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Commit 8751b3923725ca78bc593ea1bced0b9c28ff08ac

Authored by Francisco Coelho
1 parent 505cdc43
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Completing the SBF example.

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1 1 \documentclass[a4paper, 12pt]{article}
2 2  
3 3 \usepackage[
4   - bibstyle=numeric,
5   - citestyle=numeric
  4 +bibstyle=numeric,
  5 +citestyle=numeric
6 6 ]{biblatex} %Imports biblatex package
7 7 \addbibresource{zugzwang.bib} %Import the bibliography file
8 8 \usepackage[x11colors]{xcolor}
9   -%
  9 +
10 10 \usepackage{tikz}
11 11 \tikzset{
12   - event/.style={},
13   - smodel/.style={fill=gray!25},
14   - tchoice/.style={draw, circle},
15   - indep/.style={draw, dashed},
16   - proptc/.style = {-latex, dashed},
17   - propsm/.style = {-latex, thick},
18   - doubt/.style = {gray}
  12 +event/.style={},
  13 +smodel/.style={fill=gray!25},
  14 +tchoice/.style={draw, circle},
  15 +indep/.style={draw, dashed},
  16 +proptc/.style = {-latex, dashed},
  17 +propsm/.style = {-latex, thick},
  18 +doubt/.style = {gray}
19 19 }
20 20 \usetikzlibrary{calc, positioning}
21   -%
  21 +
22 22 \usepackage{hyperref}
23 23 \hypersetup{
24   - colorlinks=true,
25   - linkcolor=blue,
26   - citecolor=blue,
  24 +colorlinks=true,
  25 +linkcolor=blue,
  26 +citecolor=blue,
27 27 }
28   -%
  28 +
29 29 \usepackage{commath}
30 30 \usepackage{amsthm}
31 31 \newtheorem{assumption}{Assumption}
... ... @@ -66,12 +66,14 @@
66 66 \newcommand{\class}[1]{\ensuremath{[{#1}]_{\sim}}}
67 67 \newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}}
68 68 \newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}}
69   -\newcommand{\rep}[2]{\left\langle #1 \middle| #2 \right\rangle}
  69 +\newcommand{\rep}[2]{\ensuremath{\left\langle #1 \middle| #2 \right\rangle}}
70 70 \newcommand{\inconsistent}{\bot}
71 71 \newcommand{\given}{\ensuremath{~\middle|~}}
72 72 \newcommand{\emptyevent}{\ensuremath{\vartriangle}}
73 73 \newcommand{\indepclass}{\ensuremath{\Diamond}}
74 74 \newcommand{\probfact}[2]{\ensuremath{#1\!::\!#2}}
  75 +\newcommand{\tcgen}[1]{\ensuremath{\widehat{#1}}}
  76 +\newcommand{\lfrac}[2]{\ensuremath{{#1}/{#2}}}
75 77  
76 78 \newcommand{\todo}[1]{{\color{red!50!black}(\emph{#1})}}
77 79 \newcommand{\remark}[2]{\uwave{#1}~{\color{green!40!black}(\emph{#2})}}
... ... @@ -79,7 +81,7 @@
79 81 \newcommand{\franc}[1]{{\color{orange!60!black}#1}}
80 82 \newcommand{\bruno}{\color{red!60!blue}}
81 83 %
82   -% ACRONYMS
  84 +% Acronyms
83 85 %
84 86 \acrodef{BK}[BK]{background knowledge}
85 87 \acrodef{ASP}[ASP]{answer set program}
... ... @@ -90,34 +92,22 @@
90 92 \acrodef{SM}[SM]{stable model}
91 93 \acrodef{SC}[SC]{stable core}
92 94 \acrodef{KL}[KL]{Kullback-Leibler}
93   -%
94   -%
95   -%
  95 +
96 96 \title{Zugzwang\\\emph{Logic and Artificial Intelligence}\\{\bruno Why this title?}}
  97 +
97 98 \author{
98   - \begin{tabular}{ccc}
99   - Francisco Coelho
100   - \footnote{Universidade de Évora}
101   - & Bruno Dinis
102   - \footnote{Universidade de Évora}
103   - & Salvador Abreu
104   - \footnote{Universidade de Évora}
105   - \\
106   - \texttt{fc@uevora.pt}
107   - & \texttt{bruno.dinis@uevora.pt}
108   - & \texttt{spa@uevora.pt}
109   - % \\
110   - % \begin{minipage}{0.3\textwidth}\centering
111   - % Universidade de Évora and NOVA\textbf{LINCS}
112   - % \end{minipage}
113   - % &
114   - % \begin{minipage}{0.3\textwidth}\centering
115   - % Universidade de Évora
116   - % \end{minipage}
117   - % & \begin{minipage}{0.3\textwidth}\centering
118   - % Universidade de Évora and NOVA\textbf{LINCS}
119   - % \end{minipage}
120   - \end{tabular}
  99 +\begin{tabular}{ccc}
  100 + Francisco Coelho
  101 + \footnote{Universidade de Évora}
  102 + & Bruno Dinis
  103 + \footnote{Universidade de Évora}
  104 + & Salvador Abreu
  105 + \footnote{Universidade de Évora}
  106 + \\
  107 + \texttt{fc@uevora.pt}
  108 + & \texttt{bruno.dinis@uevora.pt}
  109 + & \texttt{spa@uevora.pt}
  110 +\end{tabular}
121 111 }
122 112  
123 113 \begin{document}
... ... @@ -133,10 +123,9 @@
133 123  
134 124 \section{Introduction and Motivation}
135 125  
136   -
137 126 \todo{Define and/or give references to all necessary concepts used in the paper}
138   -
139 127 \todo{state of the art; references}
  128 +
140 129 \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.
141 130  
142 131 \todo{references}
... ... @@ -147,9 +136,6 @@ The \ac{DS} is a key approach to extend logical representations with probabilist
147 136 \label{eq:prob.total.choice}
148 137 \end{equation}
149 138  
150   -% \todo{Insert simple example?}
151   -
152   -
153 139 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:
154 140  
155 141 \begin{enumerate}
... ... @@ -157,16 +143,13 @@ Our goal is to extend this probability, from \acp{TC}, to cover the \emph{specif
157 143 \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.
158 144 \end{enumerate}
159 145  
160   -%
161   -%\todo{Outline/Explain our idea, further developed in \cref{sec:extending.probalilities}}
162   -%
163 146 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.
