Francisco Coelho / zugzwang

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Commit 241956f9bb8224fb6e8b759f89a14ad8b2b48b68

Authored by Francisco Coelho
1 parent b43c061c
Exists in master

continuing 00_PASP

Showing 12 changed files with 344 additions and 268 deletions   Show diff stats
text/00_PASP.aux
  Diff comments   View file @241956f
... ... @@ -45,121 +45,130 @@
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... ...
text/00_PASP.fdb_latexmk
  Diff comments   View file @241956f
1 1 # Fdb version 3
2   -["xdvipdfmx"] 1667928801 "00_PASP.xdv" "00_PASP.pdf" "00_PASP" 1667928802
3   - "00_PASP.xdv" 1667928801 200932 982c61eac8b5d4d97d7ddad987e569ae "xelatex"
  2 +["xdvipdfmx"] 1667935664 "00_PASP.xdv" "00_PASP.pdf" "00_PASP" 1667935664
  3 + "00_PASP.xdv" 1667935663 214680 71a3fe8c0e42560f8631e0d09f4181a7 "xelatex"
4 4 (generated)
5 5 "00_PASP.pdf"
6   -["xelatex"] 1667928794 "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" "00_PASP.xdv" "00_PASP" 1667928802
7   - "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" 1667928794 29667 8d68ac0db717033678db2dd1c81ee58c ""
  6 +["xelatex"] 1667935657 "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" "00_PASP.xdv" "00_PASP" 1667935664
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... ... @@ -226,11 +226,11 @@
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235 235 (generated)
236 236 "00_PASP.aux"
... ...
text/00_PASP.fls
  Diff comments   View file @241956f
... ... @@ -1467,11 +1467,11 @@ INPUT /usr/share/texmf-dist/fonts/tfm/public/lm/ec-lmss8.tfm
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... ...
text/00_PASP.log
  Diff comments   View file @241956f
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... ... @@ -621,30 +621,30 @@ File: l3backend-xetex.def 2022-04-14 L3 backend support: XeTeX
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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}{11}
4   -\beamer@slide {prop:lucases.a}{11}
5   -\beamer@slide {prop:lucases.b}{11}
6   -\beamer@slide {prop:lucases.c}{11}
7   -\beamer@slide {prop:lucases.d}{11}
8   -\beamer@slide {def:w.inconsistent}{13}
9   -\beamer@slide {eq:prob.sm}{13}
10   -\beamer@slide {def:w.disj}{13}
11   -\beamer@slide {def:w.conj}{13}
12   -\beamer@slide {def:w.empty}{13}
13   -\beamer@slide {eq:def.prob}{13}
14   -\beamer@slide {eq:def.prob.event}{13}
  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}
  8 +\beamer@slide {def:w.inconsistent}{16}
  9 +\beamer@slide {eq:prob.sm}{16}
  10 +\beamer@slide {def:w.disj}{16}
  11 +\beamer@slide {def:w.conj}{16}
  12 +\beamer@slide {def:w.empty}{16}
  13 +\beamer@slide {eq:def.prob}{16}
  14 +\beamer@slide {eq:def.prob.event}{16}
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... ... @@ -49,6 +49,7 @@
49 49 \newcommand{\uset}[1]{\ensuremath{\left|{#1}\right>}}
50 50 \newcommand{\lset}[1]{\ensuremath{\left<{#1}\right|}}
51 51 \newcommand{\pr}[1]{\ensuremath{\mathrm{p}\at{#1}}}
  52 +\newcommand{\given}{\ensuremath{~\middle|~}}
52 53 %
53 54 % Identificaรงรฃo deste documento
54 55 %
... ... @@ -95,10 +96,10 @@
95 96 \item \textbf{Atoms} $\fml{A}$,
96 97 $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$,
97 98 % --------------------------------
98   - \item \textbf{Observations} $\fml{Z}$:
  99 + \item \textbf{Inputs} $\fml{Z}$:
99 100 $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$
100 101 % --------------------------------
101   - \item \textbf{Interpretations} or \textit{consistent observations} $\fml{I}$ :
  102 + \item \textbf{Interpretations} or \textit{consistent inputs} $\fml{I}$ :
102 103 $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$
103 104 % --------------------------------
104 105 \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where
... ... @@ -111,7 +112,7 @@
111 112 % --------------------------------
112 113 \item $R = R_P$ \textit{rules}.
