\documentclass[a4paper, 12pt]{article}

\usepackage[
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}
}
\usetikzlibrary{calc, positioning}

\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
citecolor=blue,
}

\usepackage{commath}
\usepackage{amsthm}
\newtheorem{assumption}{Assumption}
\newtheorem{definition}{Definition}
\newtheorem{proposition}{Proposition}
\newtheorem{example}{Example}
\newtheorem{theorem}{Theorem}
\usepackage{amssymb}
\usepackage[normalem]{ulem}
\usepackage[nice]{nicefrac}
\usepackage{stmaryrd}
\usepackage{acronym}
\usepackage{multicol}
\usepackage{cleveref}
%
% Local commands
%
\newcommand{\note}[1]{\marginpar{\scriptsize #1}}
\newcommand{\naf}{\ensuremath{\sim\!}}
\newcommand{\larr}{\ensuremath{\leftarrow}}
\newcommand{\at}[1]{\ensuremath{\!\del{#1}}}
\newcommand{\co}[1]{\ensuremath{\overline{#1}}}
\newcommand{\fml}[1]{\ensuremath{{\cal #1}}}
\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{#1^{\ast}}}
\newcommand{\lset}[1]{\ensuremath{#1_{\ast}}}
\newcommand{\yset}[1]{\ensuremath{\left\langle #1 \right\rangle}}
\newcommand{\stablecore}[1]{\ensuremath{\left\llbracket #1 \right\rrbracket}}
\newcommand{\uclass}[1]{\ensuremath{\intco{#1}}}
\newcommand{\lclass}[1]{\ensuremath{\intoc{#1}}}
\newcommand{\smclass}[1]{\ensuremath{\intcc{#1}}}
\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}}
\newcommand{\pw}[1]{\ensuremath{\mu\at{#1}}}
\newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}}
\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}}
\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}}
\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})}}
\newcommand{\replace}[2]{\sout{#1}/{\color{green!20!black}#2}}
\newcommand{\franc}[1]{{\color{orange!60!black}#1}}
\newcommand{\bruno}{\color{red!60!blue}}
%
%   Acronyms
%
\acrodef{BK}[BK]{background knowledge}
\acrodef{ASP}[ASP]{answer set program}
\acrodef{NP}[NP]{normal program}
\acrodef{DS}[DS]{distribution semantics}
\acrodef{PF}[PF]{probabilistic fact}
\acrodef{TC}[TC]{total choice}
\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}
\end{tabular}    
}

\begin{document}

\maketitle

\nocite{*}

\begin{abstract}
    \todo{rewrite}
    A major limitation of logical representations in real world applications is the implicit assumption that the \acl{BK} is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how \acl{ASP} specifications with probabilistic facts can lead to \remark{characterizations of probability functions}{Why is this important? Is this what `others in sota' are trying do to?} on the specification's domain. 
\end{abstract}

\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}
The \ac{DS} is a key approach to extend logical representations with probabilistic reasoning. \Acp{PF} are the most basic \ac{DS} stochastic primitives and take the form of logical facts, $a$, labelled with probabilities, $p$, such as $\probfact{p}{a}$; Each \ac{PF} represents a boolean random variable that is true with probability $p$ and false with probability $\co{p} = 1 - p$. A (consistent) combination of the \acp{PF} defines a \acf{TC} $t = \set{\probfact{p}{a}, \ldots}$ such that \franc{changed \acl{TC} $c$ to $t$ everywhere.}

\begin{equation}
    \pr{T = t} = \prod_{a\in t} p \prod_{a \not\in t} \co{p}.
    \label{eq:prob.total.choice}
\end{equation}

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}
    \item Support probabilistic reasoning/tasks on the specification domain.
    \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}

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 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}.}

\todo{Give an outline of the paper.}

\section{A simple but fruitful example}\label{sec:example.1}

\todo{Write an introduction to the section}

\begin{example}\label{running.example}
    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

    $$
    \begin{aligned}
        P(ab) &= 0.3 \theta,\cr
        P(ac) &= 0.3 (1 - \theta).
    \end{aligned}
    $$
\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.}

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 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}
    \todo{Expand this:} If observations are not consistent with the models of the specification, then the specification is wrong and must be changed. 
\end{quote}

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 \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 \aclp{TC} .
\end{assumption}

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}

\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) {$\co{a}bc$};
            % ----
            \draw[doubt] (a) to[bend left] (ab);
            \draw[doubt] (a) to[bend right] (ac);
            
            \draw[doubt] (ab) to[bend left] (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);
            % \draw[doubt] (ac) to[bend right] (a);
            \draw[doubt] (c) to[bend right] (bc);
            \draw[doubt] (abc) to[bend left] (bc);
            \draw[doubt] (Abc) to (bc);
            \draw[doubt] (c) to[bend right] (Ac);
        \end{tikzpicture}
    \end{center}
    
    \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}

\todo{Somewhere, we need to shift the language from extending \emph{probabilities} to extending \emph{measures}}

\note{$\emptyevent$ notation introduced in \cref{fig:running.example}.}

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}

\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} } $t \in \fml{T} \iff t = a \vee \neg a$ and \emph{\aclp{SM} } $s \in \fml{S}\subset\fml{W}$. 