164 147  
165 148 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.
166 149  
167 150 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.}
168 151  
169   -\todo{cite \cite{sympy} \franc{--- why? but cite \cite{cozman2020joy} and relate with our work.}}
  152 +\todo{cite \cite{sympy} \franc{--- why here? but cite \cite{cozman2020joy} and relate with our work.}}
170 153  
171 154 \todo{Discuss the least informed strategy and the corolary that \aclp{SM} should be conditionally independent on the \acl{TC}.}
172 155  
... ... @@ -177,21 +160,24 @@ In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model
177 160 \todo{Write an introduction to the section}
178 161  
179 162 \begin{example}\label{running.example}
180   -Consider the following specification
181   -\begin{equation}
  163 + Consider the following specification
  164 +
  165 + \begin{equation}
  166 + \begin{aligned}
  167 + \probfact{0.3}{a}&,\cr
  168 + b \vee c& \leftarrow a.
  169 + \end{aligned}
  170 + \label{eq:example.1}
  171 + \end{equation}
  172 +
  173 + 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
  174 +
  175 + $$
182 176 \begin{aligned}
183   - \probfact{0.3}{a}&,\cr
184   - b \vee c& \leftarrow a.
  177 + P(ab) &= 0.3 \theta,\cr
  178 + P(ac) &= 0.3 (1 - \theta).
185 179 \end{aligned}
186   - \label{eq:example.1}
187   -\end{equation}
188   -This specification has three stable models, $\co{a}, ab$ and $ac$ (see \cref{fig:running.example}). While it is straightforward to set $P(\co{a})=0.7$, there is no further information to assign values to $P(ab)$ and $P(ac)$. Assuming that the \acfp{SM} are (probabilistically) independent, we can use a parameter $\theta$ such that
189   -$$
190   -\begin{aligned}
191   -P(ab) &= 0.3 \theta,\cr
192   -P(ac) &= 0.3 (1 - \theta).
193   -\end{aligned}
194   -$$
  180 + $$
195 181 \end{example}
196 182  
197 183 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
199 185 In summary, if an \ac{ASP} specification is intended to describe some observable system then:
200 186  
201 187 \begin{enumerate}
  188 +
202 189 \item Observations can be used to estimate the value of the parameters (such as $\theta$ above and others entailed from further clauses).
203   - \item \todo{What about the case where we already know a distribution of $\theta$?}
  190 +
  191 + \item \todo{What about the case where we already know a distribution of $\theta$?}
  192 +
204 193 \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the specification domain.
  194 +
205 195 \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.
  196 +
206 197 \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.
  198 +
207 199 \end{enumerate}
208 200  
209 201 \begin{quote}
... ... @@ -212,13 +204,13 @@ In summary, if an \ac{ASP} specification is intended to describe some observable
212 204  
213 205 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.
214 206  
215   -The conditional independence of stable worlds asserts the least informed strategy that we discussed in the introduction and make explicit here:
  207 +The conditional independence of stable worlds asserts the \remark{least informed strategy}{references?} that we discussed in the introduction and make explicit here:
216 208  
217 209 \begin{assumption}\label{assumption:smodels.independence}
218   - \Acl{SM} are conditionally independent, given their total choices.
  210 + \Acl{SM} are conditionally independent, given their \aclp{TC} .
219 211 \end{assumption}
220 212  
221   -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}.
  213 +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}.
222 214  
223 215 \section{Extending Probabilities}\label{sec:extending.probalilities}
224 216  
... ... @@ -232,6 +224,8 @@ The stable models $ab, ac$ from \cref{running.example} result from the clause $b
232 224 \node[event, below = of ab] (b) {$b$};
233 225 \node[event, below = of ac] (c) {$c$};
234 226 \node[event, above right = of ab] (abc) {$abc$};
  227 + \node[event, above left = of ab] (abC) {$ab\co{c}$};
  228 + \node[event, above right = of ac] (aBc) {$a\co{b}c$};
235 229 \node[indep, right = of ac] (bc) {$bc$};
236 230 \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$};
237 231 \node[event, above = of A] (Ac) {$\co{a}c$};
... ... @@ -239,17 +233,20 @@ The stable models $ab, ac$ from \cref{running.example} result from the clause $b
239 233 % ----
240 234 \draw[doubt] (a) to[bend left] (ab);
241 235 \draw[doubt] (a) to[bend right] (ac);
242   -
  236 +
243 237 \draw[doubt] (ab) to[bend left] (abc);
244   - \draw[doubt] (ac) to[bend right] (abc);
  238 + \draw[doubt] (ab) to[bend right] (abC);
245 239  
  240 + \draw[doubt] (ac) to[bend right] (abc);
  241 + \draw[doubt] (ac) to[bend left] (aBc);
  242 +
246 243 \draw[doubt] (A) to (Ac);
247 244 \draw[doubt] (A) to (Abc);
248   -
  245 +
249 246 \draw[doubt] (ab) to[bend right] (E);
250 247 \draw[doubt] (ac) to[bend right] (E);
251 248 \draw[doubt] (A) to[bend left] (E);
252   -
  249 +
253 250 \draw[doubt] (ab) to (b);
254 251 \draw[doubt] (ac) to (c);
255 252 % \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
260 257 \draw[doubt] (c) to[bend right] (Ac);
261 258 \end{tikzpicture}
262 259 \end{center}
263   - \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.}
264   - % \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?}}
  260 +
  261 + \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}.}
265 262 \label{fig:running.example}
266 263 \end{figure}
267 264  
... ... @@ -269,22 +266,124 @@ The stable models $ab, ac$ from \cref{running.example} result from the clause $b
269 266  
270 267 \note{$\emptyevent$ notation introduced in \cref{fig:running.example}.}
271 268  
272   -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.
  269 +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}.
273 270  
274 271 \subsection{An Equivalence Relation}\label{subsec:equivalence.relation}
275 272  
276   -Given an ASP specification
277   -% DONE: {\bruno This should be defined somewhere (maybe in the introduction).}
278   -\remark{{\bruno Introduce also the sets mentioned below}}{how?}
279   -, 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} $s \in \fml{S}\subset\fml{W}$.
280   -
281   -% 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.
