Francisco Coelho / zugzwang

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Commit 237d62bda61e206c29f2a7e73a9a81960869bef8

Authored by Francisco Coelho
1 parent 241956f9
Exists in master

Further rewriting of 00_PASP

Showing 12 changed files with 363 additions and 352 deletions   Show diff stats
text/00_PASP.aux
  Diff comments   View file @237d62b
... ... @@ -51,119 +51,115 @@
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... ...
text/00_PASP.fdb_latexmk
  Diff comments   View file @237d62b
1 1 # Fdb version 3
2   -["xdvipdfmx"] 1667935664 "00_PASP.xdv" "00_PASP.pdf" "00_PASP" 1667935664
3   - "00_PASP.xdv" 1667935663 214680 71a3fe8c0e42560f8631e0d09f4181a7 "xelatex"
  2 +["xdvipdfmx"] 1668012215 "00_PASP.xdv" "00_PASP.pdf" "00_PASP" 1668012216
  3 + "00_PASP.xdv" 1668012215 203892 0858b8d4b4e7e8aa8085e0bea6b8815a "xelatex"
4 4 (generated)
5 5 "00_PASP.pdf"
6   -["xelatex"] 1667935657 "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" "00_PASP.xdv" "00_PASP" 1667935664
7   - "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" 1667935656 32020 e7e67df128d9c518e87afbb7c6491ef3 ""
  6 +["xelatex"] 1668012209 "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" "00_PASP.xdv" "00_PASP" 1668012216
  7 + "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" 1668012208 32880 d2ff92ff8b2ef9e5a1bd016ddcdcb92a ""
8 8 "/usr/share/texmf-dist/fonts/map/fontname/texfonts.map" 1650183167 3524 cb3e574dea2d1052e39280babc910dc8 ""
9 9 "/usr/share/texmf-dist/fonts/tfm/adobe/zapfding/pzdr.tfm" 1650183167 1528 f853c4d1b4e0550255e02831fdc8496f ""
10 10 "/usr/share/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1650183167 1004 54797486969f23fa377b128694d548df ""
... ... @@ -32,7 +32,6 @@
32 32 "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss10.tfm" 1650183167 11176 53ebf7a171df1f9447b387b178768bb5 ""
33 33 "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss12.tfm" 1650183167 11232 955a7245396175d9219648eadc654ac9 ""
34 34 "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss8.tfm" 1650183167 11180 705632ac6b4fb69204ad970192cdf4e5 ""
35   - "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmssbo10.tfm" 1650183167 11304 1bf3f99c327bf3c434fc66d6737a89a2 ""
36 35 "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmssbx10.tfm" 1650183167 11168 06d87f5698fd1b642d96449b7c8d90b0 ""
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38 37 "/usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmsso8.tfm" 1650183167 11312 6908420b41cb9ad797e206f7391721ea ""
... ... @@ -226,11 +225,11 @@
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228 227 "/var/lib/texmf/web2c/xetex/xelatex.fmt" 1667607223 5825155 acc75abcaf8e6f881414fa8770610f6c ""
229   - "00_PASP.aux" 1667935663 10150 06d1cb9804fa13f9caa5b87bee738349 "xelatex"
230   - "00_PASP.nav" 1667935663 4557 bd5b660e37644e87cd38eeeceb472239 "xelatex"
231   - "00_PASP.out" 1667935663 1097 ff89349dbd8e3874420c8d8c57eb7504 "xelatex"
232   - "00_PASP.tex" 1667935656 32020 e7e67df128d9c518e87afbb7c6491ef3 ""
233   - "00_PASP.toc" 1667935663 384 d39373a67753c9668ace5a0f122dc338 "xelatex"
  228 + "00_PASP.aux" 1668012215 10175 631506c17974104545ec9f7326871ea2 "xelatex"
  229 + "00_PASP.nav" 1668012215 4415 70ff7441078bcca01fe957123d46aa96 "xelatex"
  230 + "00_PASP.out" 1668012215 1123 b2bcf7d8938de98bdfd00e76ac45d7c1 "xelatex"
  231 + "00_PASP.tex" 1668012208 32880 d2ff92ff8b2ef9e5a1bd016ddcdcb92a ""
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234 233 "Pabc_alpha03.pdf" 1663331875 12659 e50124c327299028f63ecb501d23eb0d ""
235 234 (generated)
236 235 "00_PASP.aux"
... ...
text/00_PASP.fls
  Diff comments   View file @237d62b
... ... @@ -1471,10 +1471,6 @@ INPUT /usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss12.tfm
1471 1471 INPUT ./00_PASP.toc
1472 1472 INPUT 00_PASP.toc
1473 1473 INPUT 00_PASP.toc
1474   -INPUT /usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmssbo10.tfm
1475   -INPUT ./00_PASP.toc
1476   -INPUT 00_PASP.toc
1477   -INPUT 00_PASP.toc
1478 1474 INPUT /usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmssbx10.tfm
1479 1475 INPUT ./00_PASP.toc
1480 1476 INPUT 00_PASP.toc
... ...
text/00_PASP.log
  Diff comments   View file @237d62b
1   -This is XeTeX, Version 3.141592653-2.6-0.999994 (TeX Live 2022/Arch Linux) (preloaded format=xelatex 2022.11.5) 8 NOV 2022 19:27
  1 +This is XeTeX, Version 3.141592653-2.6-0.999994 (TeX Live 2022/Arch Linux) (preloaded format=xelatex 2022.11.5) 9 NOV 2022 16:43
2 2 entering extended mode
3 3 \write18 enabled.
4 4 file:line:error style messages enabled.
... ... @@ -608,7 +608,7 @@ LaTeX Font Info: Trying to load font information for T1+lmss on input line 11
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610 610 ))
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613 613 File: l3backend-xetex.def 2022-04-14 L3 backend support: XeTeX
614 614 \l__color_backend_stack_int=\count343
... ... @@ -621,30 +621,30 @@ File: l3backend-xetex.def 2022-04-14 L3 backend support: XeTeX
621 621 ) (./00_PASP.aux)
622 622 \openout1 = `00_PASP.aux'.
623 623  
624   -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 65.
625   -LaTeX Font Info: ... okay on input line 65.
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627   -LaTeX Font Info: ... okay on input line 65.
628   -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 65.
629   -LaTeX Font Info: ... okay on input line 65.
630   -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 65.
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632   -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 65.
633   -LaTeX Font Info: Trying to load font information for TS1+cmr on input line 65.
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  629 +LaTeX Font Info: ... okay on input line 77.
  630 +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 77.
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text/00_PASP.snm
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1 1 \beamer@slide {eq:prob.tc}{5}
2 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 8 \beamer@slide {def:w.inconsistent}{16}
9 9 \beamer@slide {eq:prob.sm}{16}
10 10 \beamer@slide {def:w.disj}{16}
11 11 \beamer@slide {def:w.conj}{16}
12 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}
