problem Statement}
  160 + How to \textit{extend} probability from total choices to stable models, interpretations and samples?
  161 + \end{block}
148 % -------------------------------- 162 % --------------------------------
149 \begin{quotation} 163 \begin{quotation}
150 There's a problem right at extending to stable models. 164 There's a problem right at extending to stable models.
@@ -179,7 +193,7 @@ @@ -179,7 +193,7 @@
179 % -------------------------------- 193 % --------------------------------
180 \item $\pr{\set{\neg a}} = 0.7$ is straightforward. 194 \item $\pr{\set{\neg a}} = 0.7$ is straightforward.
181 % -------------------------------- 195 % --------------------------------
182 - \item But, no \textit{unbiased} choice for $\alpha\in\intcc{0,1}$ in 196 + \item But, no \textit{informed} choice for $\alpha\in\intcc{0,1}$ in
183 $$ 197 $$
184 \begin{aligned} 198 \begin{aligned}
185 \pr{\set{a, b}} &= 0.3 \alpha, \cr 199 \pr{\set{a, b}} &= 0.3 \alpha, \cr
@@ -202,17 +216,17 @@ @@ -202,17 +216,17 @@
202 \pr{\set{a, c}} &= 0.3 \co{\alpha}. 216 \pr{\set{a, c}} &= 0.3 \co{\alpha}.
203 \end{aligned} 217 \end{aligned}
204 $$ 218 $$
205 - \item A \textit{random variable} captures this: 219 + \item A \textit{random variable} captures this uncertainty:
206 $$ 220 $$
207 \begin{aligned} 221 \begin{aligned}
208 - \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr  
209 - \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr  
210 - \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}. 222 + \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr
  223 + \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr
  224 + \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}.
211 \end{aligned} 225 \end{aligned}
212 $$ 226 $$
213 \item Other uncertainties lead to further parameters: 227 \item Other uncertainties lead to further parameters:
214 $$ 228 $$
215 - \pr{s \given\ A_1 = \alpha_1, \ldots, A_n = \alpha_n }. 229 + \pr{s \given A_1 = \alpha_1, \ldots, A_n = \alpha_n }.
216 $$ 230 $$
217 % -------------------------------- 231 % --------------------------------
218 \end{itemize} 232 \end{itemize}
@@ -220,14 +234,14 @@ @@ -220,14 +234,14 @@
220 % -------------------------------- 234 % --------------------------------
221 \end{frame} 235 \end{frame}
222 % ================================================================ 236 % ================================================================
223 -\begin{frame}{Main Research Question} 237 +\begin{frame}%{Main Research Question}
224 % -------------------------------- 238 % --------------------------------
225 - A \textit{random variable} captures this: 239 + A \textit{random variable} captures this uncertainty:
226 $$ 240 $$
227 \begin{aligned} 241 \begin{aligned}
228 - \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr  
229 - \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr  
230 - \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}. 242 + \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr
  243 + \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr
  244 + \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}.
231 \end{aligned} 245 \end{aligned}
232 $$ 246 $$
233 % -------------------------------- 247 % --------------------------------
@@ -237,49 +251,51 @@ @@ -237,49 +251,51 @@
237 % -------------------------------- 251 % --------------------------------
238 \begin{itemize} 252 \begin{itemize}
239 % -------------------------------- 253 % --------------------------------
240 - \item Follow ASP syntax; for each case, consider the possible uncertainty scenarios. 254 + \item Follow ASP syntax; for each case, what are the uncertainty scenarios?
  255 + % --------------------------------
  256 + \item The disjunction example illustrates one such scenario.
241 % -------------------------------- 257 % --------------------------------
242 - \item The disjunction example illustrates one such step. 258 + \item \textit{Neat} means a function $d: \fml{S} \to \intcc{0, 1}$ such that
  259 + $$
  260 + \sum_{s\in\fml{S}_\theta} d\at{s} = 1
  261 + $$
  262 + for each $\theta \in \Theta$.
243 % -------------------------------- 263 % --------------------------------
244 \end{itemize} 264 \end{itemize}
245 % -------------------------------- 265 % --------------------------------
246 \end{frame} 266 \end{frame}
247 % ================================================================ 267 % ================================================================
248 -\section{Motivation}  
249 -% ================================================================  
250 -\begin{frame}{Specification, Data \& Evaluation} 268 +\begin{frame}%{Specification, Data \& Evaluation}
251 % -------------------------------- 269 % --------------------------------
252 - Given some procedure to extend probabilities to stable models, interpretations and inputs, and given: 270 + Given a method that produces a distribution of samples, $p$, from a specification, $P$ and:
253 % -------------------------------- 271 % --------------------------------
254 \begin{itemize} 272 \begin{itemize}
255 % -------------------------------- 273 % --------------------------------
256 - \item $P$, a specification.  
