rewriting 00_PASP for scholarship project

Francisco Coelho
1 parent eb584496
Showing 16 changed files with 603 additions and 349 deletions Show diff stats
README.md
SCASP-best-practices.pdf
msc/Empty File
task_01/proposal.md
text/00_PASP.aux
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 # Probabilistic ILP
+**Check** Conformal prediction.
+
 > Fonte: [Turning 30: New Ideas in Inductive Logic Programming](https://arxiv.org/abs/2002.11002)
 ## Introduction
@@ -62,4 +64,4 @@ Recursion; Predicate Invention; Higher order, ASP Hypotheses; Optimality; Prolog
 ## Applications
-### ELearning
 \ No newline at end of file
+### ELearning
@@ -6,7 +6,7 @@
 Answer Set Programming (ASP) is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. ASP is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints.
-The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts labelled with a probability $p$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 − p$.
+The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts labeled with a probability $p$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 − p$.
 Crucially, a joint distribution of atoms derived from an ASP specification can be used to _quantitatively measure the performance of that specification_ given data observed from the system it is intended to describe. Then, given competing specifications to describe a certain system, these performance measures can be applied in various optimization techniques in order to obtain one that best describes the target system.
@@ -36,7 +36,7 @@ A team of two **(or three?)** researchers and a graduate student, working over s
 - The formalization of the methods outlined above including the parameter estimation from observations and the joint distribution extending the probabilities of the stable models.
 - Application and evaluation of this approach to well-known problems, using available software tools, such as
   - Problems: **Assim, de momento, não me lembro de nenhum!**
-  - Software tools: [`s(casp)`](https://ciao-lang.org/playground/scasp.html), Potassco suit, _etc._
+  - Software tools: [`s(casp)`](https://ciao-lang.org/playground/scasp.html), [Potassco suit](https://potassco.org/), _etc._
 ## References
@@ -44,4 +44,4 @@ A team of two **(or three?)** researchers and a graduate student, working over s
 2. Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021)
 3. Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020)
 4. Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018) 
-6. Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan & Claypool Publishers (2013)
 \ No newline at end of file
+6. Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan & Claypool Publishers (2013)
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 \documentclass{beamer}
-
+%------------------------------------------
+\usecolortheme{rose}
+%------------------------------------------
+\useinnertheme{circles}
+%------------------------------------------
 \setbeamertemplate{navigation symbols}{}
-\setbeamertemplate{itemize items}[circle]
-
-
+%------------------------------------------
+\AtBeginSection{
+    \begin{frame}<beamer>\small
+        \tableofcontents[currentsection,subsectionstyle=shaded/shaded/hide]
+    \end{frame}
+}
+%------------------------------------------
+\AtBeginSubsection{	
+    \begin{frame}<beamer>\small
+        \tableofcontents[
+            currentsection,sectionstyle=shaded/shaded,
+            currentsubsection,subsectionstyle=show/shaded/hide]
+    \end{frame}
+}
+%------------------------------------------
 \usepackage[overridenote]{pdfpc}
 \usepackage{tikz}
@@ -28,142 +44,225 @@
 \newcommand{\deft}[1]{\textbf{#1}}
 \newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}}
 \newcommand{\ent}{\ensuremath{\lhd}}
+\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}}
 \newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}}
 \newcommand{\uset}[1]{\ensuremath{\left|{#1}\right>}}
 \newcommand{\lset}[1]{\ensuremath{\left<{#1}\right|}}
+\newcommand{\pr}[1]{\ensuremath{\mathrm{p}\at{#1}}}
 %
 %	Identificação deste documento
 %
 \title{Zugzwang}
-\subtitle{Stochastic Adventures in Inductive Logic Specifications}
+\subtitle{Stochastic Adventures in Inductive Logic}
 \author{Francisco Coelho}
 \institute[\texttt{fc@uevora.pt}]{
-  Departamento de Informática\\
-  Universidade de Évora
+  Departamento de Informática, Universidade de Évora\\
+  High Performance Computing Chair\\
+  NOVA-LINCS
 }
 \begin{document}
 %
 \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 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 \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.
+        % --------------------------------
+        \item $\co{x} = 1 - x$.