  273 +\begin{figure}[t]
  274 + \begin{center}
  275 + \begin{tikzpicture}
  276 + \node[event] (E) {$\emptyevent$};
  277 + \node[tchoice, above left = of E] (a) {$a$};
  278 + \node[smodel, above left = of a] (ab) {$ab$};
  279 + \node[smodel, above right = of a] (ac) {$ac$};
  280 + \node[event, below = of ab] (b) {$b$};
  281 + \node[event, below = of ac] (c) {$c$};
  282 + \node[event, above right = of ab] (abc) {$abc$};
  283 + \node[event, above left = of ab] (abC) {$ab\co{c}$};
  284 + \node[event, above right = of ac] (aBc) {$a\co{b}c$};
  285 + \node[indep, right = of ac] (bc) {$bc$};
  286 + \node[tchoice, smodel, below right = of bc] (A) {$\co{a}$};
  287 + \node[event, above = of A] (Ac) {$\co{a}c$};
  288 + \node[event, above right = of Ac] (Abc) {$\co{a}bc$};
  289 + % ----
  290 + \path[draw, rounded corners, fill=cyan, fill opacity=0.1]
  291 + (ab.west) --
  292 + (ab.north west) --
  293 + %
  294 + (abC.south west) --
  295 + (abC.north west) --
  296 + (abC.north) --
  297 + %
  298 + (abc.north east) --
  299 + (abc.east) --
  300 + (abc.south east) --
  301 + %
  302 + (ab.north east) --
  303 + (ab.east) --
  304 + (ab.south east) --
  305 + %
  306 + (a.north east) --
  307 + %
  308 + (E.north east) --
  309 + (E.east) --
  310 + (E.south east) --
  311 + (E.south) --
  312 + (E.south west) --
  313 + %
  314 + (b.south west) --
  315 + %
  316 + (ab.west)
  317 + ;
  318 + % ----
  319 + \path[draw, rounded corners, fill=magenta, fill opacity=0.1]
  320 + (ac.south west) --
  321 + (ac.west) --
  322 + (ac.north west) --
  323 + %
  324 + (abc.south west) --
  325 + (abc.west) --
  326 + (abc.north west) --
  327 + %
  328 + (aBc.north east) --
  329 + (aBc.east) --
  330 + (aBc.south east) --
  331 + %
  332 + (ac.north east) --
  333 + %
  334 + (c.east) --
  335 + %
  336 + (E.east) --
  337 + (E.south east) --
  338 + (E.south) --
  339 + (E.south west) --
  340 + %
  341 + (a.south west) --
  342 + (a.west) --
  343 + (a.north west) --
  344 + (a.north) --
  345 + %
  346 + (ac.south west)
  347 + ;
  348 + % ----
  349 + \path[draw, rounded corners, fill=yellow, fill opacity=0.1]
  350 + % (A.north west) --
  351 + %
  352 + (Ac.north west) --
  353 + %
  354 + (Abc.north west) --
  355 + (Abc.north) --
  356 + (Abc.north east) --
  357 + (Abc.south east) --
  358 + %
  359 + % (Ac.north east) --
  360 + % (Ac.east) --
  361 + %
  362 + % (A.east) --
  363 + (A.south east) --
  364 + %
  365 + (E.south east) --
  366 + (E.south) --
  367 + (E.south west) --
  368 + (E.west) --
  369 + (E.north west) --
  370 + %
  371 + (Ac.north west)
  372 + ;
  373 + \end{tikzpicture}
  374 + \end{center}
  375 +
  376 + \caption{Classes (of consistent events) related to the \aclp{SM} of \cref{running.example} are defined through inclusions. \todo{write the caption}}
  377 + \label{fig:running.example.classes}
  378 +\end{figure}
282 379  
  380 +Given an ASP specification,
  381 +\remark{{\bruno Introduce also the sets mentioned below}}{how?}
  382 + 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} } $s \in \fml{S}\subset\fml{W}$.
283 383  
284   -Our path starts with a perspective of stable models as playing a role similar to \emph{prime} factors.
285   -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.
  384 +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}.
286 385  
287   -%\remark{\todo{Introduce a structure with worlds, events, and stable models}}{seems irrelevant}
  386 +%\remark{\todo{Introduce a structure with worlds, events, and \aclp{SM} }}{seems irrelevant}
288 387 This focus on the \acp{SM} leads to the following definition:
289 388  
290 389 \begin{definition}\label{def:stable.structure}
... ... @@ -292,108 +391,60 @@ This focus on the \acp{SM} leads to the following definition:
292 391 \end{definition}
293 392  
294 393  
295   -\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}:
  394 +\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}:
296 395  
297 396 \begin{definition}\label{def:stable.core}
298   - The \emph{\ac{SC}} of the event $e\in \fml{E}$ is
299   -
300   - % \begin{equation}
301   - % \uset{e} = \set{s \in \fml{S} \given e \subseteq s},\label{eq:uset}
302   - % \end{equation}
303   - % \begin{equation}
304   - % \lset{e} = \set{s \in \fml{S} \given e \supseteq s}, \label{eq:lset}
305   - % \end{equation}
306   - % \begin{equation}
307   - % \stablecore{e} = \uset{e} \cup \lset{e} \label{def:stable.core}
308   - % \end{equation}
  397 + The \emph{\ac{SC}} of the event $e\in \fml{E}$ is
309 398 \begin{equation}
310 399 \stablecore{e} := \set{s \in \fml{S} \given s \subseteq e \vee e \subseteq s} \label{eq:stable.core}
311 400 \end{equation}
312   -
313   - \end{definition}
314 401  
315   -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.
  402 +\end{definition}
  403 +
  404 +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}.
316 405  
317 406 \begin{definition}\label{def:equiv.rel}
318   -For a given specification, let $u, v \in \fml{E}$. The equivalence relation $\sim$ is defined by
  407 + For a given specification, let $u, v \in \fml{E}$. The equivalence relation $\sim$ is defined by
319 408 \begin{equation}
320   - 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}
  409 + 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}
321 410 \end{equation}
322 411 \end{definition}
323 412  
324   -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.