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 \aclp{SM} }}{seems irrelevant}
This focus on the \acp{SM} leads to the following definition:

\begin{definition}\label{def:stable.structure}
    A \emph{stable structure} is a pair $\del{A, S}$ where $A$ is a \remark{set of atoms}{can be extracted from $S$.} and $S$ is a set of consistent events over $A$.
\end{definition}


\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}
        \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 \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
    \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}
    \end{equation}
\end{definition}

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 \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 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}
    }
\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}}, \stablecore{e}
        & \text{\textbf{Class}}, \class{e}
        & \text{\textbf{Size}}, \# \class{e}\\
        \hline 
        %
        \inconsistent
        & a\co{a}, \ldots
        & 37
        \\
        %
        \indepclass 
        & \co{b}, \co{c}, bc, \co{b}a, \co{b}c, \co{b}\co{c}, \co{c}a, \co{c}b, \co{b}\co{c}a 
        & 9
        \\
        %
        \co{a} 
        & \co{a}, \co{a}b, \co{a}c, \co{a}\co{b}, \co{a}\co{c}, \co{a}bc, \co{a}b\co{c}, \co{a}\co{b}c, \co{a}\co{b}\co{c}
        & 9
        \\
        %
        ab
        & b, ab, ab\co{c}
        & 3
        \\
        %
        ac
        & c, ac, a\co{b}c 
        & 3
        \\
        %
        \co{a}, ab
        & \emptyset
        & 0
        \\
        %
        \co{a}, ac
        & \emptyset
        & 0
        %
        \\
        %
        ab, ac
        & a, abc
        & 2
        \\
        %
        \co{a}, ab, ac
        & \emptyevent
        & 1
        \\
        %
        \hline
        \Omega
        & \text{all events}
        & 64
    \end{array}
\end{equation*}

\begin{itemize}
    \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}:
    \[
    \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right).
    \]
    \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{enumerate}
    \item Extending the probabilities, \emph{as weights}, from the \aclp{TC} to events.
    \item Normalization of the weights.
\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 \aclp{TC}, $\pw{t} = \pr{T = t}$, 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}
    %
    \item[Total Choices.] Using \eqref{eq:prob.total.choice}, this case is given by 
    \begin{equation}
        \pw{t} := \pr{T = t}= \prod_{a\in t} p \prod_{a \not\in t} \co{p}
        \label{eq:weight.tchoice}
    \end{equation}
    %
    \item[Stable Models.] Each \acl{TC} $t$, 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 $\tcgen{t}$}{put this in the introduction, where core concepts are presented}.
    
    Given a \acl{SM} $s \in \fml{S}$, a \acl{TC} $t$, and variables/values $\theta_{s,t} \in \intcc{0, 1}$,
    \begin{equation}
        \pw{s, t} := \begin{cases}
            \theta_{s,t} & \text{if~} s \in \tcgen{t}\cr
            0&\text{otherwise}
        \end{cases}
        \label{eq:weight.stablemodel}
    \end{equation}
    such that $\sum_{s\in \tcgen{t}} \theta_{s,t} = 1$.
    %
    \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, t} := 0.
            \label{eq:weight.class.inconsistent}
        \end{equation}
        \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, t} := 0.
            \label{eq:weight.class.independent}
        \end{equation}
        \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}, t} := \sum_{k=1}^{n}\pw{s_k, t},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}.
            \label{eq:weight.class.other}
        \end{equation}
        and 
        \begin{equation}
            \pw{\class{e}} := \sum_{t \in \fml{T}} \pw{\class{e}, t}\pw{t}.
            \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 \acl{SC}, $\stablecore{e}$. So, we set:
    \begin{equation}
        \pw{e, t} := \frac{\pw{\class{e}, t}}{\# \class{e}} .
        \label{eq:weight.events}
    \end{equation}
    and 
    \begin{equation}
        \pw{e} := \sum_{t\in\fml{T}} \pw{e, t} \pw{t}.
        \label{eq:weight.events.unconditional}
    \end{equation}
    % \remark{instead of that equation}{if we set $\pw{s,t} := \theta_{s,t}$ in equation \eqref{eq:weight.stablemodel} here we do: 
    %     $$
    %     \pw{e} := \sum_{t\in\fml{T}} \pw{e, t}\pw{t}.
    %     $$
    % 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 $\pw{\inconsistent, t} = 0$ 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 $\theta_{s,t}$ 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, $\theta_{s,t}$, conditional on the \acl{TC}, $t$, 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 \aclp{SM}. 