+        % --------------------------------
+        \item \textbf{Probabilistic Atomic Choice (PAC):} $\alpha :: a$ defines $a \lor \neg a$ and probabilities $\pr{a} = \alpha, \pr{\neg a} = \co{\alpha}$. 
+        % --------------------------------
+        \item $\delta a$ denotes $a \lor \neg a$ and $\delta\! \set{\alpha :: a, a \in \fml{A}} = \set{\delta a, a \in \fml{A}}$ for a set of atoms $\fml{A}$.
+        % --------------------------------
+        \item \textbf{Closed World Assumption:} $\naf x \models \neg x$.
+        % --------------------------------
+        % \item Probabilistic choices and sub-goals are independent.
+        % --------------------------------
     \end{itemize}
-    \note{**Subgoals are IID** means that ???}
+    % --------------------------------
 \end{frame}
-
+% ================================================================
 \begin{frame}{General Setting}
-    \note{Next, we consider the following **general setting**}
-    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}
-        
-        and the sets of atoms, observations and interpretations of program $P$ are denoted $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$
+        % --------------------------------
+        \item  \textbf{Atoms} $\fml{A}$,
+        $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$,
+        % --------------------------------
+        \item \textbf{Observations} $\fml{Z}$:
+        $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$
+        % --------------------------------
+        \item \textbf{Interpretations} or \textit{consistent observations} $\fml{I}$ :
+        $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$
+        % --------------------------------
+        \item \textit{PASP Problem}  or \textbf{Specification:} $P = C \land F \land R$ where
+        % --------------------------------
+            \begin{itemize}
+                % --------------------------------
+                \item $C = C_P = \cset{\alpha_i :: a_i }{ i \in 1:n \land  a_i \in \fml{A}}$ \textit{pacs}.
+                % --------------------------------
+                \item $F = F_P$ \textit{facts}.
+                % --------------------------------
+                \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 $P$.
+            \end{itemize}
+        % --------------------------------
+        \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$.
+        % --------------------------------
+    \end{itemize}
+    % --------------------------------
+\end{frame}
+% ================================================================
+\begin{frame}{Distribution Semantics}
+    % --------------------------------
+    \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 For $s\in\fml{S}$ let $\theta_s \subseteq s$ (unique \textit{total choice})
+        %\item Define $\fml{S}_\theta = \cset{s \in \fml{S}}{\theta \subset s}$.
+        % --------------------------------
-        \item The \textbf{stable models} of $P$ are the stable models of $\delta P = \delta C + F + R$ and the respective set is denoted $\fml{S} = \fml{S}_P$.
+        % --------------------------------
+        \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}
+        \end{equation}
+        % --------------------------------
     \end{itemize}
+    % --------------------------------
+    \begin{quote}
+        This is the \emph{Distribution Semantics} as set by Sato.
+    \end{quote}
 \end{frame}
-
-\begin{frame}    
-    \note{A model x has lower and upper "bounds".}
+% ================================================================
+\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}
+    % --------------------------------
+    \begin{quotation}
+        \textbf{But} there is a problem extending probability from total choices to stable models.
+    \end{quotation}
+    % --------------------------------
+\end{frame}
+% ================================================================
+\begin{frame}{The Disjunction Case}
+    % --------------------------------    
+    \begin{exampleblock}{Disjuntion Example}
+        The specification
+        % --------------------------------
+        $$
+        \begin{aligned}
+            0.3 :: a    &,          \cr 
+            b \lor c    &\larr a    .
+        \end{aligned}
+        $$
+        % --------------------------------
+        has three stable models,
+        % --------------------------------
+        $$
+        \begin{aligned}
+            s_1 &= \set{\neg a}, & s_2 &= \set{a, b},  & s_3 &= \set{a, c}.
+        \end{aligned}
+        $$
+    \end{exampleblock}
+    % --------------------------------
+    \begin{itemize}
+        % --------------------------------
+        \item\label{prop:unique.ext.tcsm}\textit{Any stable model contains exactly one total choice.~$\blacksquare$}
+        % --------------------------------
+        \item $\pr{\set{\neg a}} = 0.7$ is straightforward.
+        % --------------------------------
+        \item But, no \textit{unbiased} choice for $\alpha\in\intcc{0,1}$ in
+        $$
+        \begin{aligned}
+            \pr{\set{a, b}} &= 0.3 \alpha, \cr
+            \pr{\set{a, c}} &= 0.3 \co{\alpha}.
