Completing the SBF example.

Francisco Coelho
1 parent 505cdc43
Showing 2 changed files with 498 additions and 455 deletions Show diff stats
text/paper_01/pre-paper.pdf
text/paper_01/pre-paper.tex
 \documentclass[a4paper, 12pt]{article}
 \usepackage[
-    bibstyle=numeric,
-    citestyle=numeric
+bibstyle=numeric,
+citestyle=numeric
 ]{biblatex} %Imports biblatex package
 \addbibresource{zugzwang.bib} %Import the bibliography file
 \usepackage[x11colors]{xcolor}
-%
+
 \usepackage{tikz}
 \tikzset{
-    event/.style={},
-    smodel/.style={fill=gray!25},
-    tchoice/.style={draw, circle},
-    indep/.style={draw, dashed},
-    proptc/.style = {-latex, dashed},
-    propsm/.style = {-latex, thick},
-    doubt/.style = {gray}
+event/.style={},
+smodel/.style={fill=gray!25},
+tchoice/.style={draw, circle},
+indep/.style={draw, dashed},
+proptc/.style = {-latex, dashed},
+propsm/.style = {-latex, thick},
+doubt/.style = {gray}
 }
 \usetikzlibrary{calc, positioning}
-%
+
 \usepackage{hyperref}
 \hypersetup{
-    colorlinks=true,
-    linkcolor=blue,
-    citecolor=blue,
+colorlinks=true,
+linkcolor=blue,
+citecolor=blue,
 }
-%
+
 \usepackage{commath}
 \usepackage{amsthm}
 \newtheorem{assumption}{Assumption}
@@ -66,12 +66,14 @@
 \newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}}
 \newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}}
 \newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}}
-\newcommand{\rep}[2]{\left\langle #1 \middle| #2 \right\rangle}
+\newcommand{\rep}[2]{\ensuremath{\left\langle #1 \middle| #2 \right\rangle}}
 \newcommand{\inconsistent}{\bot}
 \newcommand{\given}{\ensuremath{~\middle|~}}
 \newcommand{\emptyevent}{\ensuremath{\vartriangle}}
 \newcommand{\indepclass}{\ensuremath{\Diamond}}
 \newcommand{\probfact}[2]{\ensuremath{#1\!::\!#2}}
+\newcommand{\tcgen}[1]{\ensuremath{\widehat{#1}}}
+\newcommand{\lfrac}[2]{\ensuremath{{#1}/{#2}}}
 \newcommand{\todo}[1]{{\color{red!50!black}(\emph{#1})}}
 \newcommand{\remark}[2]{\uwave{#1}~{\color{green!40!black}(\emph{#2})}}
@@ -79,7 +81,7 @@
 \newcommand{\franc}[1]{{\color{orange!60!black}#1}}
 \newcommand{\bruno}{\color{red!60!blue}}
 %
-%   ACRONYMS
+%   Acronyms
 %
 \acrodef{BK}[BK]{background knowledge}
 \acrodef{ASP}[ASP]{answer set program}
@@ -90,34 +92,22 @@
 \acrodef{SM}[SM]{stable model}
 \acrodef{SC}[SC]{stable core}
 \acrodef{KL}[KL]{Kullback-Leibler}
-%
-%
-%
+
 \title{Zugzwang\\\emph{Logic and Artificial Intelligence}\\{\bruno Why this title?}}
+
 \author{
-    \begin{tabular}{ccc}
-        Francisco Coelho
-        \footnote{Universidade de Évora}
-        & Bruno Dinis
-        \footnote{Universidade de Évora}
-        & Salvador Abreu
-        \footnote{Universidade de Évora}
-        \\
-        \texttt{fc@uevora.pt}
-        & \texttt{bruno.dinis@uevora.pt}
-        & \texttt{spa@uevora.pt}
-        % \\
-        % \begin{minipage}{0.3\textwidth}\centering
-        %     Universidade de Évora and NOVA\textbf{LINCS}
-        % \end{minipage}
-        % & 
-        % \begin{minipage}{0.3\textwidth}\centering
-        %     Universidade de Évora
-        % \end{minipage}
-        % & \begin{minipage}{0.3\textwidth}\centering
-        %     Universidade de Évora and NOVA\textbf{LINCS}
-        % \end{minipage}
-    \end{tabular}    
+\begin{tabular}{ccc}
+    Francisco Coelho
+    \footnote{Universidade de Évora}
+    & Bruno Dinis
+    \footnote{Universidade de Évora}
+    & Salvador Abreu
+    \footnote{Universidade de Évora}
+    \\
+    \texttt{fc@uevora.pt}
+    & \texttt{bruno.dinis@uevora.pt}
+    & \texttt{spa@uevora.pt}
+\end{tabular}    
 }
 \begin{document}
@@ -133,10 +123,9 @@
 \section{Introduction and Motivation}
-
 \todo{Define and/or give references to all necessary concepts used in the paper}
-
 \todo{state of the art; references}
+
 \Acf{ASP} is a logic programming paradigm based on the \ac{SM} semantics of \acp{NP} that can be implemented using the latest advances in SAT solving technology. Unlike ProLog, \ac{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.
 \todo{references}
@@ -147,9 +136,6 @@ The \ac{DS} is a key approach to extend logical representations with probabilist
     \label{eq:prob.total.choice}
 \end{equation}
-% \todo{Insert simple example?}
-
-
 Our goal is to extend this probability, from \acp{TC}, to cover the \emph{specification} domain. We use the term ``specification'' as set of rules and facts, plain and probabilistic, to decouple it from any computational semantics, implied, at least implicitly, by the term ``program''. We can foresee at least two key applications of this extended probability:
 \begin{enumerate}
@@ -157,16 +143,13 @@ Our goal is to extend this probability, from \acp{TC}, to cover the \emph{specif
     \item Also, given a dataset and a divergence measure, the specification can be scored (by the divergence w.r.t.\ the \emph{empiric} distribution of the dataset), and weighted or sorted amongst other specifications. These are key ingredients in algorithms searching, for example, optimal specifications of a dataset. 
 \end{enumerate}
-%
-%\todo{Outline/Explain our idea, further developed in \cref{sec:extending.probalilities}}
-%
 Our idea to extend probabilities starts with the stance that a specification describes an \emph{observable system} and that observed events must be related with the \acp{SM} of that specification. From here, probabilities must be extended from \aclp{TC} to \acp{SM} and then from \acp{SM} to any event. 
 Extending probability from \acp{TC} to \acp{SM} faces a critical problem, illustrated by the  example in \cref{sec:example.1}, concerning situations where multiple \acp{SM}, $ab$ and $ac$, result from a single \ac{TC}, $a$, but there is not enough information (in the specification) to assign a single probability to each \ac{SM}. We propose to address this issue by using algebraic variables to describe that lack of information and then estimate the value of those variables from empirical data.