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text/00_PASP.synctex.gz
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text/00_PASP.tex
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... ... @@ -23,6 +23,18 @@
23 23 \usepackage[overridenote]{pdfpc}
24 24  
25 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 39 \usepackage{commath}
28 40 \usepackage{amssymb}
... ... @@ -96,10 +108,10 @@
96 108 \item \textbf{Atoms} $\fml{A}$,
97 109 $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$,
98 110 % --------------------------------
99   - \item \textbf{Inputs} $\fml{Z}$:
  111 + \item \textbf{Samples} $\fml{Z}$:
100 112 $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$
101 113 % --------------------------------
102   - \item \textbf{Interpretations} or \textit{consistent inputs} $\fml{I}$ :
  114 + \item \textbf{Interpretations} or \textit{consistent samples} $\fml{I}$ :
103 115 $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$
104 116 % --------------------------------
105 117 \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where
... ... @@ -112,7 +124,7 @@
112 124 % --------------------------------
113 125 \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 $P$.
116 128 \end{itemize}
117 129 % --------------------------------
118 130 \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 @@
142 154 This is the \emph{distribution semantic} as set by Sato.
143 155 \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 % --------------------------------
149 163 \begin{quotation}
150 164 There's a problem right at extending to stable models.
... ... @@ -179,7 +193,7 @@
179 193 % --------------------------------
180 194 \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 $$
184 198 \begin{aligned}
185 199 \pr{\set{a, b}} &= 0.3 \alpha, \cr
... ... @@ -202,17 +216,17 @@
202 216 \pr{\set{a, c}} &= 0.3 \co{\alpha}.
203 217 \end{aligned}
204 218 $$
205   - \item A \textit{random variable} captures this:
  219 + \item A \textit{random variable} captures this uncertainty:
206 220 $$
207 221 \begin{aligned}
208   - \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr
209   - \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr
210   - \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}.
  222 + \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr
  223 + \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr
  224 + \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}.
211 225 \end{aligned}
212 226 $$
213 227 \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 % --------------------------------
218 232 \end{itemize}
... ... @@ -220,14 +234,14 @@
220 234 % --------------------------------
221 235 \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 $$
227 241 \begin{aligned}
228   - \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr
229   - \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr
230   - \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}.
  242 + \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr
  243 + \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr
  244 + \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}.
231 245 \end{aligned}
232 246 $$
233 247 % --------------------------------
... ... @@ -237,49 +251,51 @@
237 251 % --------------------------------
238 252 \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 % --------------------------------
244 264 \end{itemize}
245 265 % --------------------------------
246 266 \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 % --------------------------------
254 272 \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 % --------------------------------
262 276 \item $e$, the respective empirical distribution.
263 277 % --------------------------------
264 278 \item $D$, some probability divergence, \textit{e.g.} Kullback-Leibler.
265 279 % --------------------------------
266 280 \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 % --------------------------------
272 286 \end{frame}
273 287 % ================================================================
274   -\section{Resolution}
  288 +\section{Extending Probability to Samples}
275 289 % ================================================================
276 290 \begin{frame}{Resolution Path}
  291 + Prior to \textit{conciliation} with data:
277 292 \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).
282 297 \end{enumerate}
  298 + Assuming a conditional probability set on stable models, \textit{How to extend it to interpretations?}
283 299 \end{frame}
284 300 \begin{frame}{Bounds of Interpretations}
285 301 % --------------------------------
... ... @@ -295,7 +311,7 @@
295 311 % --------------------------------
296 312 \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 % --------------------------------
300 316 \begin{enumerate}
301 317 % --------------------------------
... ... @@ -308,50 +324,125 @@
308 324 \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$.
309 325 % --------------------------------
310 326 \end{enumerate}
  327 + because stable models are \textit{minimal}.
311 328 % --------------------------------
312 329 \end{itemize}
313 330 % --------------------------------
314 331 \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 +% ================================================================
316 400 \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}
319 408 \end{frame}
320 409 % ================================================================
321 410 \begin{frame}
322   -
  411 + For each $z\in\fml{Z}$ only one of the following cases takes place
323 412 \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}
325 441 \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 %
356 447 \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:
357 448 \begin{enumerate}
... ... @@ -432,69 +523,6 @@
432 523 % ================================================================
433 524 \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   -
498 526 In this diagram:
499 527 \begin{itemize}
500 528 \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
  Diff comments   View file @237d62b
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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