continuing 00_PASP

Francisco Coelho
1 parent b43c061c
Showing 12 changed files with 344 additions and 268 deletions Show diff stats
text/00_PASP.aux
text/00_PASP.fdb_latexmk
text/00_PASP.fls
text/00_PASP.log
text/00_PASP.nav
text/00_PASP.out
text/00_PASP.pdf
text/00_PASP.snm
text/00_PASP.synctex.gz
text/00_PASP.tex
text/00_PASP.toc
text/00_PASP.xdv
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         $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$,
         % --------------------------------
-        \item \textbf{Observations} $\fml{Z}$:
+        \item \textbf{Inputs} $\fml{Z}$:
         $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$
         % --------------------------------
-        \item \textbf{Interpretations} or \textit{consistent observations} $\fml{I}$ :
+        \item \textbf{Interpretations} or \textit{consistent inputs} $\fml{I}$ :
         $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$
         % --------------------------------
         \item \textit{PASP Problem}  or \textbf{Specification:} $P = C \land F \land R$ where
@@ -111,7 +112,7 @@
                 % --------------------------------
                 \item $R = R_P$ \textit{rules}.
                 % --------------------------------
-                \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{observations} and \textit{interpretations} of $P$.
+                \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{inputs} and \textit{interpretations} of $P$.
             \end{itemize}
         % --------------------------------
         \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$.
@@ -124,7 +125,7 @@
     % --------------------------------
     \begin{itemize}
         % --------------------------------
-        \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 $\neg a_i$.  
+        \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 $\neg a$.  
         % --------------------------------
         %\item For $s\in\fml{S}$ let $\theta_s \subseteq s$ (unique \textit{total choice})
         %\item Define $\fml{S}_\theta = \cset{s \in \fml{S}}{\theta \subset s}$.
@@ -133,30 +134,20 @@
         % --------------------------------
         \item \textbf{Total Choice Probability:}
         \begin{equation}
-            \pr{\theta} = \prod_{a_i \in \theta}\alpha_i \prod_{\neg a_i \in \theta}\co{\alpha_i}.\label{eq:prob.tc}
+            \pr{\theta} = \prod_{a \in \theta}\alpha \prod_{\neg a \in \theta}\co{\alpha}.\label{eq:prob.tc}
         \end{equation}
         % --------------------------------
     \end{itemize}
     % --------------------------------
-    \begin{quote}
-        This is the \emph{Distribution Semantics} as set by Sato.
-    \end{quote}
+    This is the \emph{distribution semantic} as set by Sato.
 \end{frame}
 % ================================================================
 \begin{frame}{Problem Statement}
     % --------------------------------
-    \begin{quotation}
-        How to extend probability from the total choices to interpretations and observations?
-    \end{quotation}
-    % --------------------------------
-    \begin{itemize}
-        % --------------------------------
-        \item \textbf{Todo:} Extend probability to \textit{stable models}, \textit{interpretations} and \textit{observations}.
-        % --------------------------------
-    \end{itemize}
+    How to \textit{extend} probability from total choices to stable models, interpretations and inputs?
     % --------------------------------
     \begin{quotation}
-        \textbf{But} there is a problem extending probability from total choices to stable models.
+        There's a problem right at extending to stable models.
     \end{quotation}
     % --------------------------------
 \end{frame}
@@ -200,32 +191,96 @@
     % --------------------------------
 \end{frame}
 % ================================================================
+\begin{frame}{Lack of Information \& Parametrization}
+    % --------------------------------   
+    \begin{itemize}
+        % --------------------------------
+        \item The specification \textit{lacks information} to set $\alpha\in\intcc{0,1}$ in
+        $$
+        \begin{aligned}
+            \pr{\set{a, b}} &= 0.3 \alpha, \cr
+            \pr{\set{a, c}} &= 0.3 \co{\alpha}.
+        \end{aligned}
+        $$
+        \item A \textit{random variable} captures this:
+        $$
+        \begin{aligned}
+            \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr
+            \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr
+            \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}.
+        \end{aligned}
+        $$
+        \item Other uncertainties lead to further parameters:
+        $$
+        \pr{s \given\ A_1 = \alpha_1, \ldots, A_n = \alpha_n }.
+        $$ 
+        % --------------------------------
+    \end{itemize}
+    Reducing \textbf{specification uncertainty}, \textit{e.g.} setting $A = 0.21$, must result from \textbf{observations}.