 In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model uncertainty) is considered as a lack of knowledge about the underlying model, that may be mitigated via further observations. This seems to presuppose a Bayesian approach to imperfect knowledge in the sense that having further observations allows to improve/correct the model. Indeed, the approach in that work uses Beta distributions in order to be able to learn the full distribution. This approach seems to be specially fitted to being able to tell when some probability lies beneath some given value. \todo{Our approach seems to be similar in spirit. If so, we should mention this in the introduction.}
-\todo{cite \cite{sympy} \franc{--- why? but cite \cite{cozman2020joy} and relate with our work.}}
+\todo{cite \cite{sympy} \franc{--- why here? but cite \cite{cozman2020joy} and relate with our work.}}
 \todo{Discuss the least informed strategy and the corolary that \aclp{SM} should be conditionally independent on the \acl{TC}.}
@@ -177,21 +160,24 @@ In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model 
 \todo{Write an introduction to the section}
 \begin{example}\label{running.example}
-Consider the following specification
-\begin{equation}
+    Consider the following specification
+    
+    \begin{equation}
+        \begin{aligned}
+            \probfact{0.3}{a}&,\cr
+            b \vee c& \leftarrow a.
+        \end{aligned}
+        \label{eq:example.1}
+    \end{equation}
+    
+    This specification has three \aclp{SM}, $\co{a}, ab$ and $ac$ (see \cref{fig:running.example}). While it is straightforward to set $P(\co{a})=0.7$, there is no further information to assign values to $P(ab)$ and $P(ac)$. Assuming that the \acfp{SM} are (probabilistically) independent, we can use a parameter $\theta$ such that
+
+    $$
     \begin{aligned}
-        \probfact{0.3}{a}&,\cr
-        b \vee c& \leftarrow a.
+        P(ab) &= 0.3 \theta,\cr
+        P(ac) &= 0.3 (1 - \theta).
     \end{aligned}
-    \label{eq:example.1}
-\end{equation}
-This specification has three stable models, $\co{a}, ab$ and $ac$ (see \cref{fig:running.example}). While it is straightforward to set $P(\co{a})=0.7$, there is no further information to assign values to $P(ab)$ and $P(ac)$. Assuming that the \acfp{SM} are (probabilistically) independent, we can use a parameter $\theta$ such that
-$$
-\begin{aligned}
-P(ab) &= 0.3 \theta,\cr
-P(ac) &= 0.3 (1 - \theta).
-\end{aligned}
-$$
+    $$
 \end{example} 
 While uncertainty is inherent to the specification it can be mitigated with the help of a dataset: the parameter $\theta$ can be estimated from a empirical distribution \todo{or we can have a distribution of $\theta$}. \todo{point to examples of this in following sections.}
@@ -199,11 +185,17 @@ While uncertainty is inherent to the specification it can be mitigated with the 
 In summary, if an \ac{ASP} specification is intended to describe some observable system then:
 \begin{enumerate}
+    
     \item Observations can be used to estimate the value of the parameters (such as $\theta$ above and others entailed from further clauses).
-    \item \todo{What about the case where we already know a distribution of $\theta$?} 
+    
+    \item \todo{What about the case where we already know a distribution of $\theta$?}
+    
     \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the specification domain. 
+    
     \item This extended probability can then be related to the \emph{empirical distribution}, using a probability divergence, such as \ac{KL}; and the divergence value used as a \emph{performance} measure of the specification with respect to the observations.
+    
     \item If that specification is only but one of many possible candidates then that performance measure can be used, \emph{e.g.} as fitness, by algorithms searching (optimal) specifications of a dataset of observations.
+    
 \end{enumerate}
 \begin{quote}
@@ -212,13 +204,13 @@ In summary, if an \ac{ASP} specification is intended to describe some observable
 Currently, we are addressing the problem of extending a probability function (possibly using parameters such as $\theta$), defined on the \acp{SM} of a specification, to all the events of that specification. Of course, this extension must satisfy the Kolmogorov axioms of probability so that probabilistic reasoning is consistent with the \ac{ASP} specification. 
-The conditional independence of stable worlds asserts the least informed strategy that we discussed in the introduction and make explicit here:
+The conditional independence of stable worlds asserts the \remark{least informed strategy}{references?} that we discussed in the introduction and make explicit here:
 \begin{assumption}\label{assumption:smodels.independence}
-    \Acl{SM} are conditionally independent, given their total choices.
+    \Acl{SM} are conditionally independent, given their \aclp{TC} .
 \end{assumption}
-The stable models $ab, ac$ from \cref{running.example} result from the clause $b \vee c \leftarrow a$ and the total choice $a$. These formulas alone imposes no relation between $b$ and $c$ (given $a$), so none should be assumed. Dependence relations are further discussed in \cref{subsec:dependence}.
+The \aclp{SM}  $ab, ac$ from \cref{running.example} result from the clause $b \vee c \leftarrow a$ and the \acl{TC} $a$. These formulas alone imposes no relation between $b$ and $c$ (given $a$), so none should be assumed. Dependence relations are further discussed in \cref{subsec:dependence}.
 \section{Extending Probabilities}\label{sec:extending.probalilities}
@@ -232,6 +224,8 @@ The stable models $ab, ac$ from \cref{running.example} result from the clause $b
             \node[event, below = of ab] (b) {$b$};
             \node[event, below = of ac] (c) {$c$};
             \node[event, above right = of ab] (abc) {$abc$};
+            \node[event, above left = of ab] (abC) {$ab\co{c}$};
+            \node[event, above right = of ac] (aBc) {$a\co{b}c$};
             \node[indep, right = of ac] (bc) {$bc$};
             \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$};
             \node[event, above = of A] (Ac) {$\co{a}c$};
@@ -239,17 +233,20 @@ The stable models $ab, ac$ from \cref{running.example} result from the clause $b
             % ----
             \draw[doubt] (a) to[bend left] (ab);
             \draw[doubt] (a) to[bend right] (ac);
-
+            
             \draw[doubt] (ab) to[bend left] (abc);
-            \draw[doubt] (ac) to[bend right] (abc);
+            \draw[doubt] (ab) to[bend right] (abC);
+            \draw[doubt] (ac) to[bend right] (abc);
+            \draw[doubt] (ac) to[bend left] (aBc);
+            
             \draw[doubt] (A) to (Ac);
             \draw[doubt] (A) to (Abc);
-
+            
             \draw[doubt] (ab) to[bend right] (E);
             \draw[doubt] (ac) to[bend right] (E);
             \draw[doubt] (A) to[bend left] (E);
-
+            
             \draw[doubt] (ab) to (b);
             \draw[doubt] (ac) to (c);
             % \draw[doubt] (ab) to[bend left] (a);
@@ -260,8 +257,8 @@ The stable models $ab, ac$ from \cref{running.example} result from the clause $b
             \draw[doubt] (c) to[bend right] (Ac);
         \end{tikzpicture}
     \end{center}
-    \caption{Events related to the stable models of \cref{running.example}. The circle nodes are \aclp{TC} and shaded nodes are \aclp{SM}. The \emph{empty event}, with no literals, is denoted by $\emptyevent$. Notice that the event $bc$ is not related with any stable model.}
-    % \caption{Extending probabilities from total choice nodes to stable models and then to general events in a \emph{node-wise} process quickly leads to coherence problems concerning probability, with no clear systematic approach --- Instead, weight extension can be based in \emph{the relation an event has with the stable models}.{\bruno Why is this comment on the caption?}}
+    
+    \caption{Events related to the \aclp{SM}  of \cref{running.example}. The circle nodes are \aclp{TC} and shaded nodes are \aclp{SM}. The \emph{empty event}, with no literals, is denoted by $\emptyevent$. Notice that the event $bc$ is not related with any \acl{SM}.}
     \label{fig:running.example}
 \end{figure}
@@ -269,22 +266,124 @@ The stable models $ab, ac$ from \cref{running.example} result from the clause $b
 \note{$\emptyevent$ notation introduced in \cref{fig:running.example}.}
-The diagram in \cref{fig:running.example} illustrates the problem of extending probabilities from total choice nodes to stable models and then to general events in a \emph{node-wise} process. This quickly leads to coherence problems concerning probability, with no clear systematic approach --- Instead, weight extension can be based in the relation an event has with the stable models.
