rewriting 00_PASP for scholarship project

Francisco Coelho
1 parent eb584496
Showing 16 changed files with 603 additions and 349 deletions Show diff stats
README.md
SCASP-best-practices.pdf
msc/Empty File
task_01/proposal.md
text/00_PASP.aux
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 # Probabilistic ILP
  
+**Check** Conformal prediction.
+
 > Fonte: [Turning 30: New Ideas in Inductive Logic Programming](https://arxiv.org/abs/2002.11002)
  
 ## Introduction
@@ -62,4 +64,4 @@ Recursion; Predicate Invention; Higher order, ASP Hypotheses; Optimality; Prolog
  
 ## Applications
  
-### ELearning
 \ No newline at end of file
+### ELearning
@@ -6,7 +6,7 @@
  
 Answer Set Programming (ASP) is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. 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.
  
-The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts labelled with a probability $p$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 − p$.
+The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts labeled with a probability $p$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 − p$.
  
 Crucially, a joint distribution of atoms derived from an ASP specification can be used to _quantitatively measure the performance of that specification_ given data observed from the system it is intended to describe. Then, given competing specifications to describe a certain system, these performance measures can be applied in various optimization techniques in order to obtain one that best describes the target system.
  
@@ -36,7 +36,7 @@ A team of two **(or three?)** researchers and a graduate student, working over s
 - The formalization of the methods outlined above including the parameter estimation from observations and the joint distribution extending the probabilities of the stable models.
 - Application and evaluation of this approach to well-known problems, using available software tools, such as
   - Problems: **Assim, de momento, não me lembro de nenhum!**
-  - Software tools: [`s(casp)`](https://ciao-lang.org/playground/scasp.html), Potassco suit, _etc._
+  - Software tools: [`s(casp)`](https://ciao-lang.org/playground/scasp.html), [Potassco suit](https://potassco.org/), _etc._
  
 ## References
  
@@ -44,4 +44,4 @@ A team of two **(or three?)** researchers and a graduate student, working over s
 2. Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021)
 3. Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020)
 4. Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018) 
-6. Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan & Claypool Publishers (2013)
 \ No newline at end of file
+6. Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan & Claypool Publishers (2013)
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-
+%------------------------------------------
+\usecolortheme{rose}
+%------------------------------------------
+\useinnertheme{circles}
+%------------------------------------------
 \setbeamertemplate{navigation symbols}{}
-\setbeamertemplate{itemize items}[circle]
-
-
+%------------------------------------------
+\AtBeginSection{
+    \begin{frame}<beamer>\small
+        \tableofcontents[currentsection,subsectionstyle=shaded/shaded/hide]
+    \end{frame}
+}
+%------------------------------------------
+\AtBeginSubsection{	
+    \begin{frame}<beamer>\small
+        \tableofcontents[
+            currentsection,sectionstyle=shaded/shaded,
+            currentsubsection,subsectionstyle=show/shaded/hide]
+    \end{frame}
+}
+%------------------------------------------
 \usepackage[overridenote]{pdfpc}
  
 \usepackage{tikz}
@@ -28,142 +44,225 @@
 \newcommand{\deft}[1]{\textbf{#1}}
 \newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}}
 \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{\pr}[1]{\ensuremath{\mathrm{p}\at{#1}}}
 %
 %	Identificação deste documento
 %
 \title{Zugzwang}
-\subtitle{Stochastic Adventures in Inductive Logic Specifications}
+\subtitle{Stochastic Adventures in Inductive Logic}
 \author{Francisco Coelho}
 \institute[\texttt{fc@uevora.pt}]{
-  Departamento de Informática\\
-  Universidade de Évora
+  Departamento de Informática, Universidade de Évora\\
+  High Performance Computing Chair\\
+  NOVA-LINCS
 }
  
 \begin{document}
 %
 \begin{frame}[plain]
 \titlepage
-\note{The goal of this text is to **explore how ASP specifications with probabilistic facts** can lead to characterizations of the **joint distributions** of the specification's atoms.}
 \end{frame}
  
