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}
...	...	@@ -47,13 +47,14 @@
47	47	\newcommand{\ent}{\ensuremath{\lhd}}
48	48	\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}}
49	49	\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}}
50		-\newcommand{\uset}[1]{\ensuremath{\left\|{#1}\right>}}
51		-\newcommand{\lset}[1]{\ensuremath{\left<{#1}\right\|}}
	50	+\newcommand{\uset}[1]{\ensuremath{\left<{#1}\right\|}}
	51	+\newcommand{\lset}[1]{\ensuremath{\left\|{#1}\right>}}
52	52	\newcommand{\pr}[1]{\ensuremath{\mathrm{P}\at{#1}}}
53		-\newcommand{\class}[1]{\ensuremath{\nicefrac{#1}{\sim}}}
54		-\newcommand{\urep}[1]{\ensuremath{\nicefrac{#1}{\varnothing}}}
55		-\newcommand{\lrep}[1]{\ensuremath{\nicefrac{\varnothing}{#1}}}
56		-\newcommand{\rep}[2]{\nicefrac{#1}{#2}}
	53	+\newcommand{\class}[1]{\ensuremath{{#1}/_{\!\sim}}}
	54	+\newcommand{\urep}[1]{\ensuremath{\rep{#1}{}}}
	55	+\newcommand{\lrep}[1]{\ensuremath{\rep{}{#1}}}
	56	+\newcommand{\rep}[2]{\left\langle #1 \middle\| #2 \right\rangle}
	57	+\newcommand{\inconsistent}{\otimes}
57	58	\newcommand{\given}{\ensuremath{~\middle\|~}}
58	59	\newcommand{\todo}[1]{\textbf{\color{orange}~(~#1~)~}}
59	60
...	...	@@ -174,25 +175,33 @@ Given an ASP specification, we consider the \textit{atoms} $a \in \fml{A}$ and \
174	175	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}:
175	176
176	177	\begin{definition}\label{def:rel.events}
177		- Let $u,v \in \fml{E}$, and $\fml{S}, \fml{W}$ the set of stable models, resp. consistent events, of some specification. Define
	178	+ Let $e, u, v \in \fml{E}$, and $\fml{S}, \fml{W}$ the set of stable models, resp. consistent events, of some specification. Define
178	179
179	180	\begin{equation}
	181	+ \uset{e} = \set{s \in \fml{S} \given e \subseteq s},
	182	+ \end{equation}
	183	+ \begin{equation}
	184	+ \lset{e} = \set{s \in \fml{S} \given s \subseteq e}
	185	+ \end{equation}
	186	+and
	187	+ \begin{equation}
180	188	u \sim v \iff u,v \not\in\fml{W} \vee (\uset{u} = \uset{v} \wedge \lset{u} = \lset{v}).\label{eq:rel.events}
181	189	\end{equation}
182	190	\end{definition}
183	191
184	192	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
185	193	\begin{equation}
186		- \class{e} = \begin{cases}
187		- \star &\text{if~} e \in \fml{E} \setminus \fml{W}, \\
188		- \rep{\uset{e}}{\lset{e}} &\text{otherwise}.
	194	+ \class{e} =
	195	+ \begin{cases}
	196	+ \inconsistent &\text{if~} e \in \fml{E} \setminus \fml{W}, \\
	197	+ \rep{e}{e} &\text{otherwise}.
189	198	\end{cases}
190	199	\end{equation}
191	200
192	201	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
193	202	\begin{equation}
194	203	\begin{aligned}
195		- &\star, \urep{\varnothing}, \
	204	+ &\inconsistent, \rep{}{}, \
196	205	&\rep{\co{a}}{\co{a}} = \set{\co{a}}, \rep{ab}{ab} = \set{ab}, \rep{ac}{ac} = \set{ac}\\
197	206	&\urep{\co{a}}, \urep{ab}, \urep{ac}, \lrep{\co{a}}, \lrep{ab}, \lrep{ac}, \\
198	207	&\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} = \
201	210	\end{aligned}
202	211	\end{equation}
203	212
204		-For example, $\class{a} = \urep{ab, ac}$, $\class{abc} = \lrep{ab, ac}$ and $\class{bc} = \urep{\varnothing}$.
	213	+For example, $\class{a} = \urep{ab, ac}$, $\class{abc} = \lrep{ab, ac}$ and $\class{bc} = \rep{}{}$.
205	214
206	215
207	216	\begin{itemize}
208	217	\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.
209	218	\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.
210		- \item The probability events are going to be the \textit{classes}.
	219	+ \item The extended probability \textit{events} are the \textit{classes}.
211	220	\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.
212	221	\end{itemize}
213	222
...	...