+The diagram in \cref{fig:running.example} illustrates the problem of extending probabilities from \acp{TC} nodes to \acp{SM} and then to general events in a \emph{node-wise} process. This quickly leads to \remark{coherence problems}{for example?} concerning probability, with no clear systematic approach --- Instead, weight extension can be based in the relation an event has with the \aclp{SM}.
 \subsection{An Equivalence Relation}\label{subsec:equivalence.relation}
-Given an ASP specification 
-% DONE: {\bruno This should be defined somewhere (maybe in the introduction).}
-\remark{{\bruno Introduce also the sets mentioned below}}{how?}
-, we consider the \emph{atoms} $a \in \fml{A}$ and \emph{literals}, $z \in \fml{L}$, \emph{events} $e \in \fml{E} \iff e \subseteq \fml{L}$ and \emph{worlds} $w \in \fml{W}$ (consistent events), \emph{total choices} $c \in \fml{C} \iff c = a \vee \neg a$ and \emph{stable models} $s \in \fml{S}\subset\fml{W}$. 
-
-% In a statistical setting, the outcomes are the literals $x$, $\neg x$ for each atom $x$, the events express a set of possible outcomes (including $\emptyset$, $\set{a, b}$, $\set{a, \neg a, b}$, \emph{etc.}), and worlds are events with no contradictions.
+\begin{figure}[t]
+    \begin{center}
+        \begin{tikzpicture}
+            \node[event] (E) {$\emptyevent$};
+            \node[tchoice, above left = of E] (a) {$a$};
+            \node[smodel, above left = of a] (ab) {$ab$};
+            \node[smodel, above right = of a] (ac) {$ac$};
+            \node[event, below = of ab] (b) {$b$};
+            \node[event, below = of ac] (c) {$c$};
+            \node[event, above right = of ab] (abc) {$abc$};
+            \node[event, above left = of ab] (abC) {$ab\co{c}$};
+            \node[event, above right = of ac] (aBc) {$a\co{b}c$};
+            \node[indep, right = of ac] (bc) {$bc$};
+            \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$};
+            \node[event, above = of A] (Ac) {$\co{a}c$};
+            \node[event, above right = of Ac] (Abc) {$\co{a}bc$};
+            % ----
+            \path[draw, rounded corners, fill=cyan, fill opacity=0.1] 
+                (ab.west) --
+                (ab.north west) --
+                %
+                (abC.south west) --
+                (abC.north west) --
+                (abC.north) --
+                %
+                (abc.north east) --
+                (abc.east) --
+                (abc.south east) --
+                %
+                (ab.north east) --
+                (ab.east) --
+                (ab.south east) --
+                %
+                (a.north east) --
+                %
+                (E.north east) --
+                (E.east) --
+                (E.south east) --
+                (E.south) --
+                (E.south west) --
+                %
+                (b.south west) --
+                %
+                (ab.west)
+            ;
+            % ----
+            \path[draw, rounded corners, fill=magenta, fill opacity=0.1] 
+                (ac.south west) --
+                (ac.west) --
+                (ac.north west) --
+                %
+                (abc.south west) --
+                (abc.west) --
+                (abc.north west) --
+                %
+                (aBc.north east) --
+                (aBc.east) --
+                (aBc.south east) --
+                %
+                (ac.north east) --
+                %
+                (c.east) --
+                %
+                (E.east) --
+                (E.south east) --
+                (E.south) --
+                (E.south west) --
+                %
+                (a.south west) --
+                (a.west) --
+                (a.north west) --
+                (a.north) --
+                %
+                (ac.south west)
+            ;
+            % ----
+            \path[draw, rounded corners, fill=yellow, fill opacity=0.1]
+                % (A.north west) --
+                %
+                (Ac.north west) --
+                %
+                (Abc.north west) --
+                (Abc.north) --
+                (Abc.north east) --
+                (Abc.south east) --
+                %
+                % (Ac.north east) --
+                % (Ac.east) --
+                %
+                % (A.east) --
+                (A.south east) --
+                %
+                (E.south east) --
+                (E.south) --
+                (E.south west) --
+                (E.west) --
+                (E.north west) --
+                %
+                (Ac.north west)
+            ;
+        \end{tikzpicture}
+    \end{center}
+    
+    \caption{Classes (of consistent events) related to the \aclp{SM} of \cref{running.example} are defined through inclusions. \todo{write the caption}}
+    \label{fig:running.example.classes}
+\end{figure}
+Given an ASP specification, 
+\remark{{\bruno Introduce also the sets mentioned below}}{how?}
+ we consider the \emph{atoms} $a \in \fml{A}$ and \emph{literals}, $z \in \fml{L}$, \emph{events} $e \in \fml{E} \iff e \subseteq \fml{L}$ and \emph{worlds} $w \in \fml{W}$ (consistent events), \emph{\aclp{TC} } $c \in \fml{C} \iff c = a \vee \neg a$ and \emph{\aclp{SM} } $s \in \fml{S}\subset\fml{W}$. 
-Our path starts with a perspective of stable models as playing a role similar to \emph{prime} factors. 
-The stable models of a specification are the irreducible events entailed from that specification and any event must be \replace{interpreted}{considered} under its relation with the stable models.
+Our path starts with a perspective of \aclp{SM}  as playing a role similar to \emph{prime} factors.  The \aclp{SM}  of a specification are the irreducible events entailed from that specification and any event must be \replace{interpreted}{considered} under its relation with the \aclp{SM}.
-%\remark{\todo{Introduce a structure with worlds, events, and stable models}}{seems irrelevant}
+%\remark{\todo{Introduce a structure with worlds, events, and \aclp{SM} }}{seems irrelevant}
 This focus on the \acp{SM} leads to the following definition:
 \begin{definition}\label{def:stable.structure}
@@ -292,108 +391,60 @@ This focus on the \acp{SM} leads to the following definition:
 \end{definition}
-\todo{expand this text to explain how the stable models form the basis of the equivalence relation}. %This \replace{stance}{} leads to definition \ref{def:rel.events}:
+\todo{expand this text to explain how the \aclp{SM}  form the basis of the equivalence relation}. %This \replace{stance}{} leads to definition \ref{def:rel.events}:
 \begin{definition}\label{def:stable.core}
-    The \emph{\ac{SC}} of the event $e\in \fml{E}$ is
-
-    % \begin{equation}
-    %     \uset{e} = \set{s \in \fml{S} \given e \subseteq s},\label{eq:uset}
-    % \end{equation}
-    % \begin{equation}
-    %     \lset{e} = \set{s \in \fml{S} \given e \supseteq s}, \label{eq:lset}
-    % \end{equation}
-    % \begin{equation}
-    %     \stablecore{e} = \uset{e} \cup \lset{e} \label{def:stable.core}
-    % \end{equation}
+    The \emph{\ac{SC}} of the event $e\in \fml{E}$ is    
     \begin{equation}
         \stablecore{e} := \set{s \in \fml{S} \given s \subseteq e \vee e \subseteq s} \label{eq:stable.core}
     \end{equation}
-
-    \end{definition}
-We now define an equivalence relation, $\sim$, so that two events are related if either both are inconsistent or both are consistent with the same stable core.