 \section{Introduction}
  
  
 \begin{frame}{Notation and Assumptions}
-    \note{We start with **common notations and assumptions**.}
+    % --------------------------------
     \begin{itemize}
-        \item The \textbf{complement} of $x$ is $\co{x} = 1 - x$.
-        \item A \textbf{probabilistic atomic choice} $\alpha:a$ defines the disjunction $a \lor \neg a$ and assigns probabilities $P\at{a} = \alpha, P\at{\neg a} = \co{\alpha}$. 
-        \item $\delta a$ denotes the \textbf{disjunction} $a \lor \neg a$ associated to the probabilistic choice $\alpha : a$ and $\delta\! \set{\alpha: a, a \in A} = \set{\delta a, a \in A}$ for any set of atoms $A$.
-        \item We adopt the the \textbf{closed world assumption}, where $\naf x \models \neg x$.
-        \item And also assume that \textbf{probabilistic choices} and \textbf{subgoals} are iid.
+        % --------------------------------
+        \item $\co{x} = 1 - x$.
+        % --------------------------------
+        \item \textbf{Probabilistic Atomic Choice (PAC):} $\alpha :: a$ defines $a \lor \neg a$ and probabilities $\pr{a} = \alpha, \pr{\neg a} = \co{\alpha}$. 
+        % --------------------------------
+        \item $\delta a$ denotes $a \lor \neg a$ and $\delta\! \set{\alpha :: a, a \in \fml{A}} = \set{\delta a, a \in \fml{A}}$ for a set of atoms $\fml{A}$.
+        % --------------------------------
+        \item \textbf{Closed World Assumption:} $\naf x \models \neg x$.
+        % --------------------------------
+        % \item Probabilistic choices and sub-goals are independent.
+        % --------------------------------
     \end{itemize}
-    \note{**Subgoals are IID** means that ???}
+    % --------------------------------
 \end{frame}
-
+% ================================================================
 \begin{frame}{General Setting}
-    \note{Next, we consider the following **general setting**}
-    Let $\fml{A}$ be a set of \textbf{atoms}, $\overline{\fml{A}} = \set{\neg a \middle| a \in \fml{A}}$ and $\fml{Z}$ the respective set of \textbf{observations},
-    $$\fml{Z} = \set{z = \alpha \cup \beta \middle| \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$ and $\fml{I}$ the set of consistent observations or \textbf{interpretations}, $$\fml{I} = \set{z \in \fml{Z} \middle| \forall a \in \fml{A}~\abs{ \set{a, \neg a} \cap z} \leq 1}.$$
+    % --------------------------------
     \begin{itemize}    
-        \item A \textbf{PASP program} is $P = C \land F \land R$  where
-                
-        \begin{itemize}            
-            \item $C = C_P = \set{\alpha_i : a_i \middle| i = 1:n}$ is a set of probabilistic atomic choices, 
-            
-            \item $F = F_P$ is a set of (common) facts and 
-            
-            \item $R = R_P$ is a set of (common) rules.
-        \end{itemize}
-        
-        and the sets of atoms, observations and interpretations of program $P$ are denoted $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$
+        % --------------------------------
+        \item  \textbf{Atoms} $\fml{A}$,
+        $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$,
+        % --------------------------------
+        \item \textbf{Observations} $\fml{Z}$:
+        $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$
+        % --------------------------------
+        \item \textbf{Interpretations} or \textit{consistent observations} $\fml{I}$ :
+        $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$
+        % --------------------------------
+        \item \textit{PASP Problem}  or \textbf{Specification:} $P = C \land F \land R$ where
+        % --------------------------------
+            \begin{itemize}
+                % --------------------------------
+                \item $C = C_P = \cset{\alpha_i :: a_i }{ i \in 1:n \land  a_i \in \fml{A}}$ \textit{pacs}.
+                % --------------------------------
+                \item $F = F_P$ \textit{facts}.
+                % --------------------------------
+                \item $R = R_P$ \textit{rules}.
+                % --------------------------------
+                \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{observations} and \textit{interpretations} of $P$.
+            \end{itemize}
+        % --------------------------------
+        \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$.
+        % --------------------------------
+    \end{itemize}
+    % --------------------------------
+\end{frame}
+% ================================================================
+\begin{frame}{Distribution Semantics}
+    % --------------------------------
+    \begin{itemize}
+        % --------------------------------
+        \item \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are  $\theta = \set{c_1, \ldots, c_n}$ where $c_i$ is $a_i$ or $\neg a_i$.  
+        % --------------------------------
+        %\item For $s\in\fml{S}$ let $\theta_s \subseteq s$ (unique \textit{total choice})
+        %\item Define $\fml{S}_\theta = \cset{s \in \fml{S}}{\theta \subset s}$.
+        % --------------------------------
  