257 - % --------------------------------  
258 - \item $p$, the distribution of inputs from above.  
259 - % --------------------------------  
260 - \item $Z$, a dataset of inputs. 274 + \item $Z$, a dataset of samples.
261 % -------------------------------- 275 % --------------------------------
262 \item $e$, the respective empirical distribution. 276 \item $e$, the respective empirical distribution.
263 % -------------------------------- 277 % --------------------------------
264 \item $D$, some probability divergence, \textit{e.g.} Kullback-Leibler. 278 \item $D$, some probability divergence, \textit{e.g.} Kullback-Leibler.
265 % -------------------------------- 279 % --------------------------------
266 \end{itemize} 280 \end{itemize}
267 - % --------------------------------  
268 - \begin{quotation}  
269 - 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}.  
270 - \end{quotation} 281 + % --------------------------------
  282 + \begin{block}{Motivation}
  283 + 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}.
  284 + \end{block}
271 % -------------------------------- 285 % --------------------------------
272 \end{frame} 286 \end{frame}
273 % ================================================================ 287 % ================================================================
274 -\section{Resolution} 288 +\section{Extending Probability to Samples}
275 % ================================================================ 289 % ================================================================
276 \begin{frame}{Resolution Path} 290 \begin{frame}{Resolution Path}
  291 + Prior to \textit{conciliation} with data:
277 \begin{enumerate} 292 \begin{enumerate}
278 - \item \textit{Parametrize} lack of knowledge \emph{e.g.} $\alpha$ in the disjunction example.  
279 - \item This extends probability from total choices to \textit{standard models}.  
280 - \item Extend probability from standard models to \textit{interpretations}. \textbf{How?} Later.  
281 - \item Set $\pr{z} = 0$ for $z \in \fml{Z} \setminus \fml{I}$. 293 + \item Use \textit{conditional parameters} to represent lack of knowledge \emph{e.g.} $\alpha$ in the disjunction example.
  294 + \item This extends probability from total choices to \textit{standard models} --- \alert{hopefully}.
  295 + \item Assume probability set on standard models; Extend it to \textit{interpretations}. \textbf{How?} Later.
  296 + \item Set $\pr{z} = 0$ for $z \in \fml{Z} \setminus \fml{I}$ (inconsistent samples).
282 \end{enumerate} 297 \end{enumerate}
  298 + Assuming a conditional probability set on stable models, \textit{How to extend it to interpretations?}
283 \end{frame} 299 \end{frame}
284 \begin{frame}{Bounds of Interpretations} 300 \begin{frame}{Bounds of Interpretations}
285 % -------------------------------- 301 % --------------------------------
@@ -295,7 +311,7 @@ @@ -295,7 +311,7 @@
295 % -------------------------------- 311 % --------------------------------
296 \end{itemize} 312 \end{itemize}
297 % -------------------------------- 313 % --------------------------------
298 - \item\label{prop:lucases} \textbf{Proposition.} Stable models are \textit{minimal} so \textit{one} of the following cases takes place: 314 + \item\label{prop:lucases} \textbf{Proposition.} Exactly \textit{one} of the following cases takes place:
299 % -------------------------------- 315 % --------------------------------
300 \begin{enumerate} 316 \begin{enumerate}
301 % -------------------------------- 317 % --------------------------------
@@ -308,50 +324,125 @@ @@ -308,50 +324,125 @@
308 \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. 324 \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$.
309 % -------------------------------- 325 % --------------------------------
310 \end{enumerate} 326 \end{enumerate}
  327 + because stable models are \textit{minimal}.