+\end{definition}
+
+We now define an equivalence relation, $\sim$, so that two events are related if either both are inconsistent or both are consistent with the same \acl{SC}.
 \begin{definition}\label{def:equiv.rel}
-For a given specification, let $u, v \in \fml{E}$. The equivalence relation $\sim$ is defined by
+    For a given specification, let $u, v \in \fml{E}$. The equivalence relation $\sim$ is defined by
     \begin{equation}
-        u \sim v :\iff u,v \not\in\fml{W} \vee \del{u,v \in \fml{W} \wedge \stablecore{u} = \stablecore{v}}.\label{eq:equiv.rel}
+        u \sim v :\!\iff u,v \not\in\fml{W} \vee \del{u,v \in \fml{W} \wedge \stablecore{u} = \stablecore{v}}.\label{eq:equiv.rel}
     \end{equation}
 \end{definition}
-Observe that the minimality of stable models implies that, in \cref{def:stable.core}, either $e$ is a stable model or one of $s \subseteq e, e \subseteq s$ is never true. 
-%
-% \begin{definition}\label{def:smodel.events}
-%     For $\set{s_1, \ldots, s_n} \subseteq \fml{S}$ define
-%     \begin{equation}
-%         \lclass{s_1, \ldots, s_n} = \set{e\in \fml{E}\setminus \fml{S} \given \uset{e} = \set{s_1, \ldots, s_n}},
-%         \label{eq:smodel.lclass}
-%     \end{equation}
-%     \begin{equation}
-%         \uclass{s_1, \ldots, s_n} = \set{e\in \fml{E}\setminus \fml{S} \given \lset{e} = \set{s_1, \ldots, s_n}}
-%         \label{eq:smodel.uclass}
-%     \end{equation}
-%     and
-%     \begin{equation}
-%         \smclass{s_1, \ldots, s_n} = \set{s_1, \ldots, s_n}
-%         \label{eq:smodel.smclass}
-%     \end{equation}
-% \end{definition}
-%
-This relation defines a partition of the events space, where each class holds a unique relation with the stable models. In particular, we denote each class by:
+Observe that the minimality of \aclp{SM}  implies that, in \cref{def:stable.core}, either $e$ is a \acl{SM} or one of $s \subseteq e, e \subseteq s$ is never true. This relation defines a partition of the events space, where each class holds a unique relation with the \aclp{SM}. In particular, we denote each class by:
 \begin{equation}
     \class{e} =
     \begin{cases}
-        \inconsistent := \fml{E} \setminus \fml{W}   &\text{if~} e \not\in \fml{E} \setminus \fml{W}, \\
-        \set{u \in \fml{W} \given \stablecore{u} = \stablecore{e}}      &\text{if~} e \in \fml{W}, \\
-        % \lclass{\uset{e}}      &\text{if~} \uset{e} \not= \emptyset, \\
-        % \uclass{\lset{e}}      &\text{otherwise}.
+        \inconsistent := \fml{E} \setminus \fml{W}   
+            &\text{if~} e \in \fml{E} \setminus \fml{W}, \\
+        \set{u \in \fml{W} \given \stablecore{u} = \stablecore{e}}
+            &\text{if~} e \in \fml{W}, 
     \end{cases}\label{eq:event.class}
 \end{equation}
-% The stable core defines a \emph{canonical} representative of each class:
-% \begin{theorem}
-%     Let $e\in\fml{E}$ and $\stablecore{e} = \set{s_1, \ldots, s_n} \subseteq \fml{S}$. Then
-%     \begin{equation}
-%         \class{e} = \class{s_1 \cup \cdots \cup s_n}.
-%     \end{equation}
-%     We simplify the notation with $\class{s_1, \ldots, s_n} := \class{s_1 \cup \cdots \cup s_n}$. 
-%     \todo{This only works for consistent $s_1, \ldots, s_n$: $\set{\emptyevent} = \class{\co{a}, ab, ac} \not= \class{a\co{a}bc} = \inconsistent$.}
-% \end{theorem}
-% \begin{proof}
-% \todo{tbd}
-% \end{proof}
-
-The subsets of the stable models, together with $\inconsistent$, form a set of representatives. Consider again Example~\ref{running.example}. As previously mentioned, the stable models are $\fml{S} = \co{a}, ab, ac$ so the quotient set of this relation is:
+The subsets of the \aclp{SM}, together with $\inconsistent$, form a set of representatives. Consider again Example~\ref{running.example}. As previously mentioned, the \aclp{SM}  are $\fml{S} = \co{a}, ab, ac$ so the quotient set of this relation is:
 \begin{equation}
     \class{\fml{E}} = \set{
-        \inconsistent,
-        \indepclass,
-        \class{\co{a}},
-        \class{ab},
-        \class{ac},
-        \class{\co{a}, ab},
-        \class{\co{a}, ac},
-        \class{ab, ac},
-        \class{\co{a}, ab, ac}
-        }
-% \begin{aligned}
-%     & \inconsistent, \emptyset, \\
-%     & \stablecore{\co{a}}, \stablecore{ab}, \stablecore{ac}, \\
-%     & \stablecore{\co{a}, ab}, \stablecore{\co{a}, ac},  \stablecore{ab, ac}, \\
-%     & \stablecore{\co{a}, ab, ac}.
-% \end{aligned}
+    \inconsistent,
+    \indepclass,
+    \class{\co{a}},
+    \class{ab},
+    \class{ac},
+    \class{\co{a}, ab},
+    \class{\co{a}, ac},
+    \class{ab, ac},
+    \class{\co{a}, ab, ac}
+    }
 \end{equation}
 where $\indepclass$ denotes both the class of \emph{independent} events $e$ such that $\stablecore{e} = \emptyset$ and its core (which is the emptyset). We have:
 \begin{equation*}
     \begin{array}{l|lr}
-        \text{\textbf{Core}} 
-        & \text{\textbf{Class}} 
-        & \text{\textbf{Size}}\\
+        \text{\textbf{Core}}, \stablecore{e}
+        & \text{\textbf{Class}}, \class{e}
+        & \text{\textbf{Size}}, \# \class{e}\\
         \hline 
         %
         \inconsistent
-        & \text{inconsistent events}
+        & a\co{a}, \ldots
         & 37
         \\
         %
@@ -446,23 +497,25 @@ where $\indepclass$ denotes both the class of \emph{independent} events $e$ such
 \end{equation*}
 \begin{itemize}
-    \item Since all events within an equivalence class are in relation with a specific set of stable models, \emph{weights, including probability, should be constant within classes}:
+    \item Since all events within an equivalence class are in relation with a specific set of \aclp{SM}, \emph{weights, including probability, should be constant within classes}:
     \[
-    \forall u\in \class{e} \left(\pr{u} = \pr{e} \right).
+    \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right).
     \]
-    \item So, instead of dealing with $64 = 2^6$ events, we need only to \replace{handle}{handle, at most,} $9 = 2^3 + 1$ classes, well defined in terms of combinations of the stable models. \remark{This seems fine}{but, in general, we will get \emph{much more} stable models than literals, so the number of these classes is much larger than of events!} {\bruno Nevertheless, the equivalence classes alow us to propagate probabilities from total choices to events, as explained in the next subsection.}
+    \item So, instead of dealing with $64 = 2^6$ events, we consider the $9 = 2^3 + 1$ classes, well defined in terms of combinations of the \aclp{SM}. In general, we have \emph{much more} \aclp{SM} than literals. Nevertheless, the equivalence classes allow us to propagate probabilities from \aclp{TC}  to events, as explained in the next subsection.