-        \item The \textbf{stable models} of $P$ are the stable models of $\delta P = \delta C + F + R$ and the respective set is denoted $\fml{S} = \fml{S}_P$.
+        % --------------------------------
+        \item \textbf{Total Choice Probability:}
+        \begin{equation}
+            \pr{\theta} = \prod_{a_i \in \theta}\alpha_i \prod_{\neg a_i \in \theta}\co{\alpha_i}.\label{eq:prob.tc}
+        \end{equation}
+        % --------------------------------
     \end{itemize}
+    % --------------------------------
+    \begin{quote}
+        This is the \emph{Distribution Semantics} as set by Sato.
+    \end{quote}
 \end{frame}
-
-\begin{frame}    
-    \note{A model x has lower and upper "bounds".}
+% ================================================================
+\begin{frame}{Problem Statement}
+    % --------------------------------
+    \begin{quotation}
+        How to extend probability from the total choices to interpretations and observations?
+    \end{quotation}
+    % --------------------------------
+    \begin{itemize}
+        % --------------------------------
+        \item \textbf{Todo:} Extend probability to \textit{stable models}, \textit{interpretations} and \textit{observations}.
+        % --------------------------------
+    \end{itemize}
+    % --------------------------------
+    \begin{quotation}
+        \textbf{But} there is a problem extending probability from total choices to stable models.
+    \end{quotation}
+    % --------------------------------
+\end{frame}
+% ================================================================
+\begin{frame}{The Disjunction Case}
+    % --------------------------------    
+    \begin{exampleblock}{Disjuntion Example}
+        The specification
+        % --------------------------------
+        $$
+        \begin{aligned}
+            0.3 :: a    &,          \cr 
+            b \lor c    &\larr a    .
+        \end{aligned}
+        $$
+        % --------------------------------
+        has three stable models,
+        % --------------------------------
+        $$
+        \begin{aligned}
+            s_1 &= \set{\neg a}, & s_2 &= \set{a, b},  & s_3 &= \set{a, c}.
+        \end{aligned}
+        $$
+    \end{exampleblock}
+    % --------------------------------
+    \begin{itemize}
+        % --------------------------------
+        \item\label{prop:unique.ext.tcsm}\textit{Any stable model contains exactly one total choice.~$\blacksquare$}
+        % --------------------------------
+        \item $\pr{\set{\neg a}} = 0.7$ is straightforward.
+        % --------------------------------
+        \item But, no \textit{unbiased} choice for $\alpha\in\intcc{0,1}$ in
+        $$
+        \begin{aligned}
+            \pr{\set{a, b}} &= 0.3 \alpha, \cr
+            \pr{\set{a, c}} &= 0.3 \co{\alpha}.
+        \end{aligned}
+        $$
+        % --------------------------------
+    \end{itemize}
+    % --------------------------------
+\end{frame}
+% ================================================================
+\section{Motivation}
+% ================================================================
+\begin{frame}{Specification, Data \& Evaluation}
+    % --------------------------------
+    Given some procedure to assign probabilities to observations from specifications and:
+    % --------------------------------
+    \begin{itemize}
+        % --------------------------------
+        \item $P$, a specification.
+        % --------------------------------
+        \item $p$, the distribution of observations from above.
+        % --------------------------------
+        \item $Z$, a dataset of observations.
+        % --------------------------------
+        \item $e$, the respective empirical distribution.
+        % --------------------------------
+        \item $D$, some probability divergence, \textit{e.g.} Kullback-Leibler.
+        % --------------------------------
+    \end{itemize}
+    % --------------------------------
+    Given a dataset $Z$, $D\at{P} = D\at{e, p}$ is a \textit{performance} measure of $P$ and can be used, \textit{e.g.} fitness, by algorithms searching for optimal specifications of a dataset.
+    % --------------------------------
+\end{frame}
+% ================================================================