+    % --------------------------------
+\end{frame}
+% ================================================================
+\begin{frame}{Main Research Question}
+    % --------------------------------
+    A \textit{random variable} captures this:
+    $$
+    \begin{aligned}
+        \pr{\set{\neg a} \given\ A = \alpha } &= 0.7, \cr
+        \pr{\set{a, b} \given\ A = \alpha } &= 0.3 \alpha, \cr
+        \pr{\set{a, c} \given\ A = \alpha } &= 0.3 \co{\alpha}.
+    \end{aligned}
+    $$
+    % --------------------------------
+    \begin{block}{Main Research Question}
+        Can \textit{all} specification uncertainties be neatly expressed as that example?
+    \end{block}
+    % --------------------------------
+    \begin{itemize}
+        % --------------------------------
+        \item Follow ASP syntax; for each case, consider the possible uncertainty scenarios. 
+        % --------------------------------
+        \item The disjunction example illustrates one such step.
+        % --------------------------------
+    \end{itemize}    
+    % --------------------------------
+\end{frame}
+% ================================================================
 \section{Motivation}
 % ================================================================
 \begin{frame}{Specification, Data \& Evaluation}
     % --------------------------------
-    Given some procedure to assign probabilities to observations from specifications and:
+    Given some procedure to extend probabilities to stable models, interpretations and inputs, and given:
     % --------------------------------
     \begin{itemize}
         % --------------------------------
         \item $P$, a specification.
         % --------------------------------
-        \item $p$, the distribution of observations from above.
+        \item $p$, the distribution of inputs from above.
         % --------------------------------
-        \item $Z$, a dataset of observations.
+        \item $Z$, a dataset of inputs.
         % --------------------------------
         \item $e$, the respective empirical distribution.
         % --------------------------------
         \item $D$, some probability divergence, \textit{e.g.} Kullback-Leibler.
         % --------------------------------
     \end{itemize}
-    % --------------------------------
-    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.
+    % --------------------------------    
+    \begin{quotation}
+        For a dataset $Z$, $D\at{P} = D\at{e, p}$ is a \textbf{performance} measure of $P$ and can be used, \textit{e.g.} as fitness, by algorithms searching for \textbf{optimal specifications of a dataset}.
+    \end{quotation}
     % --------------------------------
 \end{frame}
 % ================================================================
 \section{Resolution}
 % ================================================================
+\begin{frame}{Resolution Path}
+    \begin{enumerate}
+        \item \textit{Parametrize} lack of knowledge \emph{e.g.} $\alpha$ in the disjunction example.
+        \item This extends probability from total choices to \textit{standard models}.
+        \item Extend probability from standard models to \textit{interpretations}. \textbf{How?} Later.
+        \item Set $\pr{z} = 0$ for $z \in \fml{Z} \setminus \fml{I}$.
+    \end{enumerate}
+\end{frame}
 \begin{frame}{Bounds of Interpretations}    
     % --------------------------------
     \begin{itemize}
@@ -294,9 +349,9 @@
             \end{equation}
         \end{enumerate}
         %
-        \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.
+        \item The last point defines a ``weight'' function on the inputs that depends not only on the total choices and stable models of a PASP but also on a certain function $d$ that must respect some conditions. To simplify the notation we use the subscript in $w_d$ only when necessary.
         %
-        \item 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 $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
+        \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an interpretation, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} input $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
         %
         \item \textbf{We haven't yet defined a probability measure.} To do so we must define a set of samples $\Omega$, a set of events $F\subseteq \pset{\Omega}$ and a function $P:F\to\intcc{0,1}$ such that:
         \begin{enumerate}
 \beamer@sectionintoc {1}{Introduction}{2}{0}{1}
-\beamer@sectionintoc {2}{Motivation}{8}{0}{2}
-\beamer@sectionintoc {3}{Resolution}{10}{0}{3}
-\beamer@sectionintoc {4}{Cases \& Examples}{14}{0}{4}
-\beamer@subsectionintoc {4}{1}{Programs with disjunctive heads}{15}{0}{4}
-\beamer@subsectionintoc {4}{2}{Non-stratified programs}{20}{0}{4}
-\beamer@sectionintoc {5}{Conclusions}{25}{0}{5}
+\beamer@sectionintoc {2}{Motivation}{10}{0}{2}
+\beamer@sectionintoc {3}{Resolution}{12}{0}{3}
+\beamer@sectionintoc {4}{Cases \& Examples}{17}{0}{4}
+\beamer@subsectionintoc {4}{1}{Programs with disjunctive heads}{18}{0}{4}
+\beamer@subsectionintoc {4}{2}{Non-stratified programs}{23}{0}{4}
+\beamer@sectionintoc {5}{Conclusions}{28}{0}{5}
1	\beamer@sectionintoc {1}{Introduction}{2}{0}{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}