113 114 % --------------------------------
114   - \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{observations} and \textit{interpretations} of $P$.
  115 + \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{inputs} and \textit{interpretations} of $P$.
115 116 \end{itemize}
116 117 % --------------------------------
117 118 \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$.
... ... @@ -124,7 +125,7 @@
124 125 % --------------------------------
125 126 \begin{itemize}
126 127 % --------------------------------
127   - \item \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are $\theta = \set{c_1, \ldots, c_n}$ where $c_i$ is $a_i$ or $\neg a_i$.
  128 + \item \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are $\theta = \cset{t_c}{c \in C}$ where $c=\alpha::a$ and $t_c$ is $a$ or $\neg a$.
128 129 % --------------------------------
129 130 %\item For $s\in\fml{S}$ let $\theta_s \subseteq s$ (unique \textit{total choice})
130 131 %\item Define $\fml{S}_\theta = \cset{s \in \fml{S}}{\theta \subset s}$.
... ... @@ -133,30 +134,20 @@
133 134 % --------------------------------
134 135 \item \textbf{Total Choice Probability:}
135 136 \begin{equation}
136   - \pr{\theta} = \prod_{a_i \in \theta}\alpha_i \prod_{\neg a_i \in \theta}\co{\alpha_i}.\label{eq:prob.tc}
  137 + \pr{\theta} = \prod_{a \in \theta}\alpha \prod_{\neg a \in \theta}\co{\alpha}.\label{eq:prob.tc}
137 138 \end{equation}
138 139 % --------------------------------
139 140 \end{itemize}
140 141 % --------------------------------
141   - \begin{quote}
142   - This is the \emph{Distribution Semantics} as set by Sato.
143   - \end{quote}
  142 + This is the \emph{distribution semantic} as set by Sato.
144 143 \end{frame}
145 144 % ================================================================
146 145 \begin{frame}{Problem Statement}
147 146 % --------------------------------
148   - \begin{quotation}
149   - How to extend probability from the total choices to interpretations and observations?
150   - \end{quotation}
151   - % --------------------------------
152   - \begin{itemize}
153   - % --------------------------------
154   - \item \textbf{Todo:} Extend probability to \textit{stable models}, \textit{interpretations} and \textit{observations}.
155   - % --------------------------------
156   - \end{itemize}
  147 + How to \textit{extend} probability from total choices to stable models, interpretations and inputs?
157 148 % --------------------------------
158 149 \begin{quotation}
159   - \textbf{But} there is a problem extending probability from total choices to stable models.
  150 + There's a problem right at extending to stable models.
160 151 \end{quotation}
161 152 % --------------------------------
162 153 \end{frame}
... ... @@ -200,32 +191,96 @@
200 191 % --------------------------------
201 192 \end{frame}
202 193 % ================================================================
  194 +\begin{frame}{Lack of Information \& Parametrization}
  195 + % --------------------------------
  196 + \begin{itemize}
  197 + % --------------------------------
  198 + \item The specification \textit{lacks information} to set $\alpha\in\intcc{0,1}$ in
  199 + $$
  200 + \begin{aligned}
  201 + \pr{\set{a, b}} &= 0.3 \alpha, \cr
  202 + \pr{\set{a, c}} &= 0.3 \co{\alpha}.
  203 + \end{aligned}
  204 + $$
  205 + \item A \textit{random variable} captures this:
  206 + $$
  207 + \begin{aligned}
  208 + \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr
  209 + \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr
  210 + \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}.
  211 + \end{aligned}
  212 + $$
  213 + \item Other uncertainties lead to further parameters:
  214 + $$
  215 + \pr{s \given\ A_1 = \alpha_1, \ldots, A_n = \alpha_n }.