325   -%
326   -% \begin{definition}\label{def:smodel.events}
327   -% For $\set{s_1, \ldots, s_n} \subseteq \fml{S}$ define
328   -% \begin{equation}
329   -% \lclass{s_1, \ldots, s_n} = \set{e\in \fml{E}\setminus \fml{S} \given \uset{e} = \set{s_1, \ldots, s_n}},
330   -% \label{eq:smodel.lclass}
331   -% \end{equation}
332   -% \begin{equation}
333   -% \uclass{s_1, \ldots, s_n} = \set{e\in \fml{E}\setminus \fml{S} \given \lset{e} = \set{s_1, \ldots, s_n}}
334   -% \label{eq:smodel.uclass}
335   -% \end{equation}
336   -% and
337   -% \begin{equation}
338   -% \smclass{s_1, \ldots, s_n} = \set{s_1, \ldots, s_n}
339   -% \label{eq:smodel.smclass}
340   -% \end{equation}
341   -% \end{definition}
342   -%
343   -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:
  413 +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:
344 414 \begin{equation}
345 415 \class{e} =
346 416 \begin{cases}
347   - \inconsistent := \fml{E} \setminus \fml{W} &\text{if~} e \not\in \fml{E} \setminus \fml{W}, \\
348   - \set{u \in \fml{W} \given \stablecore{u} = \stablecore{e}} &\text{if~} e \in \fml{W}, \\
349   - % \lclass{\uset{e}} &\text{if~} \uset{e} \not= \emptyset, \\
350   - % \uclass{\lset{e}} &\text{otherwise}.
  417 + \inconsistent := \fml{E} \setminus \fml{W}
  418 + &\text{if~} e \in \fml{E} \setminus \fml{W}, \\
  419 + \set{u \in \fml{W} \given \stablecore{u} = \stablecore{e}}
  420 + &\text{if~} e \in \fml{W},
351 421 \end{cases}\label{eq:event.class}
352 422 \end{equation}
353 423  
354   -% The stable core defines a \emph{canonical} representative of each class:
355   -% \begin{theorem}
356   -% Let $e\in\fml{E}$ and $\stablecore{e} = \set{s_1, \ldots, s_n} \subseteq \fml{S}$. Then
357   -% \begin{equation}
358   -% \class{e} = \class{s_1 \cup \cdots \cup s_n}.
359   -% \end{equation}
360   -% We simplify the notation with $\class{s_1, \ldots, s_n} := \class{s_1 \cup \cdots \cup s_n}$.
361   -% \todo{This only works for consistent $s_1, \ldots, s_n$: $\set{\emptyevent} = \class{\co{a}, ab, ac} \not= \class{a\co{a}bc} = \inconsistent$.}
362   -% \end{theorem}
363   -% \begin{proof}
364   -% \todo{tbd}
365   -% \end{proof}
366   -
367   -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:
  424 +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:
368 425 \begin{equation}
369 426 \class{\fml{E}} = \set{
370   - \inconsistent,
371   - \indepclass,
372   - \class{\co{a}},
373   - \class{ab},
374   - \class{ac},
375   - \class{\co{a}, ab},
376   - \class{\co{a}, ac},
377   - \class{ab, ac},
378   - \class{\co{a}, ab, ac}
379   - }
380   -% \begin{aligned}
381   -% & \inconsistent, \emptyset, \\
382   -% & \stablecore{\co{a}}, \stablecore{ab}, \stablecore{ac}, \\
383   -% & \stablecore{\co{a}, ab}, \stablecore{\co{a}, ac}, \stablecore{ab, ac}, \\
384   -% & \stablecore{\co{a}, ab, ac}.
385   -% \end{aligned}
  427 + \inconsistent,
  428 + \indepclass,
  429 + \class{\co{a}},
  430 + \class{ab},
  431 + \class{ac},
  432 + \class{\co{a}, ab},
  433 + \class{\co{a}, ac},
  434 + \class{ab, ac},
  435 + \class{\co{a}, ab, ac}
  436 + }
386 437 \end{equation}
387 438 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:
388 439 \begin{equation*}
389 440 \begin{array}{l|lr}
390   - \text{\textbf{Core}}
391   - & \text{\textbf{Class}}
392   - & \text{\textbf{Size}}\\
  441 + \text{\textbf{Core}}, \stablecore{e}
  442 + & \text{\textbf{Class}}, \class{e}
  443 + & \text{\textbf{Size}}, \# \class{e}\\
393 444 \hline
394 445 %
395 446 \inconsistent
396   - & \text{inconsistent events}
  447 + & a\co{a}, \ldots
397 448 & 37
398 449 \\
399 450 %
... ... @@ -446,23 +497,25 @@ where $\indepclass$ denotes both the class of \emph{independent} events $e$ such
446 497 \end{equation*}
447 498  
448 499 \begin{itemize}
449   - \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}:
  500 + \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}:
450 501 \[
451   - \forall u\in \class{e} \left(\pr{u} = \pr{e} \right).
  502 + \forall u\in \class{e} \left(\mu\at{u} = \mu\at{e} \right).
452 503 \]
453   - \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.}
  504 + \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.
454 505 % \item The extended probability \emph{events} are the \emph{classes}.
455 506 \end{itemize}
456 507  
  508 +
  509 +
457 510 \subsection{From Total Choices to Events}\label{subsec:from.tchoices.to.events}
458 511  
459 512 \todo{Check adaptation} Our path to set a probability measure on $\fml{E}$ has two phases:
460   -\begin{itemize}
461   - \item Extending the probabilities, \emph{as weights}, of the total choices to events.
  513 +\begin{enumerate}
  514 + \item Extending the probabilities, \emph{as weights}, from the \aclp{TC} to events.
462 515 \item Normalization of the weights.
463   -\end{itemize}
  516 +\end{enumerate}
464 517  
465   -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.
  518 +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.
466 519  
467 520 \begin{description}
468 521 %
... ... @@ -472,72 +525,78 @@ The ``extension'' phase, traced by equations (\ref{eq:prob.total.choice}) and (\
472 525 \label{eq:weight.tchoice}
473 526 \end{equation}
474 527 %
475   - \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 $S_c$.
  528 + \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}.
476 529  
477   - Given a stable model $s \in \fml{S}$, a total choice $c$, and variables/values $\theta_{s,c} \in \intcc{0, 1}$,
  530 + Given a \acl{SM} $s \in \fml{S}$, a \acl{TC} $c$, and variables/values $\theta_{s,c} \in \intcc{0, 1}$,
478 531 \begin{equation}
479 532 \pw{s, c} := \begin{cases}
480   - \theta_{s,c}\remark{\pw{c}}{\text{maybe not!}} & \text{if~} s \in S_c\cr
  533 + \theta_{s,c} & \text{if~} s \in \tcgen{c}\cr
481 534 0&\text{otherwise}
482 535 \end{cases}
483 536 \label{eq:weight.stablemodel}
484 537 \end{equation}
485   - such that $\sum_{s\in S_c} \theta_{s,c} = 1$.
