From 0ca7041569dd6f83cc1ee12d062950e7ed5475bb Mon Sep 17 00:00:00 2001 From: Francisco Coelho Date: Mon, 10 Jul 2023 02:43:46 +0100 Subject: [PATCH] sync --- text/paper_01/pre-paper.pdf | Bin 118580 -> 0 bytes text/paper_01/pre-paper.tex | 63 +++++++++++++++++++++++++++++++++------------------------------ 2 files changed, 33 insertions(+), 30 deletions(-) diff --git a/text/paper_01/pre-paper.pdf b/text/paper_01/pre-paper.pdf index ef4f5ed..1d29bc9 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 11e88bf..fdc8c24 100644 --- a/text/paper_01/pre-paper.tex +++ b/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'' 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'' 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} -- libgit2 0.21.2