+\section{Resolution}
+% ================================================================
+\begin{frame}{Bounds of Interpretations}    
+    % --------------------------------
     \begin{itemize}
-        \item \textbf{Proposition.} Let $x\in\fml{I}$ be an interpretation.
+        % --------------------------------
+        \item For $x\in\fml{I}$:
+        % --------------------------------
         \begin{itemize}
-            \item[Lower Models] $\lset{x} = \set{s\in \fml{S} \middle| s \subseteq x}$.
-            
-            \item[Upper Models] $\uset{x} = \set{s\in \fml{S} \middle| x \subseteq s}$.
+            % --------------------------------
+            \item \textbf{Lower Models:} $\lset{x} = \cset{s\in \fml{S} }{ s \subseteq x}$.
+            % --------------------------------
+            \item \textbf{Upper Models:} $\uset{x} = \cset{s\in \fml{S} }{ x \subseteq s}$.
+            % --------------------------------
         \end{itemize}
-        \note{If $a$ is a lower model and $b$ an upper model, since stable models are minimal, must be $a = b = x$.}
-        
-        \item Exactly one of the following cases takes place: 
+        % --------------------------------
+        \item\label{prop:lucases} \textbf{Proposition.} Stable models are \textit{minimal} so \textit{one} of the following cases takes place: 
+        % --------------------------------
         \begin{enumerate}
-            %
+            % --------------------------------
             \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$ and $x$ is a stable model.
-            %
+            % --------------------------------
             \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$. 
-            %
+            % --------------------------------
             \item\label{prop:lucases.c} $\lset{x} = \emptyset \land \uset{x} \neq \emptyset$.
-            %
+            % --------------------------------
             \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. 
+            % --------------------------------
         \end{enumerate}
+        % --------------------------------
     \end{itemize}
+    % --------------------------------
 \end{frame}
-
-\begin{frame}
-    \note{Total choice are key to define probability of a clause.}
-    \begin{itemize}
-        \item The probabilistic facts $C$ define a set $\Theta = \Theta_C$ of \textbf{total choices}, with $2^n$ elements, each one a set $\theta = \set{c_1, \ldots, c_n}$ where $c_i$ is either $a_i$ or $\neg a_i$.  
-        
-        \item For each stable model $s\in\fml{S}$ let $\theta_s$ be the unique \textbf{total choice} contained in $s$ and $\fml{S}_\theta \subseteq \fml{S}$ the set of stable models that contains $\theta$.
-        
-        \item Define 
-        \begin{equation}
-            p\at{\theta} = \prod_{a_i \in \theta}\alpha_i \prod_{\neg a_i \in \theta}\co{\alpha_i}.\label{eq:prob.tc}
-        \end{equation}
-    \end{itemize}
-\end{frame}
-
-
-\begin{frame}
-    \note{Relate stable models with Sato's probabilistic semantics}
-    \begin{quotation}
-        The problem we address is how to \textbf{assign probabilities to observations} given that a total choice might entail zero or many stable models \emph{i.e.} How to assign probabilities to the stable models of $\fml{S}_\theta$ when $\envert{\fml{S}_\theta} \not= 1$?
-    \end{quotation}
-\end{frame}
-
-\begin{frame}
-    \note{There are some problems}
-    
-    As it turns out, it is quite easy to come out with a program from which result no single probability distribution. For example
-    $$
-    \begin{aligned}
-        0.3:a,& \cr 
-        b \lor c \larr& a.
-    \end{aligned}
-    $$
-    has three stable models
-    $$
-    \begin{aligned}
-        s_1 &= \set{\neg a} \cr
-        s_2 &= \set{a, b} \cr
-        s_3 &= \set{a, c}
-    \end{aligned}
-    $$
-    and while $p\at{\set{\neg a}} = 0.7$ is quite natural, we have no further information to support the choice of a singular $\alpha\in\intcc{0,1}$ in the assignment 
-    $$
-    \begin{aligned}
-        p\at{\set{a, b}} &= 0.3 \alpha \cr
-        p\at{\set{a, c}} &= 0.3 \co{\alpha}
-    \end{aligned}
-    $$
-\end{frame}
-
+% ================================================================
 \begin{frame}
  