311 % -------------------------------- 328 % --------------------------------
312 \end{itemize} 329 \end{itemize}
313 % -------------------------------- 330 % --------------------------------
314 \end{frame} 331 \end{frame}
315 % ================================================================ 332 % ================================================================
  333 +\begin{frame}{Disjunction Example | The Interpretation's Lattice}
  334 + \begin{center}
  335 + \begin{tikzpicture}
  336 + %
  337 + \draw [help lines, color=gray!20] grid (11,7);
  338 + %
  339 + \node [fill=gray!50] (sm) at (1, 7) {sm};
  340 + \node [draw,rrect] (sub) at (2, 7) {sub};
  341 + \node [draw, fill=gray!10] (sup) at (3, 7) {sup};
  342 + \node (ind) at (4, 7) {ind};
  343 + %
  344 + \node [draw, rrect] (E) at (5.5,0) {$\emptyset$};
  345 + %
  346 + \node [draw, rrect] (a) at (1.5,1.5) {$a$};
  347 + \node [draw, rrect] (b) at (0,1.5) {$b$};
  348 + \node [draw, rrect] (c) at (4.5,1.5) {$c$};
  349 + \node [fill=gray!50] (A) at (8.5,1.5) {$\co{a}$};
  350 + \node (B) at (9.5,1.5) {$\co{b}$};
  351 + \node (C) at (10.5,1.5) {$\co{c}$};
  352 + %
  353 + \node [fill=gray!50] (ab) at (0,4) {$ab$};
  354 + \node [fill=gray!50] (ac) at (3,4) {$ac$};
  355 + \node (aB) at (1,4) {$a\co{b}$};
  356 + \node (aC) at (2,4) {$a\co{c}$};
  357 + \node [draw, fill=gray!10] (Ab) at (4,4) {$\co{a}b$};
  358 + \node [draw, fill=gray!10] (Ac) at (5,4) {$\co{a}c$};
  359 + \node [draw, fill=gray!10] (AB) at (6,4) {$\co{a}\co{b}$};
  360 + \node [draw, fill=gray!10] (AC) at (7,4) {$\co{a}\co{c}$};
  361 + \node [fill=white] (bc) at (10,4) {$bc$};
  362 + \node [fill=white] (bC) at (11,4) {$b\co{c}$};
  363 + \node [fill=white] (Bc) at (12,4) {$\co{b}c$};
  364 + \node [fill=white] (BC) at (13,4) {$\co{b}\co{c}$};
  365 + %
  366 + \node [draw, fill=gray!10] (abc) at (1.5,6) {$abc$};
  367 + \node (abC) at (3,6) {$ab\co{c}$};
  368 + \node (aBc) at (4,6) {$a\co{b}c$};
  369 + \node (aBC) at (5,6) {$a\co{b}\co{c}$};
  370 + \node [draw, fill=gray!10] (Abc) at (7,6) {$\co{a}bc$};
  371 + \node [draw, fill=gray!10] (AbC) at (8,6) {$\co{a}b\co{c}$};
  372 + \node [draw, fill=gray!10] (ABc) at (9,6) {$\co{a}\co{b}c$};
  373 + \node [draw, fill=gray!10] (ABC) at (10,6) {$\co{a}\co{b}\co{c}$};
  374 + %
  375 + \draw [->] (ab) to [out=270,in=180] (E);
  376 + \draw [->] (ab) to [out=270,in=90] (a);
  377 + \draw [->] (ab) to [out=270,in=90] (b);
  378 + \draw [->] (ab) to [out=90,in=270] (abc);
  379 + %
  380 + \draw [->] (ac) to [out=270,in=180] (E);
  381 + \draw [->] (ac) to [out=270,in=90] (a);
  382 + \draw [->] (ac) to [out=270,in=90] (c);
  383 + \draw [->] (ac) to [out=90,in=270] (abc);
  384 + %
  385 + \draw [->] (A) to [out=270,in=0] (E);
  386 + %
  387 + \draw [->] (A) to [out=90,in=270] (Abc);
  388 + \draw [->] (A) to [out=90,in=270] (AbC);
  389 + \draw [->] (A) to [out=90,in=270] (ABc);
  390 + \draw [->] (A) to [out=90,in=270] (ABC);
  391 + %
  392 + \draw [->] (A) to [out=90,in=270] (Ab);
  393 + \draw [->] (A) to [out=90,in=270] (Ac);
  394 + \draw [->] (A) to [out=90,in=270] (AB);
  395 + \draw [->] (A) to [out=90,in=270] (AC);
  396 + \end{tikzpicture}
  397 + \end{center}
  398 +\end{frame}
  399 +% ================================================================
316 \begin{frame} 400 \begin{frame}
317 401
318 - 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 $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$. 402 +
  403 + \begin{itemize}
  404 + \item Consider the ASP program $P = C \land F \land R$ with total choices $\Theta $ and stable models $\fml{S}$.
  405 +
  406 + \item Let $d : \fml{S} \to \intcc{0,1}$ such that $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$ for each $\theta \in \Theta$.
