Commit 237d62bda61e206c29f2a7e73a9a81960869bef8
1 parent
241956f9
Exists in
master
Further rewriting of 00_PASP
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363 additions
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352 deletions
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text/00_PASP.aux
@@ -51,119 +51,115 @@ | @@ -51,119 +51,115 @@ | ||
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text/00_PASP.fdb_latexmk
1 | # Fdb version 3 | 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 | (generated) | 4 | (generated) |
5 | "00_PASP.pdf" | 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 "" | ||
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text/00_PASP.fls
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text/00_PASP.log
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839 | File: Pabc_alpha03.pdf Graphic file (type pdf) | 836 | File: Pabc_alpha03.pdf Graphic file (type pdf) |
840 | <use Pabc_alpha03.pdf> | 837 | <use Pabc_alpha03.pdf> |
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882 | File: se-utf32be.def 2019/11/29 v1.12 stringenc: UTF-32BE | 879 | File: se-utf32be.def 2019/11/29 v1.12 stringenc: UTF-32BE |
883 | ) (./00_PASP.aux) | 880 | ) (./00_PASP.aux) |
884 | Package rerunfilecheck Info: File `00_PASP.out' has not changed. | 881 | Package rerunfilecheck Info: File `00_PASP.out' has not changed. |
885 | -(rerunfilecheck) Checksum: FF89349DBD8E3874420C8D8C57EB7504;1097. | 882 | +(rerunfilecheck) Checksum: B2BCF7D8938DE98BDFD00E76AC45D7C1;1123. |
886 | ) | 883 | ) |
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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
@@ -19,68 +19,65 @@ | @@ -19,68 +19,65 @@ | ||
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73 | -\headcommand {\sectionentry {5}{Conclusions}{28}{Conclusions}{0}} | ||
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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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