     % \item The extended probability \emph{events} are the \emph{classes}.
 \end{itemize}
+
+
 \subsection{From Total Choices to Events}\label{subsec:from.tchoices.to.events}
 \todo{Check adaptation} Our path to set a probability measure on $\fml{E}$ has two phases: 
-\begin{itemize}
-    \item Extending the probabilities, \emph{as weights}, of the total choices to events.
+\begin{enumerate}
+    \item Extending the probabilities, \emph{as weights}, from the \aclp{TC} to events.
     \item Normalization of the weights.
-\end{itemize}
+\end{enumerate}
-The ``extension'' phase, traced by equations (\ref{eq:prob.total.choice}) and (\ref{eq:weight.tchoice} --- \ref{eq:weight.events}), starts with the weight (probability) of total choices, $\pw{c} = \pr{C = c}$, expands it to stable models, $\pw{s}$, and then, within the equivalence relation from Equation \eqref{eq:equiv.rel}, to (general) events, $\pw{e}$, including  (consistent) worlds.
+The ``extension'' phase, traced by equations (\ref{eq:prob.total.choice}) and (\ref{eq:weight.tchoice} --- \ref{eq:weight.events}), starts with the weight (probability) of \aclp{TC}, $\pw{c} = \pr{C = c}$, expands it to \aclp{SM}, $\pw{s}$, and then, within the equivalence relation from \cref{eq:equiv.rel}, to (general) events, $\pw{e}$, including  (consistent) worlds.
 \begin{description}
     %
@@ -472,72 +525,78 @@ The ``extension&#39;&#39; phase, traced by equations (\ref{eq:prob.total.choice}) and (\
         \label{eq:weight.tchoice}
     \end{equation}
     %
-    \item[Stable Models.] Each total choice $c$, together with the rules and the other facts of a specification, defines a set of stable models associated with that choice, that we denote by $S_c$.
+    \item[Stable Models.] Each \acl{TC} $c$, together with the rules and the other facts of a specification, defines a set of \aclp{SM}  associated with that choice, that \remark{we denote by $\tcgen{c}$}{put this in the introduction, where core concepts are presented}.
-    Given a stable model $s \in \fml{S}$, a total choice $c$, and variables/values $\theta_{s,c} \in \intcc{0, 1}$,
+    Given a \acl{SM} $s \in \fml{S}$, a \acl{TC} $c$, and variables/values $\theta_{s,c} \in \intcc{0, 1}$,
     \begin{equation}
         \pw{s, c} := \begin{cases}
-            \theta_{s,c}\remark{\pw{c}}{\text{maybe not!}} & \text{if~} s \in S_c\cr
+            \theta_{s,c} & \text{if~} s \in \tcgen{c}\cr
             0&\text{otherwise}
         \end{cases}
         \label{eq:weight.stablemodel}
     \end{equation}
-    such that $\sum_{s\in S_c} \theta_{s,c} = 1$.
+    such that $\sum_{s\in \tcgen{c}} \theta_{s,c} = 1$.
     %
-    \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of  stable models.
-    \begin{itemize}
-        \item \textbf{Inconsistent Class.} The inconsistent class contains events that are logically inconsistent. Since these events should never be observed:
+    \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of \aclp{SM}.
+    \begin{description}
+        \item[Inconsistent Class.] The inconsistent class contains events that are logically inconsistent, thus should never be observed:
         \begin{equation}
             \pw{\inconsistent, c} := 0.
             \label{eq:weight.class.inconsistent}
         \end{equation}
-        \item \textbf{Independent Class.} A world that neither contains nor is contained in a stable model describes a case that, according to the specification, should never be observed. So the respective weight is set to zero:
+        \item[Independent Class.] A world that neither contains nor is contained in a \acl{SM} describes a case that, according to the specification, shouldn't exist. So the respective weight is set to zero:
         \begin{equation}
             \pw{\indepclass, c} := 0.
             \label{eq:weight.class.independent}
         \end{equation}
-        \item \textbf{Other Classes.} The extension must be constant within a class, its value should result from the elements in the stable core, and respect the assumption \ref{assumption:smodels.independence} (stable models independence):
+        \item[Other Classes.] The extension must be constant within a class, its value should result from the elements in the \acl{SC}, and respect the assumption \ref{assumption:smodels.independence} (\aclp{SM}  independence):
         \begin{equation}
-            \pw{\class{e}, c} := \prod_{k=1}^{n}\pw{s_k, c},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}.
+            \pw{\class{e}, c} := \sum_{k=1}^{n}\pw{s_k, c},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}.
             \label{eq:weight.class.other}
         \end{equation}
-    \end{itemize} 
+        and 
+        \begin{equation}
+            \pw{\class{e}} := \sum_{c \in \fml{C}} \pw{\class{e}, c}\pw{c}.
+            \label{eq:weight.class.unconditional}
+        \end{equation}
+        \remark{}{Changed from $\prod$ to $\sum$ to represent ``either'' instead of ``both'' since the later is not consistent with the ``only one stable model at a time'' assumption.}
+    \end{description} 
     %
-    \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its stable core, $\stablecore{e}$. So, we set:
+    \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, we set:
     \begin{equation}
-        \pw{e, c} := \pw{\class{e}, c}.
+        \pw{e, c} := \frac{\pw{\class{e}, c}}{\# \class{e}} .
         \label{eq:weight.events}
     \end{equation}
     and 
     \begin{equation}
-        \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}.
+        \pw{e} := \sum_{c\in\fml{C}} \pw{e, c} \pw{c}.
         \label{eq:weight.events.unconditional}
     \end{equation}
-    \remark{instead of that equation}{if we set $\pw{s,c} := \theta_{s,c}$ in equation \eqref{eq:weight.stablemodel} here we do: 
-        $$
-        \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\pw{c}.
-        $$
-    By the way, this is the \emph{marginalization + bayes theorem} in statistics:
-    $$
-    P(A) = \sum_b P(A | B=b)P(B=b) 
-    $$
-    }
+    % \remark{instead of that equation}{if we set $\pw{s,c} := \theta_{s,c}$ in equation \eqref{eq:weight.stablemodel} here we do: 
+    %     $$
+    %     \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\pw{c}.
+    %     $$
+    % By the way, this is the \emph{marginalization + bayes theorem} in statistics:
+    % $$
+    % P(A) = \sum_b P(A | B=b)P(B=b) 
+    % $$
+    % }
 \end{description}
 % PARAMETERS FOR UNCERTAINTY
 \begin{itemize}
-    \item \todo{Remark that $\pw{\inconsistent, c} = 0$ is independent of the total choice.}
-    \item \todo{Remark the example $bc$ for equation \ref{eq:weight.class.independent}.}
+    \item \todo{Remark that $\pw{\inconsistent, c} = 0$ is independent of the \acl{TC}.}
+    \item Consider the event $bc$. Since $\class{bc} = \indepclass$, from \cref{eq:weight.class.independent} we get $\mu\at{bc} = 0$.