+        \end{aligned}
+        $$
+        % --------------------------------
+    \end{itemize}
+    % --------------------------------
+\end{frame}
+% ================================================================
+\section{Motivation}
+% ================================================================
+\begin{frame}{Specification, Data \& Evaluation}
+    % --------------------------------
+    Given some procedure to assign probabilities to observations from specifications and:
+    % --------------------------------
+    \begin{itemize}
+        % --------------------------------
+        \item $P$, a specification.
+        % --------------------------------
+        \item $p$, the distribution of observations from above.
+        % --------------------------------
+        \item $Z$, a dataset of observations.
+        % --------------------------------
+        \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.
+    % --------------------------------
+\end{frame}
+% ================================================================
+\section{Resolution}
+% ================================================================
+\begin{frame}{Bounds of Interpretations}    
+    % --------------------------------
     \begin{itemize}
-        \item \textbf{Proposition.} Let $x\in\fml{I}$ be an interpretation.
+        % --------------------------------
+        \item For $x\in\fml{I}$:
+        % --------------------------------
         \begin{itemize}
-            \item[Lower Models] $\lset{x} = \set{s\in \fml{S} \middle| s \subseteq x}$.
-            
-            \item[Upper Models] $\uset{x} = \set{s\in \fml{S} \middle| x \subseteq s}$.
+            % --------------------------------
+            \item \textbf{Lower Models:} $\lset{x} = \cset{s\in \fml{S} }{ s \subseteq x}$.
+            % --------------------------------
+            \item \textbf{Upper Models:} $\uset{x} = \cset{s\in \fml{S} }{ x \subseteq s}$.
+            % --------------------------------
         \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: 
+        % --------------------------------
+        \item\label{prop:lucases} \textbf{Proposition.} Stable models are \textit{minimal} so \textit{one} of the following cases takes place: 
+        % --------------------------------
         \begin{enumerate}
-            %
+            % --------------------------------
             \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$ and $x$ is a stable model.
-            %
+            % --------------------------------
             \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$. 
-            %
+            % --------------------------------
             \item\label{prop:lucases.c} $\lset{x} = \emptyset \land \uset{x} \neq \emptyset$.
-            %
+            % --------------------------------
             \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. 
+            % --------------------------------
         \end{enumerate}
+        % --------------------------------
     \end{itemize}
+    % --------------------------------
 \end{frame}
-
-\begin{frame}
-    \note{Total choice are key to define probability of a clause.}
-    \begin{itemize}
-        \item The probabilistic facts $C$ define a set $\Theta = \Theta_C$ of \textbf{total choices}, with $2^n$ elements, each one a set $\theta = \set{c_1, \ldots, c_n}$ where $c_i$ is either $a_i$ or $\neg a_i$.  
-        
-        \item For each stable model $s\in\fml{S}$ let $\theta_s$ be the unique \textbf{total choice} contained in $s$ and $\fml{S}_\theta \subseteq \fml{S}$ the set of stable models that contains $\theta$.
-        
-        \item Define 
-        \begin{equation}
-            p\at{\theta} = \prod_{a_i \in \theta}\alpha_i \prod_{\neg a_i \in \theta}\co{\alpha_i}.\label{eq:prob.tc}
-        \end{equation}
-    \end{itemize}
-\end{frame}
-
-
-\begin{frame}
-    \note{Relate stable models with Sato's probabilistic semantics}
-    \begin{quotation}
-        The problem we address is how to \textbf{assign probabilities to observations} given that a total choice might entail zero or many stable models \emph{i.e.} How to assign probabilities to the stable models of $\fml{S}_\theta$ when $\envert{\fml{S}_\theta} \not= 1$?
-    \end{quotation}
-\end{frame}
-
-\begin{frame}
-    \note{There are some problems}
-    
-    As it turns out, it is quite easy to come out with a program from which result no single probability distribution. For example
-    $$
-    \begin{aligned}
-        0.3:a,& \cr 
-        b \lor c \larr& a.