  407 + \end{itemize}
319 \end{frame} 408 \end{frame}
320 % ================================================================ 409 % ================================================================
321 \begin{frame} 410 \begin{frame}
322 - 411 + For each $z\in\fml{Z}$ only one of the following cases takes place
323 \begin{enumerate} 412 \begin{enumerate}
324 - \item For each $z\in\fml{Z}$ only one of the following cases takes place 413 + \item $z$ is inconsistent. Then \textbf{define}
  414 + \begin{equation}
  415 + w_d\at{x} = 0.\label{def:w.inconsistent}
  416 + \end{equation}
  417 + %
  418 + \item $z$ is an interpretation and $\lset{z} = \set{z} = \uset{z}$. Then $z$ is a stable model and \textbf{define}
  419 + \begin{equation}
  420 + w_d\at{z} = w\at{z} = d\at{z} \pr{\theta_z}.\label{eq:prob.sm}
  421 + \end{equation}
  422 + %
  423 + \item $z$ is an interpretation and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define}
  424 + \begin{equation}
  425 + w_d\at{z} = \sum_{s \in \lset{z}} w_d\at{s}.\label{def:w.disj}
  426 + \end{equation}
  427 + %
  428 + \item $z$ is an interpretation and $\lset{z} = \emptyset \land \uset{z} \neq \emptyset$. Then \textbf{define}
  429 + \begin{equation}
  430 + w_d\at{z} = \prod_{s \in \uset{z}} w_d\at{s}.\label{def:w.conj}
  431 + \end{equation}
  432 + %
  433 + \item $z$ is an interpretation and $\lset{z} = \emptyset \land \uset{z} = \emptyset$. Then \textbf{define}
  434 + \begin{equation}
  435 + w_d\at{z} = 0.\label{def:w.empty}
  436 + \end{equation}
  437 + \end{enumerate}
  438 + \end{frame}
  439 + % ================================================================
  440 + \begin{frame}
325 \begin{enumerate} 441 \begin{enumerate}
326 - \item $z$ is inconsistent. Then \textbf{define}  
327 - \begin{equation}  
328 - w_d\at{x} = 0.\label{def:w.inconsistent}  
329 - \end{equation}  
330 - %  
331 - \item $z$ is an interpretation and $\lset{z} = \set{z} = \uset{x}$. Then $z = s$ is a stable model and \textbf{define}  
332 - \begin{equation}  
333 - w_d\at{z} = w\at{s} = d\at{s} \pr{\theta_s}.\label{eq:prob.sm}  
334 - \end{equation}  
335 - %  
336 - \item $z$ is an interpretation and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define}  
337 - \begin{equation}  
338 - w_d\at{z} = \sum_{s \in \lset{z}} w_d\at{s}.\label{def:w.disj}  
339 - \end{equation}  
340 - %  
341 - \item $z$ is an interpretation and $\lset{z} = \emptyset \land \uset{z} \neq \emptyset$. Then \textbf{define}  
342 - \begin{equation}  
343 - w_d\at{z} = \prod_{s \in \uset{z}} w_d\at{s}.\label{def:w.conj}  
344 - \end{equation}  
345 - %  
346 - \item $z$ is an interpretation and $\lset{z} = \emptyset \land \uset{z} = \emptyset$. Then \textbf{define}  
347 - \begin{equation}  
348 - w_d\at{z} = 0.\label{def:w.empty}  
349 - \end{equation}  
350 - \end{enumerate}  
351 % 442 %
352 - \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. 443 + \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.