     \item \todo{Remark that equation \eqref{eq:weight.events.unconditional}, together with observations, can be used to learn about the \emph{initial} probabilities of the atoms, in the specification.}
 \end{itemize}
-The $\theta_{s,c}$ parameters in equation \eqref{eq:weight.stablemodel} express the \emph{specification's} lack of knowledge about the weight assignment, when a single total choice entails more than one stable model. In that case, how to distribute the respective weights? Our proposal to address this problem consists in assigning an unknown weight, $\theta_{s,c}$, conditional on the total choice, $c$, to each stable model $s$. This approach allows the expression of an unknown quantity and future estimation, given observed data.
+The $\theta_{s,c}$ parameters in equation \eqref{eq:weight.stablemodel} express the \emph{specification's} lack of knowledge about the weight assignment, when a single \acl{TC} entails more than one \acl{SM}. In that case, how to distribute the respective weights? Our proposal to address this problem consists in assigning an unknown weight, $\theta_{s,c}$, conditional on the \acl{TC}, $c$, to each \acl{SM} $s$. This approach allows the expression of an unknown quantity and future estimation, given observed data.
 % SUPERSET
-Equation \eqref{eq:weight.class.other} results from conditional independence of stable models. 
+Equation \eqref{eq:weight.class.other} results from conditional independence of \aclp{SM}. 
+
-    
 \section{Developed Examples}
 \subsection{The SBF Example}
@@ -546,180 +605,164 @@ We continue with the specification from Equation \eqref{eq:example.1}.
 \begin{description}
     %
-    \item[Total Choices.] The total choices, and respective stable models, are
+    \item[\Aclp{TC}.] The \aclp{TC}, and respective \aclp{SM}, are
     \begin{center}
-        \begin{tabular}{llr}
-            \textbf{Total Choice} & \textbf{Stable Models} & \textbf{$\pw{\cdot}$}\\
+        \begin{tabular}{ll|r}
+            \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pw{c}$}\\
             \hline 
             $a$ & $ab, ac$ & $0.3$\\
             $\co{a} = \neg a$ & $\co{a}$ & $\co{0.3} = 0.7$
         \end{tabular}
     \end{center}
     %
-    \item[Stable Models.] \todo{Enter the $\theta$ parameters.}
-    \begin{equation*}
-        \begin{array}{ll|r}
-            \text{\textbf{Stable Model}}
-            & \text{\textbf{Total Choice}} 
-            & \pw{s,c}
-            \\
-            \hline
-            \co{a}
-            & \co{a}
-            & 1.0
-            \\
-            ab
-            & a
-            & \theta
-            \\
-            ac
-            & a
-            & \co{\theta}
-        \end{array}
-    \end{equation*}
-    \item[Classes.] Following the definitions in \cref{eq:stable.core,eq:equiv.rel,eq:event.class} and in \cref{eq:weight.class.inconsistent,eq:weight.class.independent,eq:weight.class.other} we get the following quotient set, and weights:
+    \item[\Aclp{SM}.] The $\theta_{s,c}$ parameters in this example are
+    $$
+    \theta_{ab,\co{a}} = \theta_{ac,\co{a}} = \theta_{\co{a}, a} = 0
+    $$
+    and
+    $$
+    \theta_{\co{a}, \co{a}} = 1, \theta_{ab, a} = \theta, \theta_{ac, a} = \co{\theta}
+    $$
+    with $\theta \in \intcc{0, 1}$.
+    \item[Classes.] Following the definitions in \cref{eq:stable.core,eq:equiv.rel,eq:event.class} and in \cref{eq:weight.class.inconsistent,eq:weight.class.independent,eq:weight.class.other} we get the following quotient set (ignoring $\inconsistent$ and $\indepclass$), and weights:
     \begin{equation*}
-        \begin{array}{l|rrr|r}
-            \text{\textbf{Core}} 
-            & \pw{c} 
-            & \pw{s,c} 
-            & \pw{\class{e}, c}
-            & \pw{e}
+        \begin{array}{l|ll|r}
+            \stablecore{e}
+            & \pw{s_k, c= \co{a}}
+            & \pw{s_k, c= a}
+            & \pw{\class{e}}=\sum_{c}\pw{\class{e},c}\pw{c}
             \\
             \hline 
-            %
-            \inconsistent
-            &
-            & 0.0
-            & 0.0
-            & 0.0
-            \\
-            %
-            \indepclass
-            &
-            & 0.0
-            & 0.0
-            & 0.0
-            \\
-            %
             \co{a} 
-            & 0.7 
-            & 1.0
-            & 1.0
+            & 1
+            &
             & 0.7
             \\
             %
             ab
-            & 0.3
-            & \theta
+            &
             & \theta
             & 0.3\theta
             \\
             %
             ac
-            & 0.3
-            & \co{\theta}
+            &  
             & \co{\theta}
             & 0.3\co{\theta}
             \\
             %
             \co{a}, ab
-            & 0.7, 0.3
-            & 1.0, \theta
-            & 1.0, \theta
+            & 1, 0
+            & 0, \theta
             & 0.7 + 0.3\theta
             \\
             %
             \co{a}, ac
-            & 0.7, 0.3
-            & 1.0, \co{\theta}
-            & 1.0, \co{\theta}
+            & 1, 0
+            & 0, \co{\theta}
             & 0.7 + 0.3\co{\theta}
             \\
             %
             ab, ac
-            & 0.3, 0.3
+            &
             & \theta, \co{\theta}
-            & \theta\co{\theta}
-            & 0.3\theta\co{\theta}
+            & 0.3
             \\
             %
             \co{a}, ab, ac
-            & 0.7, 0.3, 0.3
-            & 1.0, \theta, \co{\theta}
-            & 1.0, \theta\co{\theta}
-            & 0.7 + 0.3\theta\co{\theta}
+            & 1, 0, 0
+            & 0, \theta, \co{\theta}
+            & 1
         \end{array}
     \end{equation*}
-    \item[Normalization.] To get a weight that sums up to one, we compute the \emph{normalization factor}
+    \item[Normalization.] To get a weight that sums up to one, we compute the \emph{normalization factor}. Since $\pw{\cdot}$ is constant on classes we have
     \begin{equation*}
         Z := \sum_{e\in\fml{E}} \pw{e}
-        = \sum_{e\in\fml{E}} \sum_{c\in\fml{C}}\pw{e,c}\pw{c}
-        = \sum_{e\in\fml{E}} \sum_{c\in\fml{C}} \del{\prod_{s\in\stablecore{e}}\pw{s,c}}\pw{c}
+        = \sum_{\class{e} \in\fml{E}/\sim} \frac{\pw{\class{e}}}{\#\class{e}},
     \end{equation*}
-    to divide the weight function into a normalized weight:
+    that divides the weight function into a normalized weight:
     \begin{equation*}
-        \nu\at{e} := \frac{\pw{e}}{Z}.
+        \pr{e} := \frac{\pw{e}}{Z}.
