presentation, as a beamer document

Francisco Coelho
1 parent e70aa13e
Showing 7 changed files with 127 additions and 104 deletions Show diff stats
text/00_PASP.fdb_latexmk
text/00_PASP.fls
text/00_PASP.log
text/00_PASP.pdf
text/00_PASP.synctex.gz
text/00_PASP.tex
text/00_PASP.xdv
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@@ -31,52 +31,75 @@
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 \newcommand{\uset}[1]{\ensuremath{\left|{#1}\right>}}
 \newcommand{\lset}[1]{\ensuremath{\left<{#1}\right|}}
+%
+%	Identificação deste documento
+%
+\title{Zugzwang}
+\subtitle{Stochastic Adventures in Inductive Logic Specifications}
+\author{Francisco Coelho}
+\institute[\texttt{fc@uevora.pt}]{
+  Departamento de Informática\\
+  Universidade de Évora
+}
 \begin{document}
-
-\begin{frame}
-The goal of this text is to \textbf{explore how ASP programs with probabilistic facts} can lead to characterizations of the \textbf{joint distributions} of the program's atoms.
+%
+\begin{frame}[plain]
+\titlepage
+\note{The goal of this text is to **explore how ASP specifications with probabilistic facts** can lead to characterizations of the **joint distributions** of the specification's atoms.}
 \end{frame}
 \section{Introduction}
-\begin{frame}{Notation}
-    \note{
-- We start with **common notations and assumptions** such as:\\
-- Hi\\
-- There\\
-- And another note}
+\begin{frame}{Notation and Assumptions}
+    \note{We start with **common notations and assumptions**.}
     \begin{itemize}
         \item The \textbf{complement} of $x$ is $\co{x} = 1 - x$.
         \item A \textbf{probabilistic atomic choice} $\alpha:a$ defines the disjunction $a \lor \neg a$ and assigns probabilities $P\at{a} = \alpha, P\at{\neg a} = \co{\alpha}$. 
-        \item $\delta a$ denotes the disjunction $a \lor \neg a$ associated to the probabilistic choice $\alpha : a$ and $\delta\! \set{\alpha: a, a \in A} = \set{\delta a, a \in A}$ for any set of atoms $A$.
-        \item Start with the \textbf{closed world assumption}, where $\naf x \models \neg x$.
-        \item Also assume that \textbf{probabilistic choices} and \textbf{subgoals} are iid.
+        \item $\delta a$ denotes the \textbf{disjunction} $a \lor \neg a$ associated to the probabilistic choice $\alpha : a$ and $\delta\! \set{\alpha: a, a \in A} = \set{\delta a, a \in A}$ for any set of atoms $A$.
+        \item We adopt the the \textbf{closed world assumption}, where $\naf x \models \neg x$.
+        \item And also assume that \textbf{probabilistic choices} and \textbf{subgoals} are iid.
     \end{itemize}
+    \note{**Subgoals are IID** means that ???}
 \end{frame}
-\begin{frame}    
+\begin{frame}{General Setting}
     \note{Next, we consider the following **general setting**}
-    \begin{itemize}        
-        \item  Let $\fml{A}$ be a set of \textbf{atoms}, $\fml{Z}$ the respective set of \textbf{observations},
-        $\fml{Z} = \set{z = \alpha \cup \nu \middle| \alpha \subseteq \fml{A} \land \nu \subseteq \set{\neg a \middle| a \in \fml{A}} }$ and $\fml{I}$ the set of consistent observations or \textbf{interpretations}, $\fml{I} = \set{z \in \fml{Z} \middle| \forall a \in \fml{A}~\abs{ \set{a, \neg a} \cap z} \leq 1}$.
-        
-        \item A \textbf{PASP program} is $P = C \land F \land R$ and the sets of atoms, observations and interpretations of program $P$ are denoted $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$.
+    Let $\fml{A}$ be a set of \textbf{atoms}, $\overline{\fml{A}} = \set{\neg a \middle| a \in \fml{A}}$ and $\fml{Z}$ the respective set of \textbf{observations},
+    $$\fml{Z} = \set{z = \alpha \cup \beta \middle| \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$ and $\fml{I}$ the set of consistent observations or \textbf{interpretations}, $$\fml{I} = \set{z \in \fml{Z} \middle| \forall a \in \fml{A}~\abs{ \set{a, \neg a} \cap z} \leq 1}.$$
+    \begin{itemize}    
+        \item A \textbf{PASP program} is $P = C \land F \land R$  where
+                
+        \begin{itemize}            
+            \item $C = C_P = \set{\alpha_i : a_i \middle| i = 1:n}$ is a set of probabilistic atomic choices, 
+            
+            \item $F = F_P$ is a set of (common) facts and 
+            
+            \item $R = R_P$ is a set of (common) rules.
+        \end{itemize}
-        \item $C = C_P = \set{\alpha_i : a_i \middle| i = 1:n}$ is a set of probabilistic atomic choices and $\delta C_P$ the set of associated disjunctions, $F = F_P$ is a set of (common) facts and $R = R_P$ is a set of (common) rules.
+        and the sets of atoms, observations and interpretations of program $P$ are denoted $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$
-        \item The \textbf{stable models} of $P = C \land F \land R$ are the stable models of $\delta P = \delta C + F + R$ and denoted $\fml{S} = \fml{S}_P$.
+        \item The \textbf{stable models} of $P$ are the stable models of $\delta P = \delta C + F + R$ and the respective set is denoted $\fml{S} = \fml{S}_P$.
     \end{itemize}
 \end{frame}
 \begin{frame}    
     \note{A model x has lower and upper "bounds".}
     \begin{itemize}
-        \item \textbf{Proposition.} Let $x\in\fml{I}$ be an interpretation and $\lset{x} = \set{s\in \fml{S} \middle| s \subseteq x}$ and $\uset{x} = \set{s\in \fml{S} \middle| x \subseteq s}$. Exactly one of the following cases takes place 
+        \item \textbf{Proposition.} Let $x\in\fml{I}$ be an interpretation.
+        \begin{itemize}
+            \item[Lower Models] $\lset{x} = \set{s\in \fml{S} \middle| s \subseteq x}$.
+            
+            \item[Upper Models] $\uset{x} = \set{s\in \fml{S} \middle| x \subseteq s}$.
+        \end{itemize}
+        \note{If $a$ is a lower model and $b$ an upper model, since stable models are minimal, must be $a = b = x$.}
+        
+        \item Exactly one of the following cases takes place: 
         \begin{enumerate}
             %
-            \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$. If $a \in \lset{x}$ and $b \in \uset{x}$ then $a \subseteq b$. Since stable models are minimal must be $a = b = x$ and $x$ is a stable model.
+            \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$ and $x$ is a stable model.
             %
             \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$. 
             %