progress pre-paper, eq.rel defn, examples

Francisco Coelho
1 parent 87b4cf2d
Showing 2 changed files with 22 additions and 13 deletions Show diff stats
text/paper_01/pre-paper.pdf
text/paper_01/pre-paper.tex
@@ -47,13 +47,14 @@
 \newcommand{\ent}{\ensuremath{\lhd}}
 \newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}}
 \newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}}
-\newcommand{\uset}[1]{\ensuremath{\left|{#1}\right>}}
-\newcommand{\lset}[1]{\ensuremath{\left<{#1}\right|}}
+\newcommand{\uset}[1]{\ensuremath{\left<{#1}\right|}}
+\newcommand{\lset}[1]{\ensuremath{\left|{#1}\right>}}
 \newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}}
-\newcommand{\class}[1]{\ensuremath{\nicefrac{#1}{\sim}}}
-\newcommand{\urep}[1]{\ensuremath{\nicefrac{#1}{\varnothing}}}
-\newcommand{\lrep}[1]{\ensuremath{\nicefrac{\varnothing}{#1}}}
-\newcommand{\rep}[2]{\nicefrac{#1}{#2}}
+\newcommand{\class}[1]{\ensuremath{{#1}/_{\!\sim}}}
+\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}}
+\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}}
+\newcommand{\rep}[2]{\left\langle #1 \middle| #2 \right\rangle}
+\newcommand{\inconsistent}{\otimes}
 \newcommand{\given}{\ensuremath{~\middle|~}}
 \newcommand{\todo}[1]{\textbf{\color{orange}~(~#1~)~}}
@@ -174,25 +175,33 @@ Given an ASP specification, we consider the \textit{atoms} $a \in \fml{A}$ and \
 Out path starts with a perspective of stable models as playing a role similar to \textit{prime} factors. The stable models of specification are the irreducible events entailed from that specification and any event must be interpreted under its relation with the stable models. This stance leads to definition \ref{def:rel.events}:
 \begin{definition}\label{def:rel.events}
-    Let $u,v \in \fml{E}$, and $\fml{S}, \fml{W}$ the set of stable models, resp. consistent events, of some specification. Define
+    Let $e, u, v \in \fml{E}$, and $\fml{S}, \fml{W}$ the set of stable models, resp. consistent events, of some specification. Define
     \begin{equation}
+        \uset{e} = \set{s \in \fml{S} \given e \subseteq s},
+    \end{equation}
+    \begin{equation}
+        \lset{e} = \set{s \in \fml{S} \given s \subseteq e}
+    \end{equation}
+and
+    \begin{equation}
         u \sim v \iff u,v \not\in\fml{W} \vee (\uset{u} = \uset{v} \wedge \lset{u} = \lset{v}).\label{eq:rel.events}
     \end{equation}
 \end{definition}
 This equivalence relation defines a partition of the events space, where each class holds a unique relation with the stable models. In particular, we can denote each class by
 \begin{equation}
-    \class{e} = \begin{cases}
-        \star &\text{if~} e \in \fml{E} \setminus \fml{W}, \\
-        \rep{\uset{e}}{\lset{e}} &\text{otherwise}.
+    \class{e} =
+    \begin{cases}
+        \inconsistent   &\text{if~} e \in \fml{E} \setminus \fml{W}, \\
+        \rep{e}{e}      &\text{otherwise}.
     \end{cases}
 \end{equation}
 Consider the example from \ref{eq:example.1}. The stable models are $\fml{S} = \co{a}, ab, ac$ so the quotient set of this relation is
 \begin{equation}
 \begin{aligned}
-    &\star, \urep{\varnothing}, \
+    &\inconsistent, \rep{}{}, \
     &\rep{\co{a}}{\co{a}} = \set{\co{a}}, \rep{ab}{ab} = \set{ab}, \rep{ac}{ac} = \set{ac}\\
     &\urep{\co{a}},  \urep{ab}, \urep{ac}, \lrep{\co{a}},  \lrep{ab}, \lrep{ac}, \\
     &\urep{\co{a}, ab}, \urep{\co{a}, ac},  \urep{ab, ac}, 
@@ -201,13 +210,13 @@ Consider the example from \ref{eq:example.1}. The stable models are $\fml{S} = \
 \end{aligned}
 \end{equation}
-For example, $\class{a} = \urep{ab, ac}$, $\class{abc} = \lrep{ab, ac}$ and $\class{bc} = \urep{\varnothing}$.
+For example, $\class{a} = \urep{ab, ac}$, $\class{abc} = \lrep{ab, ac}$ and $\class{bc} = \rep{}{}$.
 \begin{itemize}
     \item Since all events within a equivalence class have the same relation with the stable models, probability assignment should be constant for the elements of that class.
     \item So, instead of dealing with $2^6$ events, we need only to handle $19$ classes, well defined in terms of combinations of the stable models. 
-    \item The probability events are going to be the \textit{classes}.
+    \item The extended probability \textit{events} are the \textit{classes}.
     \item The physical system might have \textit{latent} variables, possibly also represented in the specification. These variables are never observed, so observations should be concentrated in the $\nicefrac{\uset{e}}{\varnothing}$ classes.
 \end{itemize}