     \end{equation*}
-    Since $\pw{\cdot}$ is constant on classes we have:
+    
+    For the SBF example,
     \begin{equation*}
-        \begin{array}{lr|r}
-            \text{\textbf{Core}} 
-            & \text{\textbf{Size}} 
-            & \pw{\class{\cdot}} 
+        \begin{array}{lr|r|rr}
+            \stablecore{e} 
+            & \# \class{e} 
+            & \pw{\class{e}} 
+            & \pw{e}
+            & \pr{e}
             \\
             \hline 
             %
             \inconsistent
             & 37
-            & 0.0
+            & 0
+            & 0
+            & 0
             \\
             %
             \indepclass 
             & 9
-            & 0.0
+            & 0
+            & 0
+            & 0
             \\
             %
             \co{a}
             & 9
-            & 0.7
+            & \lfrac{7}{10}
+            & \lfrac{7}{90}
+            & \lfrac{7}{792}
             \\
             %
             ab
             & 3
-            & 0.3\theta
+            & \lfrac{3\theta}{10}
+            & \lfrac{\theta}{10}
+            & \lfrac{\theta}{88}
             \\
             %
             ac
             & 3
-            & 0.3\co{\theta}
+            & \lfrac{3\co{\theta}}{10}
+            & \lfrac{\co{\theta}}{10}
+            & \lfrac{\co{\theta}}{88}
             \\
             %
             \co{a}, ab
             & 0
-            & 0.3\theta + 0.7
+            & 
+            &
+            & 
             \\
             %
             \co{a}, ac
             & 0
-            & 0.3\co{\theta} + 0.7
+            & 
+            &
+            & 
             %
             \\
             %
             ab, ac
             & 2
-            & 0.3\theta\co{\theta}
+            & \lfrac{3}{10}
+            & \lfrac{3}{20}
+            & \lfrac{3}{176}
             \\
             %
             \co{a}, ab, ac
             & 1
-            & 0.3\theta\co{\theta} + 0.7
+            & 1
+            & 1
+            & \lfrac{5}{176}
             \\
             %
             \hline
-            Z
             &
-            & 0.9\theta\co{\theta} + 7.9
+            & Z = 8.8
         \end{array}
     \end{equation*}
 \end{description}
@@ -751,71 +794,71 @@ Considering the probabilities given in \cref{Figure_Alarm} we obtain the followi
 \begin{figure}
-\begin{center}
-\begin{tikzpicture}[node distance=2.5cm]
-
-% Nodes
-\node[smodel, circle] (A) {A};
-\node[tchoice, above right of=A] (B) {B};
-\node[tchoice, above left of=A] (E) {E};
-\node[tchoice, below left of=A] (M) {M};
-\node[tchoice, below right of=A] (J) {J};
-
-% Edges
-\draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$P(B)=0,001$}} ;
-\draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$P(E)=0,002$}} ;
-\draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$P(M|A)$}};
-\draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$P(J|A)$}} ;
-\end{tikzpicture}
-\end{center}
-
-\begin{multicols}{3}
-
-\footnotesize{
- \begin{equation*}
- \begin{split}
- &P(M|A)\\
-      &  \begin{array}{c|cc}
-        A & T & F \\
-            \hline 
-        T & 0,9 & 0,1\\
-        F& 0,05 & 0,95    
-        \end{array}
-        \end{split}
-\end{equation*}   
-}
-
-\footnotesize{
- \begin{equation*}
- \begin{split}
- &P(J|A)\\
-      &  \begin{array}{c|cc}
-        A & T & F \\
-            \hline 
-        T & 0,7 & 0,3\\
-        F& 0,01 & 0,99    
-        \end{array}
-        \end{split}
-\end{equation*}   
-}
-\footnotesize{
- \begin{equation*}
- \begin{split}
- P(A|B \vee E)\\
-        \begin{array}{c|c|cc}
-         B & E& T & F \\
-            \hline 
-        T & T & 0,95 & 0,05\\
-        T & F & 0,94 & 0,06\\
-        F & T & 0,29 & 0,71\\
-        F & F & 0,001 & 0,999     
-        \end{array}
-\end{split}
-\end{equation*} 
-}
-\end{multicols}
-\caption{The Earthquake, Burglary, Alarm model}
-\label{Figure_Alarm}
+    \begin{center}
+        \begin{tikzpicture}[node distance=2.5cm]
+            
+            % Nodes
+            \node[smodel, circle] (A) {A};
+            \node[tchoice, above right of=A] (B) {B};
+            \node[tchoice, above left of=A] (E) {E};
+            \node[tchoice, below left of=A] (M) {M};
+            \node[tchoice, below right of=A] (J) {J};
+            
+            % Edges
+            \draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$P(B)=0,001$}} ;
+            \draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$P(E)=0,002$}} ;
+            \draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$P(M|A)$}};
+            \draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$P(J|A)$}} ;
+        \end{tikzpicture}
+    \end{center}
+    
+    \begin{multicols}{3}
+        
+        \footnotesize{
+        \begin{equation*}
+            \begin{split}
+                &P(M|A)\\
+                &  \begin{array}{c|cc}
+                    A & T & F \\
+                    \hline 
+                    T & 0,9 & 0,1\\
+                    F& 0,05 & 0,95    
+                \end{array}
+            \end{split}
+        \end{equation*}   
+        }
+        
+        \footnotesize{
+        \begin{equation*}
+            \begin{split}
+                &P(J|A)\\
+                &  \begin{array}{c|cc}
+                    A & T & F \\
+                    \hline 
+                    T & 0,7 & 0,3\\
+                    F& 0,01 & 0,99    
+                \end{array}
+            \end{split}
+        \end{equation*}   
+        }
+        \footnotesize{
+        \begin{equation*}
+            \begin{split}
+                P(A|B \vee E)\\
+                \begin{array}{c|c|cc}
+                    B & E& T & F \\
+                    \hline 
+                    T & T & 0,95 & 0,05\\
+                    T & F & 0,94 & 0,06\\
+                    F & T & 0,29 & 0,71\\
+                    F & F & 0,001 & 0,999     
+                \end{array}
+            \end{split}
+        \end{equation*} 
+        }
+    \end{multicols}
+    \caption{The Earthquake, Burglary, Alarm model}
+    \label{Figure_Alarm}
 \end{figure}
@@ -830,111 +873,111 @@ Considering the probabilities given in \cref{Figure_Alarm} we obtain the followi
 %
 % My first guess was
 % \begin{equation*}
-%     \pr{W  = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c}.
-% \end{equation*}
-%
-% $\pr{W = w \given C = c}$ already separates $\pr{W}$ into \textbf{disjoint} events!
-%
-% Also, I am assuming that stable models are independent. 
-%
-% This would entail $p(w) = p(s_1) + p(s_2) - p(s_1)p(s_2)$ \emph{if I'm bound to set inclusion}. But I'm not. I'm defining a relation
-%
-% Also, if I set $p(w) =  p(s_1) + p(s_2)$ and respect the laws of probability, this entails $p(s_1)p(s_2) = 0$.
-%
-% So, maybe what I want is (1) to define the cover $\hat{w} = \cup_{s \supset w} s$
-%
-% \begin{equation*}
-%     \pr{W  = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c} - \pr{W = \hat{w} \given C = c}.
-% \end{equation*}
-%
-% But this doesn't works, because we'd get $\pr{W = a \given C = a} < 1$.
-% %
-%
-% %
-% \bigskip
-% \hrule
-%
-% INDEPENDENCE
-%
-%, per equation (\ref{eq:weight.class.independent}).
-%
-%   ================================================================
-%
-\subsection{Dependence}
-\label{subsec:dependence}
-
-Our basic assertion about dependence relations between atoms of the underlying system is that they can be \emph{explicitly expressed in the specification}. And, in that case, they should be. 
-
-For example, a dependence relation between $b$ and $c$ can be expressed by $b \leftarrow c \wedge d$, where $d$ is an atomic choice that explicitly expresses the dependence between $b$ and $c$. One would get, for example, a specification such as
-$$
-\probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d.
-$$
-with stable models
-$
-\co{ad}, \co{a}d, a\co{d}b, a\co{d}c, adb
-$. 