  216 + $$
  217 + % --------------------------------
  218 + \end{itemize}
  219 + Reducing \textbf{specification uncertainty}, \textit{e.g.} setting $A = 0.21$, must result from \textbf{observations}.
  220 + % --------------------------------
  221 +\end{frame}
  222 +% ================================================================
  223 +\begin{frame}{Main Research Question}
  224 + % --------------------------------
  225 + A \textit{random variable} captures this:
  226 + $$
  227 + \begin{aligned}
  228 + \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr
  229 + \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr
  230 + \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}.
  231 + \end{aligned}
  232 + $$
  233 + % --------------------------------
  234 + \begin{block}{Main Research Question}
  235 + Can \textit{all} specification uncertainties be neatly expressed as that example?
  236 + \end{block}
  237 + % --------------------------------
  238 + \begin{itemize}
  239 + % --------------------------------
  240 + \item Follow ASP syntax; for each case, consider the possible uncertainty scenarios.
  241 + % --------------------------------
  242 + \item The disjunction example illustrates one such step.
  243 + % --------------------------------
  244 + \end{itemize}
  245 + % --------------------------------
  246 +\end{frame}
  247 +% ================================================================
203 248 \section{Motivation}
204 249 % ================================================================
205 250 \begin{frame}{Specification, Data \& Evaluation}
206 251 % --------------------------------
207   - Given some procedure to assign probabilities to observations from specifications and:
  252 + Given some procedure to extend probabilities to stable models, interpretations and inputs, and given:
208 253 % --------------------------------
209 254 \begin{itemize}
210 255 % --------------------------------
211 256 \item $P$, a specification.
212 257 % --------------------------------
213   - \item $p$, the distribution of observations from above.
  258 + \item $p$, the distribution of inputs from above.
214 259 % --------------------------------
215   - \item $Z$, a dataset of observations.
  260 + \item $Z$, a dataset of inputs.
216 261 % --------------------------------
217 262 \item $e$, the respective empirical distribution.
218 263 % --------------------------------
219 264 \item $D$, some probability divergence, \textit{e.g.} Kullback-Leibler.
220 265 % --------------------------------
221 266 \end{itemize}
222   - % --------------------------------
223   - Given a dataset $Z$, $D\at{P} = D\at{e, p}$ is a \textit{performance} measure of $P$ and can be used, \textit{e.g.} fitness, by algorithms searching for optimal specifications of a dataset.
  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}
224 271 % --------------------------------
225 272 \end{frame}
226 273 % ================================================================
227 274 \section{Resolution}
228 275 % ================================================================
  276 +\begin{frame}{Resolution Path}
  277 + \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 $z \in \fml{Z} \setminus \fml{I}$.
  282 + \end{enumerate}
  283 +\end{frame}
229 284 \begin{frame}{Bounds of Interpretations}
230 285 % --------------------------------
231 286 \begin{itemize}
... ... @@ -294,9 +349,9 @@
294 349 \end{equation}
295 350 \end{enumerate}
296 351 %
297   - \item The last point defines a ``weight'' function on the observations that depends not only on the total choices and stable models of a PASP but also on a certain function $d$ that must respect some conditions. To simplify the notation we use the subscript in $w_d$ only when necessary.
  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.
298 353 %
299   - \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an interpretation, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} observation $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
  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 $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
300 355 %
301 356 \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:
302 357 \begin{enumerate}
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text/00_PASP.toc
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1 1 \beamer@sectionintoc {1}{Introduction}{2}{0}{1}
2   -\beamer@sectionintoc {2}{Motivation}{8}{0}{2}
3   -\beamer@sectionintoc {3}{Resolution}{10}{0}{3}
4   -\beamer@sectionintoc {4}{Cases \& Examples}{14}{0}{4}
5   -\beamer@subsectionintoc {4}{1}{Programs with disjunctive heads}{15}{0}{4}
6   -\beamer@subsectionintoc {4}{2}{Non-stratified programs}{20}{0}{4}
7   -\beamer@sectionintoc {5}{Conclusions}{25}{0}{5}
  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}
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