  538 + such that $\sum_{s\in \tcgen{c}} \theta_{s,c} = 1$.
486 539 %
487   - \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of stable models.
488   - \begin{itemize}
489   - \item \textbf{Inconsistent Class.} The inconsistent class contains events that are logically inconsistent. Since these events should never be observed:
  540 + \item[Classes.] \label{item:class.cases} Each class is either the inconsistent class, $\inconsistent$, or is represented by some set of \aclp{SM}.
  541 + \begin{description}
  542 + \item[Inconsistent Class.] The inconsistent class contains events that are logically inconsistent, thus should never be observed:
490 543 \begin{equation}
491 544 \pw{\inconsistent, c} := 0.
492 545 \label{eq:weight.class.inconsistent}
493 546 \end{equation}
494   - \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:
  547 + \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:
495 548 \begin{equation}
496 549 \pw{\indepclass, c} := 0.
497 550 \label{eq:weight.class.independent}
498 551 \end{equation}
499   - \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):
  552 + \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):
500 553 \begin{equation}
501   - \pw{\class{e}, c} := \prod_{k=1}^{n}\pw{s_k, c},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}.
  554 + \pw{\class{e}, c} := \sum_{k=1}^{n}\pw{s_k, c},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}.
502 555 \label{eq:weight.class.other}
503 556 \end{equation}
504   - \end{itemize}
  557 + and
  558 + \begin{equation}
  559 + \pw{\class{e}} := \sum_{c \in \fml{C}} \pw{\class{e}, c}\pw{c}.
  560 + \label{eq:weight.class.unconditional}
  561 + \end{equation}
  562 + \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.}
  563 + \end{description}
505 564 %
506   - \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its stable core, $\stablecore{e}$. So, we set:
  565 + \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, we set:
507 566 \begin{equation}
508   - \pw{e, c} := \pw{\class{e}, c}.
  567 + \pw{e, c} := \frac{\pw{\class{e}, c}}{\# \class{e}} .
509 568 \label{eq:weight.events}
510 569 \end{equation}
511 570 and
512 571 \begin{equation}
513   - \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}.
  572 + \pw{e} := \sum_{c\in\fml{C}} \pw{e, c} \pw{c}.
514 573 \label{eq:weight.events.unconditional}
515 574 \end{equation}
516   - \remark{instead of that equation}{if we set $\pw{s,c} := \theta_{s,c}$ in equation \eqref{eq:weight.stablemodel} here we do:
517   - $$
518   - \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\pw{c}.
519   - $$
520   - By the way, this is the \emph{marginalization + bayes theorem} in statistics:
521   - $$
522   - P(A) = \sum_b P(A | B=b)P(B=b)
523   - $$
524   - }
  575 + % \remark{instead of that equation}{if we set $\pw{s,c} := \theta_{s,c}$ in equation \eqref{eq:weight.stablemodel} here we do:
  576 + % $$
  577 + % \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\pw{c}.
  578 + % $$
  579 + % By the way, this is the \emph{marginalization + bayes theorem} in statistics:
  580 + % $$
  581 + % P(A) = \sum_b P(A | B=b)P(B=b)
  582 + % $$
  583 + % }
525 584 \end{description}
526 585  
527 586 % PARAMETERS FOR UNCERTAINTY
528 587 \begin{itemize}
529   - \item \todo{Remark that $\pw{\inconsistent, c} = 0$ is independent of the total choice.}
530   - \item \todo{Remark the example $bc$ for equation \ref{eq:weight.class.independent}.}
  588 + \item \todo{Remark that $\pw{\inconsistent, c} = 0$ is independent of the \acl{TC}.}
  589 + \item Consider the event $bc$. Since $\class{bc} = \indepclass$, from \cref{eq:weight.class.independent} we get $\mu\at{bc} = 0$.
531 590 \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.}
532 591 \end{itemize}
533 592  
534 593  
535   -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.
  594 +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.
536 595  
537 596 % SUPERSET
538   -Equation \eqref{eq:weight.class.other} results from conditional independence of stable models.
  597 +Equation \eqref{eq:weight.class.other} results from conditional independence of \aclp{SM}.
  598 +
539 599  
540   -
541 600 \section{Developed Examples}
542 601  
543 602 \subsection{The SBF Example}
... ... @@ -546,180 +605,164 @@ We continue with the specification from Equation \eqref{eq:example.1}.
546 605  
547 606 \begin{description}
548 607 %
549   - \item[Total Choices.] The total choices, and respective stable models, are
  608 + \item[\Aclp{TC}.] The \aclp{TC}, and respective \aclp{SM}, are
550 609 \begin{center}
551   - \begin{tabular}{llr}
552   - \textbf{Total Choice} & \textbf{Stable Models} & \textbf{$\pw{\cdot}$}\\
  610 + \begin{tabular}{ll|r}
  611 + \textbf{\Acl{TC}} & \textbf{\Aclp{SM}} & \textbf{$\pw{c}$}\\
553 612 \hline
554 613 $a$ & $ab, ac$ & $0.3$\\
555 614 $\co{a} = \neg a$ & $\co{a}$ & $\co{0.3} = 0.7$
556 615 \end{tabular}
557 616 \end{center}
558 617 %
559   - \item[Stable Models.] \todo{Enter the $\theta$ parameters.}
560   - \begin{equation*}
561   - \begin{array}{ll|r}
562   - \text{\textbf{Stable Model}}
563   - & \text{\textbf{Total Choice}}
564   - & \pw{s,c}
565   - \\
566   - \hline
567   - \co{a}
568   - & \co{a}
569   - & 1.0
570   - \\
571   - ab
572   - & a
573   - & \theta
574   - \\
575   - ac
576   - & a
577   - & \co{\theta}
578   - \end{array}
579   - \end{equation*}
580   - \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:
  618 + \item[\Aclp{SM}.] The $\theta_{s,c}$ parameters in this example are
  619 + $$
  620 + \theta_{ab,\co{a}} = \theta_{ac,\co{a}} = \theta_{\co{a}, a} = 0
  621 + $$
  622 + and
  623 + $$
  624 + \theta_{\co{a}, \co{a}} = 1, \theta_{ab, a} = \theta, \theta_{ac, a} = \co{\theta}
  625 + $$
  626 + with $\theta \in \intcc{0, 1}$.