-
-
-The interesting case is the subtree of the total choice $ad$. Notice that no stable model $s$ contains $adc$ because $(i)$  $adb$ is a stable model and $(ii)$ if $adc \subset s$ then $b \in s$ so $adb \subset s$. 
-
-Following equations \eqref{eq:world.fold.stablemodel} and \eqref{eq:world.fold.independent}  {\bruno What are these equations?} this entails
-\begin{equation*}
-    \begin{cases}
-        \pr{W = adc \given C = ad} = 0,\cr
-        \pr{W = adb \given C = ad} = 1
-    \end{cases}
-\end{equation*}
-which concentrates all probability mass from the total choice $ad$ in the $adb$ branch, including the node $W = adbc$. This leads to the following cases:
-$$
-\begin{array}{l|c}
-    x & \pr{W = x \given C = ad}\\
-    \hline
-    ad & 1 \\
-    adb & 1\\
-    adc & 0\\
-    adbc & 1    
-\end{array}
-$$
-so, for $C = ad$, 
-$$
-\begin{aligned}
-    \pr{W = b} &= \frac{2}{4} \cr
-    \pr{W = c} &= \frac{1}{4} \cr
-    \pr{W = bc} &= \frac{1}{4} \cr
-                &\not= \pr{W = b}\pr{W = c}
-\end{aligned}
-$$
-\emph{i.e.} the events $W = b$ and $W = c$ are dependent and that dependence results directly from the segment $\probfact{0.2}{d}, b \leftarrow c \wedge d$ in the specification.
-
-{\bruno Why does this not contradict Assumption 1?}
-
-%
-
-%
-\hrule
-\begin{quotation}\note{Todo}
-    
-    Prove the four world cases (done), support the product (done) and sum (tbd) options, with the independence assumptions.
-\end{quotation}
-
-\section{Final Remarks}
-
-\todo{develop this section.}
-
-\begin{itemize}
-    \item The measure of the inconsistent events doesn't need to be set to $0$ and, maybe, in some cases, it shouldn't.
-    \item The physical system might have \emph{latent} variables, possibly also represented in the specification. These variables are never observed, so observations should be concentrated \emph{somewhere else}.
-    \item Comment on the possibility of extending equation \eqref{eq:weight.events.unconditional} with parameters expressing further uncertainties, enabling a tuning of the model's total choices, given observations.
-    \begin{equation*}
-        \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\theta_c.
-    \end{equation*}
-\end{itemize}
-
-
-\section*{Acknowledgements}
-
-This work is supported by NOVA\textbf{LINCS} (UIDB/04516/2020) with the financial support of FCT.IP.
-
-\printbibliography
-
-\end{document}
 \ No newline at end of file
+    %     \pr{W  = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c}.
+    % \end{equation*}
+    %
+    % $\pr{W = w \given C = c}$ already separates $\pr{W}$ into \textbf{disjoint} events!
+    %
+    % Also, I am assuming that \aclp{SM}  are independent. 
+    %
+    % This would entail $p(w) = p(s_1) + p(s_2) - p(s_1)p(s_2)$ \emph{if I'm bound to set inclusion}. But I'm not. I'm defining a relation
+    %
+    % Also, if I set $p(w) =  p(s_1) + p(s_2)$ and respect the laws of probability, this entails $p(s_1)p(s_2) = 0$.
+    %
+    % So, maybe what I want is (1) to define the cover $\hat{w} = \cup_{s \supset w} s$
+    %
+    % \begin{equation*}
+        %     \pr{W  = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c} - \pr{W = \hat{w} \given C = c}.
+        % \end{equation*}
+        %
+        % But this doesn't works, because we'd get $\pr{W = a \given C = a} < 1$.
+        % %
+        %
+        % %
+        % \bigskip
+        % \hrule
+        %
+        % INDEPENDENCE
+        %
+        %, per equation (\ref{eq:weight.class.independent}).
+        %
+        %   ================================================================
+        %
+        \subsection{Dependence}
+        \label{subsec:dependence}
+        
+        Our basic assertion about dependence relations between atoms of the underlying system is that they can be \emph{explicitly expressed in the specification}. And, in that case, they should be. 
+        
+        For example, a dependence relation between $b$ and $c$ can be expressed by $b \leftarrow c \wedge d$, where $d$ is an atomic choice that explicitly expresses the dependence between $b$ and $c$. One would get, for example, a specification such as
+        $$
+        \probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d.
+        $$
+        with \aclp{SM} 
+        $
+        \co{ad}, \co{a}d, a\co{d}b, a\co{d}c, adb
+        $. 
+        
+        
+        The interesting case is the subtree of the \acl{TC} $ad$. Notice that no \acl{SM} $s$ contains $adc$ because $(i)$  $adb$ is a \acl{SM} and $(ii)$ if $adc \subset s$ then $b \in s$ so $adb \subset s$. 
+        
+        Following equations \eqref{eq:world.fold.stablemodel} and \eqref{eq:world.fold.independent}  {\bruno What are these equations?} this entails
+        \begin{equation*}
+            \begin{cases}
+                \pr{W = adc \given C = ad} = 0,\cr
+                \pr{W = adb \given C = ad} = 1
+            \end{cases}
+        \end{equation*}
+        which concentrates all probability mass from the \acl{TC} $ad$ in the $adb$ branch, including the node $W = adbc$. This leads to the following cases:
+        $$
+        \begin{array}{l|c}
+            x & \pr{W = x \given C = ad}\\
+            \hline
+            ad & 1 \\
+            adb & 1\\
+            adc & 0\\
+            adbc & 1    
+        \end{array}
+        $$
+        so, for $C = ad$, 
+        $$
+        \begin{aligned}
+            \pr{W = b} &= \frac{2}{4} \cr
+            \pr{W = c} &= \frac{1}{4} \cr
+            \pr{W = bc} &= \frac{1}{4} \cr
+            &\not= \pr{W = b}\pr{W = c}
+        \end{aligned}
+        $$
+        \emph{i.e.} the events $W = b$ and $W = c$ are dependent and that dependence results directly from the segment $\probfact{0.2}{d}, b \leftarrow c \wedge d$ in the specification.
+        
+        {\bruno Why does this not contradict Assumption 1?}
+        
+        %
+        
+        %
+        \hrule
+        \begin{quotation}\note{Todo}
+            
+            Prove the four world cases (done), support the product (done) and sum (tbd) options, with the independence assumptions.
+        \end{quotation}
+        
+        \section{Final Remarks}
+        
+        \todo{develop this section.}
+        
+        \begin{itemize}
+            \item The measure of the inconsistent events doesn't need to be set to $0$ and, maybe, in some cases, it shouldn't.
+            \item The physical system might have \emph{latent} variables, possibly also represented in the specification. These variables are never observed, so observations should be concentrated \emph{somewhere else}.
+            \item Comment on the possibility of extending equation \eqref{eq:weight.events.unconditional} with parameters expressing further uncertainties, enabling a tuning of the model's \aclp{TC}, given observations.
+            \begin{equation*}
+                \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\theta_c.
+            \end{equation*}
+        \end{itemize}
+        
+        
+        \section*{Acknowledgements}
+        
+        This work is supported by NOVA\textbf{LINCS} (UIDB/04516/2020) with the financial support of FCT.IP.
+        
+        \printbibliography
+        
+        \end{document}
 \ No newline at end of file