sync

Francisco Coelho
1 parent 8751b392
Showing 2 changed files with 33 additions and 30 deletions Show diff stats
text/paper_01/pre-paper.pdf
text/paper_01/pre-paper.tex
@@ -129,10 +129,10 @@ citecolor=blue,
 \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} $c = \set{\probfact{p}{a}, \ldots}$ such that
+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{C = c} = \prod_{a\in c} p \prod_{a \not\in c} \co{p}.
+    \pr{T = t} = \prod_{a\in t} p \prod_{a \not\in t} \co{p}.
     \label{eq:prob.total.choice}
 \end{equation}
@@ -379,7 +379,7 @@ The diagram in \cref{fig:running.example} illustrates the problem of extending p
 Given an ASP specification, 
 \remark{{\bruno Introduce also the sets mentioned below}}{how?}
- we consider the \emph{atoms} $a \in \fml{A}$ and \emph{literals}, $z \in \fml{L}$, \emph{events} $e \in \fml{E} \iff e \subseteq \fml{L}$ and \emph{worlds} $w \in \fml{W}$ (consistent events), \emph{\aclp{TC} } $c \in \fml{C} \iff c = a \vee \neg a$ and \emph{\aclp{SM} } $s \in \fml{S}\subset\fml{W}$. 
+ 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}.
@@ -515,48 +515,48 @@ where $\indepclass$ denotes both the class of \emph{independent} events $e$ such
     \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}, $\pw{c} = \pr{C = c}$, 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.
+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{c} := \pr{C = c}= \prod_{a\in c} p \prod_{a \not\in c} \co{p}
+        \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} $c$, 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}.
+    \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} $c$, and variables/values $\theta_{s,c} \in \intcc{0, 1}$,
+    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, c} := \begin{cases}
-            \theta_{s,c} & \text{if~} s \in \tcgen{c}\cr
+        \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 $\sum_{s\in \tcgen{c}} \theta_{s,c} = 1$.
+    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, c} := 0.
+            \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, c} := 0.
+            \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}, c} := \sum_{k=1}^{n}\pw{s_k, c},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}.
+            \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_{c \in \fml{C}} \pw{\class{e}, c}\pw{c}.
+            \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.}
@@ -564,17 +564,17 @@ The ``extension&#39;&#39; phase, traced by equations (\ref{eq:prob.total.choice}) and (\
     %
     \item[Events.] \label{item:event.cases} Each (general) event $e$ is in the class defined by its \acl{SC}, $\stablecore{e}$. So, we set:
     \begin{equation}
-        \pw{e, c} := \frac{\pw{\class{e}, c}}{\# \class{e}} .
+        \pw{e, t} := \frac{\pw{\class{e}, t}}{\# \class{e}} .
         \label{eq:weight.events}
     \end{equation}
     and 
     \begin{equation}
-        \pw{e} := \sum_{c\in\fml{C}} \pw{e, c} \pw{c}.
+        \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 $\pw{s,c} := \theta_{s,c}$ in equation \eqref{eq:weight.stablemodel} here we do: 
+    % \remark{instead of that equation}{if we set $\pw{s,t} := \theta_{s,t}$ in equation \eqref{eq:weight.stablemodel} here we do: 
     %     $$
-    %     \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\pw{c}.
+    %     \pw{e} := \sum_{t\in\fml{T}} \pw{e, t}\pw{t}.
     %     $$
     % By the way, this is the \emph{marginalization + bayes theorem} in statistics:
     % $$
@@ -585,13 +585,13 @@ The ``extension&#39;&#39; phase, traced by equations (\ref{eq:prob.total.choice}) and (\
 % PARAMETERS FOR UNCERTAINTY
 \begin{itemize}
-    \item \todo{Remark that $\pw{\inconsistent, c} = 0$ is independent of the \acl{TC}.}
+    \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,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.
+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}. 
@@ -608,14 +608,14 @@ We continue with the specification from Equation \eqref{eq:example.1}.
     \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{c}$}\
+            \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,c}$ parameters in this example are
+    \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
     $$
@@ -628,9 +628,9 @@ We continue with the specification from Equation \eqref{eq:example.1}.
     \begin{equation*}
         \begin{array}{l|ll|r}
             \stablecore{e}
-            & \pw{s_k, c= \co{a}}
-            & \pw{s_k, c= a}
-            & \pw{\class{e}}=\sum_{c}\pw{\class{e},c}\pw{c}
+            & \pw{s_k, t= \co{a}}
+            & \pw{s_k, t= a}
+            & \pw{\class{e}}=\sum_{t}\pw{\class{e},t}\pw{t}
             \\
             \hline 
             \co{a} 
@@ -675,16 +675,19 @@ We continue with the specification from Equation \eqref{eq:example.1}.
             & 1
         \end{array}
     \end{equation*}
-    \item[Normalization.] To get a weight that sums up to one, we compute the \emph{normalization factor}. Since $\pw{\cdot}$ is constant on classes we have
+    \item[Normalization.] To get a weight that sums up to one, we compute the \emph{normalization factor}. Since $\pw{\cdot}$ is constant on classes,
     \begin{equation*}
         Z := \sum_{e\in\fml{E}} \pw{e}
-        = \sum_{\class{e} \in\fml{E}/\sim} \frac{\pw{\class{e}}}{\#\class{e}},
+        = \sum_{\class{e} \in\class{\fml{E}}} \frac{\pw{\class{e}}}{\#\class{e}},
     \end{equation*}
-    that divides the weight function into a normalized weight:
+    that divides the weight function into a normalized weight
     \begin{equation*}
         \pr{e} := \frac{\pw{e}}{Z}.
     \end{equation*}
-    
+    such that
+    $$
+    \sum_{e \in \fml{E}} \pr{e} = 1.
+    $$
     For the SBF example,
     \begin{equation*}
         \begin{array}{lr|r|rr}
@@ -969,7 +972,7 @@ Considering the probabilities given in \cref{Figure_Alarm} we obtain the followi
             \item The physical system might have \emph{latent} variables, possibly also represented in the specification. These variables are never observed, so observations should be concentrated \emph{somewhere else}.
             \item Comment on the possibility of extending equation \eqref{eq:weight.events.unconditional} with parameters expressing further uncertainties, enabling a tuning of the model's \aclp{TC}, given observations.
             \begin{equation*}
-                \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\theta_c.
+                \pw{e} := \sum_{c\in\fml{T}} \pw{e, c}\theta_c.
             \end{equation*}
         \end{itemize}