  627 + \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:
581 628 \begin{equation*}
582   - \begin{array}{l|rrr|r}
583   - \text{\textbf{Core}}
584   - & \pw{c}
585   - & \pw{s,c}
586   - & \pw{\class{e}, c}
587   - & \pw{e}
  629 + \begin{array}{l|ll|r}
  630 + \stablecore{e}
  631 + & \pw{s_k, c= \co{a}}
  632 + & \pw{s_k, c= a}
  633 + & \pw{\class{e}}=\sum_{c}\pw{\class{e},c}\pw{c}
588 634 \\
589 635 \hline
590   - %
591   - \inconsistent
592   - &
593   - & 0.0
594   - & 0.0
595   - & 0.0
596   - \\
597   - %
598   - \indepclass
599   - &
600   - & 0.0
601   - & 0.0
602   - & 0.0
603   - \\
604   - %
605 636 \co{a}
606   - & 0.7
607   - & 1.0
608   - & 1.0
  637 + & 1
  638 + &
609 639 & 0.7
610 640 \\
611 641 %
612 642 ab
613   - & 0.3
614   - & \theta
  643 + &
615 644 & \theta
616 645 & 0.3\theta
617 646 \\
618 647 %
619 648 ac
620   - & 0.3
621   - & \co{\theta}
  649 + &
622 650 & \co{\theta}
623 651 & 0.3\co{\theta}
624 652 \\
625 653 %
626 654 \co{a}, ab
627   - & 0.7, 0.3
628   - & 1.0, \theta
629   - & 1.0, \theta
  655 + & 1, 0
  656 + & 0, \theta
630 657 & 0.7 + 0.3\theta
631 658 \\
632 659 %
633 660 \co{a}, ac
634   - & 0.7, 0.3
635   - & 1.0, \co{\theta}
636   - & 1.0, \co{\theta}
  661 + & 1, 0
  662 + & 0, \co{\theta}
637 663 & 0.7 + 0.3\co{\theta}
638 664 \\
639 665 %
640 666 ab, ac
641   - & 0.3, 0.3
  667 + &
642 668 & \theta, \co{\theta}
643   - & \theta\co{\theta}
644   - & 0.3\theta\co{\theta}
  669 + & 0.3
645 670 \\
646 671 %
647 672 \co{a}, ab, ac
648   - & 0.7, 0.3, 0.3
649   - & 1.0, \theta, \co{\theta}
650   - & 1.0, \theta\co{\theta}
651   - & 0.7 + 0.3\theta\co{\theta}
  673 + & 1, 0, 0
  674 + & 0, \theta, \co{\theta}
  675 + & 1
652 676 \end{array}
653 677 \end{equation*}
654   - \item[Normalization.] To get a weight that sums up to one, we compute the \emph{normalization factor}
  678 + \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
655 679 \begin{equation*}
656 680 Z := \sum_{e\in\fml{E}} \pw{e}
657   - = \sum_{e\in\fml{E}} \sum_{c\in\fml{C}}\pw{e,c}\pw{c}
658   - = \sum_{e\in\fml{E}} \sum_{c\in\fml{C}} \del{\prod_{s\in\stablecore{e}}\pw{s,c}}\pw{c}
  681 + = \sum_{\class{e} \in\fml{E}/\sim} \frac{\pw{\class{e}}}{\#\class{e}},
659 682 \end{equation*}
660   - to divide the weight function into a normalized weight:
  683 + that divides the weight function into a normalized weight:
661 684 \begin{equation*}
662   - \nu\at{e} := \frac{\pw{e}}{Z}.
  685 + \pr{e} := \frac{\pw{e}}{Z}.
663 686 \end{equation*}
664   - Since $\pw{\cdot}$ is constant on classes we have:
  687 +
  688 + For the SBF example,
665 689 \begin{equation*}
666   - \begin{array}{lr|r}
667   - \text{\textbf{Core}}
668   - & \text{\textbf{Size}}
669   - & \pw{\class{\cdot}}
  690 + \begin{array}{lr|r|rr}
  691 + \stablecore{e}
  692 + & \# \class{e}
  693 + & \pw{\class{e}}
  694 + & \pw{e}
  695 + & \pr{e}
670 696 \\
671 697 \hline
672 698 %
673 699 \inconsistent
674 700 & 37
675   - & 0.0
  701 + & 0
  702 + & 0
  703 + & 0
676 704 \\
677 705 %
678 706 \indepclass
679 707 & 9
680   - & 0.0
  708 + & 0
  709 + & 0
  710 + & 0
681 711 \\
682 712 %
683 713 \co{a}
684 714 & 9
685   - & 0.7
  715 + & \lfrac{7}{10}
  716 + & \lfrac{7}{90}
  717 + & \lfrac{7}{792}
686 718 \\
687 719 %
688 720 ab
689 721 & 3
690   - & 0.3\theta
  722 + & \lfrac{3\theta}{10}
  723 + & \lfrac{\theta}{10}
  724 + & \lfrac{\theta}{88}
691 725 \\
692 726 %
693 727 ac
694 728 & 3
695   - & 0.3\co{\theta}
  729 + & \lfrac{3\co{\theta}}{10}
  730 + & \lfrac{\co{\theta}}{10}
  731 + & \lfrac{\co{\theta}}{88}
696 732 \\
697 733 %
698 734 \co{a}, ab
699 735 & 0
700   - & 0.3\theta + 0.7
  736 + &
  737 + &
  738 + &
701 739 \\
702 740 %
703 741 \co{a}, ac
704 742 & 0
705   - & 0.3\co{\theta} + 0.7
  743 + &
  744 + &
  745 + &
706 746 %
707 747 \\
708 748 %
709 749 ab, ac
710 750 & 2
711   - & 0.3\theta\co{\theta}
  751 + & \lfrac{3}{10}
  752 + & \lfrac{3}{20}
  753 + & \lfrac{3}{176}
712 754 \\
713 755 %
714 756 \co{a}, ab, ac
715 757 & 1
716   - & 0.3\theta\co{\theta} + 0.7
  758 + & 1
  759 + & 1
  760 + & \lfrac{5}{176}
717 761 \\
718 762 %
719 763 \hline
720   - Z
721 764 &
722   - & 0.9\theta\co{\theta} + 7.9
  765 + & Z = 8.8
723 766 \end{array}
724 767 \end{equation*}
725 768 \end{description}
... ... @@ -751,71 +794,71 @@ Considering the probabilities given in \cref{Figure_Alarm} we obtain the followi
751 794  
752 795  
753 796 \begin{figure}
754   -\begin{center}
755   -\begin{tikzpicture}[node distance=2.5cm]
756   -
757   -% Nodes
758   -\node[smodel, circle] (A) {A};
759   -\node[tchoice, above right of=A] (B) {B};
760   -\node[tchoice, above left of=A] (E) {E};
761   -\node[tchoice, below left of=A] (M) {M};