-    \end{aligned}
-    $$
-    has three stable models
-    $$
-    \begin{aligned}
-        s_1 &= \set{\neg a} \cr
-        s_2 &= \set{a, b} \cr
-        s_3 &= \set{a, c}
-    \end{aligned}
-    $$
-    and while $p\at{\set{\neg a}} = 0.7$ is quite natural, we have no further information to support the choice of a singular $\alpha\in\intcc{0,1}$ in the assignment 
-    $$
-    \begin{aligned}
-        p\at{\set{a, b}} &= 0.3 \alpha \cr
-        p\at{\set{a, c}} &= 0.3 \co{\alpha}
-    \end{aligned}
-    $$
-\end{frame}
-
+% ================================================================
 \begin{frame}
-    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 $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$. 
+    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 $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$. 
 \end{frame}
-
+% ================================================================
 \begin{frame}
     \begin{enumerate}
@@ -176,7 +275,7 @@
             %
             \item $z$ is an interpretation and $\lset{z} = \set{z} = \uset{x}$. Then $z = s$ is a stable model and \textbf{define}
             \begin{equation}
-                w_d\at{z} = w\at{s} = d\at{s} p\at{\theta_s}.\label{eq:prob.sm}
+                w_d\at{z} = w\at{s} = d\at{s} \pr{\theta_s}.\label{eq:prob.sm}
             \end{equation}
             %
             \item $z$ is an interpretation and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define} 
@@ -197,36 +296,36 @@
         %
         \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 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 $\top = \fml{A} \cup \set{\neg a \middle| a \in \fml{A}}$.
+        \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} observation $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
         %
         \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}
-            \item $P\at{E} \in \intcc{0, 1}$ for any $E \in F$.
-            \item $P\at{\Omega} = 1$.
-            \item if $E_1 \cap E_2 = \emptyset$ then $P\at{E_1 \cup E_2} = P\at{E_1} + P\at{E_2}$. 
+            \item $\pr{E} \in \intcc{0, 1}$ for any $E \in F$.
+            \item $\pr{\Omega} = 1$.
+            \item if $E_1 \cap E_2 = \emptyset$ then $\pr{E_1 \cup E_2} = \pr{E_1} + \pr{E_2}$. 
         \end{enumerate} 
         %
         \item In the following, assume that the stable models are iid.
         %
         \item Let the sample space $\Omega = \fml{Z}$ and the event space $F = \pset{\Omega}$. Define $Z = \sum_{\zeta\in\fml{Z}} w\at{\zeta}$ and
         \begin{equation}
-            P\at{z} = \frac{w\at{z}}{Z}, z \in \Omega \label{eq:def.prob}
+            \pr{z} = \frac{w\at{z}}{Z}, z \in \Omega \label{eq:def.prob}
         \end{equation}
         and
         \begin{equation}
-            P\at{E} = \sum_{x\in E} P\at{x}, E \subseteq \Omega. \label{eq:def.prob.event}
+            \pr{E} = \sum_{x\in E} \pr{x}, E \subseteq \Omega. \label{eq:def.prob.event}
         \end{equation}
         Now:
         \begin{enumerate}
             \item $P(E) \in \intcc{0,1}$ results directly from the definitions of $P$ and $w$.
-            \item $P\at{\Omega} = 1$ also results directly from the definitions.
+            \item $\pr{\Omega} = 1$ also results directly from the definitions.
             \item Consider two disjunct events $A, B \subset \Omega \land A \cap B = \emptyset$. Then 
             $$
             \begin{aligned}
-                P\at{A \cup B} &= \sum_{x \in A \cup B} P\at{x} \cr
-                    &= \sum_{x \in A} P\at{x} + \sum_{x \in B} P\at{x} - \sum_{x \in A \cap B} P\at{x}  \cr
-                    &= \sum_{x \in A} P\at{x} + \sum_{x \in B} P\at{x} &\text{because}~A\cap B = \emptyset \cr
-                    &= P\at{A} + P\at{B}.
+                \pr{A \cup B} &= \sum_{x \in A \cup B} \pr{x} \cr
+                    &= \sum_{x \in A} \pr{x} + \sum_{x \in B} \pr{x} - \sum_{x \in A \cap B} \pr{x}  \cr
+                    &= \sum_{x \in A} \pr{x} + \sum_{x \in B} \pr{x} &\text{because}~A\cap B = \emptyset \cr
+                    &= \pr{A} + \pr{B}.