\section{Developed Examples}

\subsection{The SBF Example}

We continue with the specification from Equation \eqref{eq:example.1}. 

\begin{description}
    %
    \item[\Aclp{TC}.] The \aclp{TC}, and respective \aclp{SM}, are
    \begin{center}
        \begin{tabular}{ll|r}
            \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pw{t}$}\\
            \hline 
            $a$ & $ab, ac$ & $0.3$\\
            $\co{a} = \neg a$ & $\co{a}$ & $\co{0.3} = 0.7$
        \end{tabular}
    \end{center}
    %
    \item[\Aclp{SM}.] The $\theta_{s,t}$ 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|ll|r}
            \stablecore{e}
            & \pw{s_k, t= \co{a}}
            & \pw{s_k, t= a}
            & \pw{\class{e}}=\sum_{t}\pw{\class{e},t}\pw{t}
            \\
            \hline 
            \co{a} 
            & 1
            &
            & 0.7
            \\
            %
            ab
            &
            & \theta
            & 0.3\theta
            \\
            %
            ac
            &  
            & \co{\theta}
            & 0.3\co{\theta}
            \\
            %
            \co{a}, ab
            & 1, 0
            & 0, \theta
            & 0.7 + 0.3\theta
            \\
            %
            \co{a}, ac
            & 1, 0
            & 0, \co{\theta}
            & 0.7 + 0.3\co{\theta}
            \\
            %
            ab, ac
            &
            & \theta, \co{\theta}
            & 0.3
            \\
            %
            \co{a}, ab, ac
            & 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}. Since $\pw{\cdot}$ is constant on classes,
    \begin{equation*}
        Z := \sum_{e\in\fml{E}} \pw{e}
        = \sum_{\class{e} \in\class{\fml{E}}} \frac{\pw{\class{e}}}{\#\class{e}},
    \end{equation*}
    that divides the weight function into a normalized weight
    \begin{equation*}
        \pr{e} := \frac{\pw{e}}{Z}.
    \end{equation*}
    such that
    $$
    \sum_{e \in \fml{E}} \pr{e} = 1.
    $$
    For the SBF example,
    \begin{equation*}
        \begin{array}{lr|r|rr}
            \stablecore{e} 
            & \# \class{e} 
            & \pw{\class{e}} 
            & \pw{e}
            & \pr{e}
            \\
            \hline 
            %
            \inconsistent
            & 37
            & 0
            & 0
            & 0
            \\
            %
            \indepclass 
            & 9
            & 0
            & 0
            & 0
            \\
            %
            \co{a}
            & 9
            & \lfrac{7}{10}
            & \lfrac{7}{90}
            & \lfrac{7}{792}
            \\
            %
            ab
            & 3
            & \lfrac{3\theta}{10}
            & \lfrac{\theta}{10}
            & \lfrac{\theta}{88}
            \\
            %
            ac
            & 3
            & \lfrac{3\co{\theta}}{10}
            & \lfrac{\co{\theta}}{10}
            & \lfrac{\co{\theta}}{88}
            \\
            %
            \co{a}, ab
            & 0
            & 
            &
            & 
            \\
            %
            \co{a}, ac
            & 0
            & 
            &
            & 
            %
            \\
            %
            ab, ac
            & 2
            & \lfrac{3}{10}
            & \lfrac{3}{20}
            & \lfrac{3}{176}
            \\
            %
            \co{a}, ab, ac
            & 1
            & 1
            & 1
            & \lfrac{5}{176}
            \\
            %
            \hline
            &
            & Z = 8.8
        \end{array}
    \end{equation*}
\end{description}
%
%
%
\subsection{A Not So Simple Example}

\todo{this subsection}

In this section we see how the classical example of the Burglary, Earthquake, Alarm \cite{Judea88} works in our setting. This example is a commonly used example in Bayesian networks because it illustrates reasoning under uncertainty.  The gist of example is given in \cref{Figure_Alarm}. It involves a simple network of events and conditional probabilities. 

The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) and John calls ($J$). The initial events $B$ and $E$ are assumed to be independent events that occur with probabilities $P(B)$ and $P(E)$, respectively. There is an alarm system that can be triggered by either of the initial events $B$ and $E$. The probability of the alarm going off is a conditional probability depending on whether $B$, or $E$ as occurred. One denotes these probabilities, as per usual,  by $P(A|B)$, and $P(A|E)$. There are two neighbours, Mary and John who have agreed to call if they hear the alarm. The probability that they do actually call is also a conditional probability denoted by $P(M|A)$ and $P(J|A)$, respectively.

Considering the probabilities given in \cref{Figure_Alarm} we obtain the following specification

\begin{equation*}
    \begin{aligned}
        \probfact{0.001}{B}&,\cr
        \probfact{0.002}{E}&,\cr
        \probfact{??}{A}&,\cr
        ... &. \cr
    \end{aligned}
    \label{eq:not_so_simple_example}
\end{equation*}





\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}
\end{figure}



\section{Discussion}

% % SUBSET
% \hrule
%
% \bigskip
% I'm not sure about what to say here.\marginpar{todo}
%
% 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 \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{T}} \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}