762   -\node[tchoice, below right of=A] (J) {J};
763   -
764   -% Edges
765   -\draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$P(B)=0,001$}} ;
766   -\draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$P(E)=0,002$}} ;
767   -\draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$P(M|A)$}};
768   -\draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$P(J|A)$}} ;
769   -\end{tikzpicture}
770   -\end{center}
771   -
772   -\begin{multicols}{3}
773   -
774   -\footnotesize{
775   - \begin{equation*}
776   - \begin{split}
777   - &P(M|A)\\
778   - & \begin{array}{c|cc}
779   - A & T & F \\
780   - \hline
781   - T & 0,9 & 0,1\\
782   - F& 0,05 & 0,95
783   - \end{array}
784   - \end{split}
785   -\end{equation*}
786   -}
787   -
788   -\footnotesize{
789   - \begin{equation*}
790   - \begin{split}
791   - &P(J|A)\\
792   - & \begin{array}{c|cc}
793   - A & T & F \\
794   - \hline
795   - T & 0,7 & 0,3\\
796   - F& 0,01 & 0,99
797   - \end{array}
798   - \end{split}
799   -\end{equation*}
800   -}
801   -\footnotesize{
802   - \begin{equation*}
803   - \begin{split}
804   - P(A|B \vee E)\\
805   - \begin{array}{c|c|cc}
806   - B & E& T & F \\
807   - \hline
808   - T & T & 0,95 & 0,05\\
809   - T & F & 0,94 & 0,06\\
810   - F & T & 0,29 & 0,71\\
811   - F & F & 0,001 & 0,999
812   - \end{array}
813   -\end{split}
814   -\end{equation*}
815   -}
816   -\end{multicols}
817   -\caption{The Earthquake, Burglary, Alarm model}
818   -\label{Figure_Alarm}
  797 + \begin{center}
  798 + \begin{tikzpicture}[node distance=2.5cm]
  799 +
  800 + % Nodes
  801 + \node[smodel, circle] (A) {A};
  802 + \node[tchoice, above right of=A] (B) {B};
  803 + \node[tchoice, above left of=A] (E) {E};
  804 + \node[tchoice, below left of=A] (M) {M};
  805 + \node[tchoice, below right of=A] (J) {J};
  806 +
  807 + % Edges
  808 + \draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$P(B)=0,001$}} ;
  809 + \draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$P(E)=0,002$}} ;
  810 + \draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$P(M|A)$}};
  811 + \draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$P(J|A)$}} ;
  812 + \end{tikzpicture}
  813 + \end{center}
  814 +
  815 + \begin{multicols}{3}
  816 +
  817 + \footnotesize{
  818 + \begin{equation*}
  819 + \begin{split}
  820 + &P(M|A)\\
  821 + & \begin{array}{c|cc}
  822 + A & T & F \\
  823 + \hline
  824 + T & 0,9 & 0,1\\
  825 + F& 0,05 & 0,95
  826 + \end{array}
  827 + \end{split}
  828 + \end{equation*}
  829 + }
  830 +
  831 + \footnotesize{
  832 + \begin{equation*}
  833 + \begin{split}
  834 + &P(J|A)\\
  835 + & \begin{array}{c|cc}
  836 + A & T & F \\
  837 + \hline
  838 + T & 0,7 & 0,3\\
  839 + F& 0,01 & 0,99
  840 + \end{array}
  841 + \end{split}
  842 + \end{equation*}
  843 + }
  844 + \footnotesize{
  845 + \begin{equation*}
  846 + \begin{split}
  847 + P(A|B \vee E)\\
  848 + \begin{array}{c|c|cc}
  849 + B & E& T & F \\
  850 + \hline
  851 + T & T & 0,95 & 0,05\\
  852 + T & F & 0,94 & 0,06\\
  853 + F & T & 0,29 & 0,71\\
  854 + F & F & 0,001 & 0,999
  855 + \end{array}
  856 + \end{split}
  857 + \end{equation*}
  858 + }
  859 + \end{multicols}
  860 + \caption{The Earthquake, Burglary, Alarm model}
  861 + \label{Figure_Alarm}
819 862 \end{figure}
820 863  
821 864  
... ... @@ -830,111 +873,111 @@ Considering the probabilities given in \cref{Figure_Alarm} we obtain the followi
830 873 %
831 874 % My first guess was
832 875 % \begin{equation*}
833   -% \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c}.
834   -% \end{equation*}
835   -%
836   -% $\pr{W = w \given C = c}$ already separates $\pr{W}$ into \textbf{disjoint} events!
837   -%
838   -% Also, I am assuming that stable models are independent.
839   -%
840   -% 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
841   -%
842   -% Also, if I set $p(w) = p(s_1) + p(s_2)$ and respect the laws of probability, this entails $p(s_1)p(s_2) = 0$.
843   -%
844   -% So, maybe what I want is (1) to define the cover $\hat{w} = \cup_{s \supset w} s$
845   -%
846   -% \begin{equation*}
847   -% \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c} - \pr{W = \hat{w} \given C = c}.
848   -% \end{equation*}
849   -%
850   -% But this doesn't works, because we'd get $\pr{W = a \given C = a} < 1$.
851   -% %
852   -%
853   -% %
854   -% \bigskip
855   -% \hrule
856   -%
857   -% INDEPENDENCE
858   -%
859   -%, per equation (\ref{eq:weight.class.independent}).
860   -%
861   -% ================================================================
862   -%
863   -\subsection{Dependence}
864   -\label{subsec:dependence}
865   -
866   -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.