             \end{aligned}
             $$
             \item So $\del{\Omega = \fml{Z}, F = \pset{\Omega}, P}$ is a probability space. {$\blacksquare$}
@@ -234,10 +333,11 @@
     \end{enumerate}
 \end{frame}
-
+% ================================================================
 \section{Cases \& Examples}
+% ================================================================
 \subsection{Programs with disjunctive heads}
-
+% ================================================================
 \begin{frame}
     Consider the program:
@@ -263,10 +363,9 @@
     \end{aligned}
     $$
 \end{frame}
-
-
+% ================================================================
 \begin{frame}
-    Suppose that we add an annotation $\alpha:a$, which entails $\co{\alpha}:\neg a$. This is enough to get $w\at{s_1} = \co{\alpha}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $\alpha$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get:
+    Suppose that we add an annotation $\alpha :: a$, which entails $\co{\alpha} :: \neg a$. This is enough to get $w\at{s_1} = \co{\alpha}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $\alpha$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get:
     $$
     \begin{cases}
         w\at{s_1 } = &\co{\alpha}\cr
@@ -275,7 +374,7 @@
     \end{cases}
     $$
 \end{frame}
-
+% ================================================================
 \begin{frame}
     Now consider all the interpretations for this program:
@@ -285,11 +384,11 @@
         %
         \node [draw, circle] (E) at (5.5,0) {$\emptyset$};
         %
-        \node [draw, circle] (a) at (2,2) {$a$};
-        \node [draw, circle] (b) at (3,2) {$b$};
+        \node [draw, circle] (a) at (3,2) {$a$};
+        \node [draw, circle] (b) at (2,2) {$b$};
         \node [draw, circle] (c) at (4,2) {$c$};
-        \node [fill=gray!50] (A) at (9,2) {$\co{a}$};
-        \node (B) at (8,2) {$\co{b}$};
+        \node [fill=gray!50] (A) at (8,2) {$\co{a}$};
+        \node (B) at (9,2) {$\co{b}$};
         \node (C) at (7,2) {$\co{c}$};
         %
         \node [fill=gray!50] (ab) at (0,4) {$ab$};
@@ -338,7 +437,7 @@
     \end{tikzpicture}    
     \end{center}
 \end{frame}
-
+% ================================================================
 \begin{frame}
     In this diagram:
@@ -378,7 +477,7 @@
         $$ 
         \item Now some statistics are possible. For example we get 
         $$
-        P\at{abc \mid \alpha = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9}
+        \pr{abc \mid \alpha = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9}
         $$. 
         \item This expression can be plotted for $d\in\intcc{0,1}$
@@ -386,10 +485,10 @@
             \includegraphics[height=15em]{Pabc_alpha03.pdf}
         \end{center} 
-        \item If a data set $E$ entails \emph{e.g.} $P\at{abc \mid E} = 0.0015$ we can numerically solve
+        \item If a data set $E$ entails \emph{e.g.} $\pr{abc \mid E} = 0.0015$ we can numerically solve
         $$
         \begin{aligned}
-            P\at{abc \mid \alpha = 0.3} &= P\at{abc \mid E} \cr
+            \pr{abc \mid \alpha = 0.3} &= \pr{abc \mid E} \cr
             \iff\cr
             \frac{0.09 d \del{d - 1}}{0.09 d^{2} - 0.69 d - 7.9} &= 0.0015
         \end{aligned}
@@ -397,10 +496,9 @@
         which has two solutions, $d \approx 0.15861$ or $d \approx 0.83138$.
     \end{itemize}
 \end{frame}
-
+% ================================================================
 \subsection{Non-stratified programs}
-
-
+% ================================================================
 \begin{frame}
     The following LP is non-stratified, because has a cycle with negated arcs:
     $$
@@ -429,13 +527,13 @@
     }.
     $$
 \end{frame}
-
+% ================================================================
 \begin{frame}
-    Looking into probabilistic interpretations of the program and/or its models, we define $\alpha = P\at{\Theta = \theta_1}\in\intcc{0, 1}$ and $P\at{\Theta = \theta_2} = \co{\alpha}$.
+    Looking into probabilistic interpretations of the program and/or its models, we define $\alpha = \pr{\Theta = \theta_1}\in\intcc{0, 1}$ and $\pr{\Theta = \theta_2} = \co{\alpha}$.