-    Next we try to formalize the possible configurations of this scenario. Consider the ASP program $P = C \land F \land R$ with total choices $\Theta $ and stable models $\fml{S}$. Let $d : \fml{S} \to \intcc{0,1}$ such that $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$. 
+    Next we try to formalize the possible configurations of this scenario. Consider the ASP program $P = C \land F \land R$ with total choices $\Theta $ and stable models $\fml{S}$. Let $d  ::  \fml{S} \to \intcc{0,1}$ such that $\sum_{s\in\fml{S}_\theta} d\at{s} = 1$. 
 \end{frame}
-
+% ================================================================
 \begin{frame}
  
     \begin{enumerate}
@@ -176,7 +275,7 @@
             %
             \item $z$ is an interpretation and $\lset{z} = \set{z} = \uset{x}$. Then $z = s$ is a stable model and \textbf{define}
             \begin{equation}
-                w_d\at{z} = w\at{s} = d\at{s} p\at{\theta_s}.\label{eq:prob.sm}
+                w_d\at{z} = w\at{s} = d\at{s} \pr{\theta_s}.\label{eq:prob.sm}
             \end{equation}
             %
             \item $z$ is an interpretation and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define} 
@@ -197,36 +296,36 @@
         %
         \item The last point defines a ``weight'' function on the observations that depends not only on the total choices and stable models of a PASP but also on a certain function $d$ that must respect some conditions. To simplify the notation we use the subscript in $w_d$ only when necessary.
         %
-        \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an interpretation, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} observation $\top = \fml{A} \cup \set{\neg a \middle| a \in \fml{A}}$.
+        \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an interpretation, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} observation $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
         %
         \item \textbf{We haven't yet defined a probability measure.} To do so we must define a set of samples $\Omega$, a set of events $F\subseteq \pset{\Omega}$ and a function $P:F\to\intcc{0,1}$ such that:
         \begin{enumerate}
-            \item $P\at{E} \in \intcc{0, 1}$ for any $E \in F$.
-            \item $P\at{\Omega} = 1$.
-            \item if $E_1 \cap E_2 = \emptyset$ then $P\at{E_1 \cup E_2} = P\at{E_1} + P\at{E_2}$. 
+            \item $\pr{E} \in \intcc{0, 1}$ for any $E \in F$.
+            \item $\pr{\Omega} = 1$.
+            \item if $E_1 \cap E_2 = \emptyset$ then $\pr{E_1 \cup E_2} = \pr{E_1} + \pr{E_2}$. 
         \end{enumerate} 
         %
         \item In the following, assume that the stable models are iid.
         %
         \item Let the sample space $\Omega = \fml{Z}$ and the event space $F = \pset{\Omega}$. Define $Z = \sum_{\zeta\in\fml{Z}} w\at{\zeta}$ and
         \begin{equation}
-            P\at{z} = \frac{w\at{z}}{Z}, z \in \Omega \label{eq:def.prob}
+            \pr{z} = \frac{w\at{z}}{Z}, z \in \Omega \label{eq:def.prob}
         \end{equation}
         and
         \begin{equation}
-            P\at{E} = \sum_{x\in E} P\at{x}, E \subseteq \Omega. \label{eq:def.prob.event}
+            \pr{E} = \sum_{x\in E} \pr{x}, E \subseteq \Omega. \label{eq:def.prob.event}
         \end{equation}
         Now:
         \begin{enumerate}
             \item $P(E) \in \intcc{0,1}$ results directly from the definitions of $P$ and $w$.
-            \item $P\at{\Omega} = 1$ also results directly from the definitions.
+            \item $\pr{\Omega} = 1$ also results directly from the definitions.
             \item Consider two disjunct events $A, B \subset \Omega \land A \cap B = \emptyset$. Then 
             $$
             \begin{aligned}
-                P\at{A \cup B} &= \sum_{x \in A \cup B} P\at{x} \cr
-                    &= \sum_{x \in A} P\at{x} + \sum_{x \in B} P\at{x} - \sum_{x \in A \cap B} P\at{x}  \cr
-                    &= \sum_{x \in A} P\at{x} + \sum_{x \in B} P\at{x} &\text{because}~A\cap B = \emptyset \cr
-                    &= P\at{A} + P\at{B}.
+                \pr{A \cup B} &= \sum_{x \in A \cup B} \pr{x} \cr
+                    &= \sum_{x \in A} \pr{x} + \sum_{x \in B} \pr{x} - \sum_{x \in A \cap B} \pr{x}  \cr
+                    &= \sum_{x \in A} \pr{x} + \sum_{x \in B} \pr{x} &\text{because}~A\cap B = \emptyset \cr
+                    &= \pr{A} + \pr{B}.
             \end{aligned}
             $$
             \item So $\del{\Omega = \fml{Z}, F = \pset{\Omega}, P}$ is a probability space. {$\blacksquare$}
@@ -234,10 +333,11 @@
     \end{enumerate}
  