867   -
868   -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
869   -$$
870   -\probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d.
871   -$$
872   -with stable models
873   -$
874   -\co{ad}, \co{a}d, a\co{d}b, a\co{d}c, adb
875   -$.
876   -
877   -
878   -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 $adb \subset s$.
879   -
880   -Following equations \eqref{eq:world.fold.stablemodel} and \eqref{eq:world.fold.independent} {\bruno What are these equations?} this entails
881   -\begin{equation*}
882   - \begin{cases}
883   - \pr{W = adc \given C = ad} = 0,\cr
884   - \pr{W = adb \given C = ad} = 1
885   - \end{cases}
886   -\end{equation*}
887   -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:
888   -$$
889   -\begin{array}{l|c}
890   - x & \pr{W = x \given C = ad}\\
891   - \hline
892   - ad & 1 \\
893   - adb & 1\\
894   - adc & 0\\
895   - adbc & 1
896   -\end{array}
897   -$$
898   -so, for $C = ad$,
899   -$$
900   -\begin{aligned}
901   - \pr{W = b} &= \frac{2}{4} \cr
902   - \pr{W = c} &= \frac{1}{4} \cr
903   - \pr{W = bc} &= \frac{1}{4} \cr
904   - &\not= \pr{W = b}\pr{W = c}
905   -\end{aligned}
906   -$$
907   -\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.
908   -
909   -{\bruno Why does this not contradict Assumption 1?}
910   -
911   -%
912   -
913   -%
914   -\hrule
915   -\begin{quotation}\note{Todo}
916   -
917   - Prove the four world cases (done), support the product (done) and sum (tbd) options, with the independence assumptions.
918   -\end{quotation}
919   -
920   -\section{Final Remarks}
921   -
922   -\todo{develop this section.}
923   -
924   -\begin{itemize}
925   - \item The measure of the inconsistent events doesn't need to be set to $0$ and, maybe, in some cases, it shouldn't.
926   - \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}.
927   - \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.
928   - \begin{equation*}
929   - \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\theta_c.
930   - \end{equation*}
931   -\end{itemize}
932   -
933   -
934   -\section*{Acknowledgements}
935   -
936   -This work is supported by NOVA\textbf{LINCS} (UIDB/04516/2020) with the financial support of FCT.IP.
937   -
938   -\printbibliography
939   -
940   -\end{document}
941 876 \ No newline at end of file
  877 + % \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c}.
  878 + % \end{equation*}
  879 + %
  880 + % $\pr{W = w \given C = c}$ already separates $\pr{W}$ into \textbf{disjoint} events!
  881 + %
  882 + % Also, I am assuming that \aclp{SM} are independent.
  883 + %
  884 + % 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
  885 + %
  886 + % Also, if I set $p(w) = p(s_1) + p(s_2)$ and respect the laws of probability, this entails $p(s_1)p(s_2) = 0$.
  887 + %
  888 + % So, maybe what I want is (1) to define the cover $\hat{w} = \cup_{s \supset w} s$
  889 + %
  890 + % \begin{equation*}
  891 + % \pr{W = w \given C = c} = \sum_{s \supset w}\pr{S = s \given C = c} - \pr{W = \hat{w} \given C = c}.
  892 + % \end{equation*}
  893 + %
  894 + % But this doesn't works, because we'd get $\pr{W = a \given C = a} < 1$.
  895 + % %
  896 + %
  897 + % %
  898 + % \bigskip
  899 + % \hrule
  900 + %
  901 + % INDEPENDENCE
  902 + %
  903 + %, per equation (\ref{eq:weight.class.independent}).
  904 + %
  905 + % ================================================================
  906 + %
  907 + \subsection{Dependence}
  908 + \label{subsec:dependence}
  909 +
  910 + 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.
  911 +
  912 + 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
  913 + $$
  914 + \probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d.
  915 + $$
  916 + with \aclp{SM}
  917 + $
  918 + \co{ad}, \co{a}d, a\co{d}b, a\co{d}c, adb
  919 + $.
  920 +
  921 +
  922 + 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 $adb \subset s$.
  923 +
  924 + Following equations \eqref{eq:world.fold.stablemodel} and \eqref{eq:world.fold.independent} {\bruno What are these equations?} this entails
  925 + \begin{equation*}
  926 + \begin{cases}
  927 + \pr{W = adc \given C = ad} = 0,\cr
  928 + \pr{W = adb \given C = ad} = 1
  929 + \end{cases}
  930 + \end{equation*}
  931 + 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:
  932 + $$
  933 + \begin{array}{l|c}
  934 + x & \pr{W = x \given C = ad}\\
  935 + \hline
  936 + ad & 1 \\
  937 + adb & 1\\
  938 + adc & 0\\
  939 + adbc & 1
  940 + \end{array}
  941 + $$
  942 + so, for $C = ad$,
  943 + $$
  944 + \begin{aligned}
  945 + \pr{W = b} &= \frac{2}{4} \cr
  946 + \pr{W = c} &= \frac{1}{4} \cr
  947 + \pr{W = bc} &= \frac{1}{4} \cr
  948 + &\not= \pr{W = b}\pr{W = c}
  949 + \end{aligned}
  950 + $$
  951 + \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.
  952 +
  953 + {\bruno Why does this not contradict Assumption 1?}
  954 +
  955 + %
  956 +
  957 + %
  958 + \hrule
  959 + \begin{quotation}\note{Todo}
  960 +
  961 + Prove the four world cases (done), support the product (done) and sum (tbd) options, with the independence assumptions.
  962 + \end{quotation}
  963 +
  964 + \section{Final Remarks}
  965 +
  966 + \todo{develop this section.}
  967 +
  968 + \begin{itemize}
  969 + \item The measure of the inconsistent events doesn't need to be set to $0$ and, maybe, in some cases, it shouldn't.
  970 + \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}.
  971 + \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.
  972 + \begin{equation*}
  973 + \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\theta_c.
  974 + \end{equation*}
  975 + \end{itemize}
  976 +
  977 +
  978 + \section*{Acknowledgements}
  979 +
  980 + This work is supported by NOVA\textbf{LINCS} (UIDB/04516/2020) with the financial support of FCT.IP.
  981 +
  982 + \printbibliography
  983 +
  984 + \end{document}
942 985 \ No newline at end of file
... ...