From 9b8ebd89ce46b3662f1538805d0c9d2184f02459 Mon Sep 17 00:00:00 2001
From: Francisco Coelho <fc@uevora.pt>
Date: Wed, 7 Dec 2022 14:13:09 +0000
Subject: [PATCH] progress pre-paper, eq.rel defn, examples

---
 text/paper_01/pre-paper.pdf | Bin 77797 -> 0 bytes
 text/paper_01/pre-paper.tex | 35 ++++++++++++++++++++++-------------
 2 files changed, 22 insertions(+), 13 deletions(-)

diff --git a/text/paper_01/pre-paper.pdf b/text/paper_01/pre-paper.pdf
index c4dbe6c..1b59b86 100644
Binary files a/text/paper_01/pre-paper.pdf and b/text/paper_01/pre-paper.pdf differ
diff --git a/text/paper_01/pre-paper.tex b/text/paper_01/pre-paper.tex
index 1a88d78..15111b0 100644
--- a/text/paper_01/pre-paper.tex
+++ b/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}
 
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