353 % 444 %
354 - \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 $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$. 445 + \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 $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
355 % 446 %
356 \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: 447 \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:
357 \begin{enumerate} 448 \begin{enumerate}
@@ -432,69 +523,6 @@ @@ -432,69 +523,6 @@
432 % ================================================================ 523 % ================================================================
433 \begin{frame} 524 \begin{frame}
434 525
435 - Now consider all the interpretations for this program:  
436 - \begin{center}  
437 - \begin{tikzpicture}  
438 - %\draw [help lines] grid (11,3);  
439 - %  
440 - \node [draw, circle] (E) at (5.5,0) {$\emptyset$};  
441 - %  
442 - \node [draw, circle] (a) at (3,2) {$a$};  
443 - \node [draw, circle] (b) at (2,2) {$b$};  
444 - \node [draw, circle] (c) at (4,2) {$c$};  
445 - \node [fill=gray!50] (A) at (8,2) {$\co{a}$};  
446 - \node (B) at (9,2) {$\co{b}$};  
447 - \node (C) at (7,2) {$\co{c}$};  
448 - %  
449 - \node [fill=gray!50] (ab) at (0,4) {$ab$};  
450 - \node [fill=gray!50] (ac) at (1,4) {$ac$};  
451 - \node (aB) at (2,4) {$a\co{b}$};  
452 - \node (aC) at (3,4) {$a\co{c}$};  
453 - \node [draw] (Ab) at (4,4) {$\co{a}b$};  
454 - \node [draw] (Ac) at (5,4) {$\co{a}c$};  
455 - \node [draw] (AB) at (6,4) {$\co{a}\co{b}$};  
456 - \node [draw] (AC) at (7,4) {$\co{a}\co{c}$};  
457 - \node [fill=white] (bc) at (10,4) {$bc$};  
458 - \node [fill=white] (bC) at (11,4) {$b\co{c}$};  
459 - \node [fill=white] (Bc) at (12,4) {$\co{b}c$};  
460 - \node [fill=white] (BC) at (13,4) {$\co{b}\co{c}$};  
461 - %  
462 - \node [draw] (abc) at (0.5,6) {$abc$};  
463 - \node (abC) at (3,6) {$ab\co{c}$};  
464 - \node (aBc) at (4,6) {$a\co{b}c$};  
465 - \node (aBC) at (5,6) {$a\co{b}\co{c}$};  
466 - \node [draw] (Abc) at (7,6) {$\co{a}bc$};  
467 - \node [draw] (AbC) at (8,6) {$\co{a}b\co{c}$};  
468 - \node [draw] (ABc) at (9,6) {$\co{a}\co{b}c$};  
469 - \node [draw] (ABC) at (10,6) {$\co{a}\co{b}\co{c}$};  
470 - %  
471 - \draw [->] (ab) to [out=270,in=180] (E);  
472 - \draw [->] (ab) to [out=270,in=90] (a);  
473 - \draw [->] (ab) to [out=270,in=90] (b);  
474 - \draw [->] (ab) to [out=90,in=270] (abc);  
475 - %  
476 - \draw [->] (ac) to [out=270,in=180] (E);  
477 - \draw [->] (ac) to [out=270,in=90] (a);  
478 - \draw [->] (ac) to [out=270,in=90] (c);  
479 - \draw [->] (ac) to [out=90,in=270] (abc);  
480 - %  
481 - \draw [->] (A) to [out=270,in=0] (E);  
482 - %  
483 - \draw [->] (A) to [out=90,in=270] (Abc);  
484 - \draw [->] (A) to [out=90,in=270] (AbC);  
485 - \draw [->] (A) to [out=90,in=270] (ABc);  
486 - \draw [->] (A) to [out=90,in=270] (ABC);  
487 - %  
488 - \draw [->] (A) to [out=90,in=270] (Ab);  
489 - \draw [->] (A) to [out=90,in=270] (Ac);  
490 - \draw [->] (A) to [out=90,in=270] (AB);  
491 - \draw [->] (A) to [out=90,in=270] (AC);  
492 - \end{tikzpicture}  
493 - \end{center}  
494 -\end{frame}  
495 -% ================================================================  
496 -\begin{frame}  
497 -  
498 In this diagram: 526 In this diagram:
499 \begin{itemize} 527 \begin{itemize}
500 \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] {$ab$}}. 528 \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] {$ab$}}.
text/00_PASP.toc
1 \beamer@sectionintoc {1}{Introduction}{2}{0}{1} 1 \beamer@sectionintoc {1}{Introduction}{2}{0}{1}
2 -\beamer@sectionintoc {2}{Motivation}{10}{0}{2}  
3 -\beamer@sectionintoc {3}{Resolution}{12}{0}{3}  
4 -\beamer@sectionintoc {4}{Cases \& Examples}{17}{0}{4}  
5 -\beamer@subsectionintoc {4}{1}{Programs with disjunctive heads}{18}{0}{4}  
6 -\beamer@subsectionintoc {4}{2}{Non-stratified programs}{23}{0}{4}  
7 -\beamer@sectionintoc {5}{Conclusions}{28}{0}{5} 2 +\beamer@sectionintoc {2}{Extending Probability to Samples}{11}{0}{2}
  3 +\beamer@sectionintoc {3}{Cases \& Examples}{18}{0}{3}
  4 +\beamer@subsectionintoc {3}{1}{Programs with disjunctive heads}{19}{0}{3}
  5 +\beamer@subsectionintoc {3}{2}{Non-stratified programs}{23}{0}{3}
  6 +\beamer@sectionintoc {4}{Conclusions}{28}{0}{4}
text/00_PASP.xdv
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