-    Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $P\at{ s_1 } = P\at{\Theta = \theta_1}  = \alpha$. However, there is no clear way to assign $P\at{s_2}, P\at{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly, 
-    $$P\at{s_2 \mid \Theta} + P\at{s_3 \mid \Theta} =
+    Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $\pr{ s_1 } = \pr{\Theta = \theta_1}  = \alpha$. However, there is no clear way to assign $\pr{s_2}, \pr{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly, 
+    $$\pr{s_2 \mid \Theta} + \pr{s_3 \mid \Theta} =
     \begin{cases}
         0 & \text{if}~\Theta = \theta_1\cr
         1 & \text{if}~\Theta = \theta_2
@@ -444,16 +542,15 @@
     but further assumptions are not supported \emph{a priori}. So let's \textbf{parameterize} the equation above,
     $$
     \begin{cases}
-        P\at{s_2 \mid \Theta = \theta_2} = &\beta \in \intcc{0, 1} \cr
-        P\at{s_3 \mid \Theta = \theta_2} = &\co{\beta},
+        \pr{s_2 \mid \Theta = \theta_2} = &\beta \in \intcc{0, 1} \cr
+        \pr{s_3 \mid \Theta = \theta_2} = &\co{\beta},
     \end{cases}
     $$
     in order to explicit our knowledge, or lack of, with numeric values and relations.
 \end{frame}
-
-
+% ================================================================
 \begin{frame}
-    Now we are able to define the \textbf{joint distribution} of the boolean random variables $A,B,C$, :
+    Now we are able to define the \textbf{joint distribution} of the boolean random variables $A,B,C$:
     $$
     \begin{array}{cc|l}
@@ -467,10 +564,9 @@
     $$
     where $\alpha, \beta\in\intcc{0,1}$.
 \end{frame}
-
+% ================================================================
 \section{Conclusions}
-
-
+% ================================================================
 \begin{frame}
     \begin{itemize}
         \item We can use the basics of probability theory and logic programming to assign explicit \emph{parameterized} probabilities to the (stable) models of a program.
@@ -479,10 +575,9 @@
         \item However, it is non-restrictive since \emph{no unusual assumptions are made}.
     \end{itemize}
 \end{frame}
-
+% ================================================================
 \section*{ASP \& related definitions}
-
-
+% ================================================================
 \begin{frame}
     \begin{itemize}
@@ -536,4 +631,5 @@
         \end{itemize}
     \end{itemize}
 \end{frame}
+% ================================================================
 \end{document}
 \ No newline at end of file
 \beamer@sectionintoc {1}{Introduction}{2}{0}{1}
-\beamer@sectionintoc {2}{Cases \& Examples}{10}{0}{2}
-\beamer@subsectionintoc {2}{1}{Programs with disjunctive heads}{10}{0}{2}
-\beamer@subsectionintoc {2}{2}{Non-stratified programs}{14}{0}{2}
-\beamer@sectionintoc {3}{Conclusions}{18}{0}{3}
+\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}
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1	-\beamer@slide {prop:lucases.a}{4}
2	-\beamer@slide {prop:lucases.b}{4}
3	-\beamer@slide {prop:lucases.c}{4}
4	-\beamer@slide {prop:lucases.d}{4}
5	\beamer@slide {eq:prob.tc}{5}	1	\beamer@slide {eq:prob.tc}{5}
6	-\beamer@slide {def:w.inconsistent}{9}
7	-\beamer@slide {eq:prob.sm}{9}
8	-\beamer@slide {def:w.disj}{9}
9	-\beamer@slide {def:w.conj}{9}
10	-\beamer@slide {def:w.empty}{9}
11	-\beamer@slide {eq:def.prob}{9}
12	-\beamer@slide {eq:def.prob.event}{9}	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}
1	\beamer@sectionintoc {1}{Introduction}{2}{0}{1}	1	\beamer@sectionintoc {1}{Introduction}{2}{0}{1}
2	-\beamer@sectionintoc {2}{Cases \& Examples}{10}{0}{2}
3	-\beamer@subsectionintoc {2}{1}{Programs with disjunctive heads}{10}{0}{2}
4	-\beamer@subsectionintoc {2}{2}{Non-stratified programs}{14}{0}{2}
5	-\beamer@sectionintoc {3}{Conclusions}{18}{0}{3}	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}