 \end{frame}
-
+% ================================================================
 \section{Cases \& Examples}
+% ================================================================
 \subsection{Programs with disjunctive heads}
-
+% ================================================================
 \begin{frame}
  
     Consider the program:
@@ -263,10 +363,9 @@
     \end{aligned}
     $$
 \end{frame}
-
-
+% ================================================================
 \begin{frame}
-    Suppose that we add an annotation $\alpha:a$, which entails $\co{\alpha}:\neg a$. This is enough to get $w\at{s_1} = \co{\alpha}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $\alpha$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get:
+    Suppose that we add an annotation $\alpha :: a$, which entails $\co{\alpha} :: \neg a$. This is enough to get $w\at{s_1} = \co{\alpha}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $\alpha$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get:
     $$
     \begin{cases}
         w\at{s_1 } = &\co{\alpha}\cr
@@ -275,7 +374,7 @@
     \end{cases}
     $$
 \end{frame}
-
+% ================================================================
 \begin{frame}
  
     Now consider all the interpretations for this program:
@@ -285,11 +384,11 @@
         %
         \node [draw, circle] (E) at (5.5,0) {$\emptyset$};
         %
-        \node [draw, circle] (a) at (2,2) {$a$};
-        \node [draw, circle] (b) at (3,2) {$b$};
+        \node [draw, circle] (a) at (3,2) {$a$};
+        \node [draw, circle] (b) at (2,2) {$b$};
         \node [draw, circle] (c) at (4,2) {$c$};
-        \node [fill=gray!50] (A) at (9,2) {$\co{a}$};
-        \node (B) at (8,2) {$\co{b}$};
+        \node [fill=gray!50] (A) at (8,2) {$\co{a}$};
+        \node (B) at (9,2) {$\co{b}$};
         \node (C) at (7,2) {$\co{c}$};
         %
         \node [fill=gray!50] (ab) at (0,4) {$ab$};
@@ -338,7 +437,7 @@
     \end{tikzpicture}    
     \end{center}
 \end{frame}
-
+% ================================================================
 \begin{frame}
  
     In this diagram:
@@ -378,7 +477,7 @@
         $$ 
         \item Now some statistics are possible. For example we get 
         $$
-        P\at{abc \mid \alpha = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9}
+        \pr{abc \mid \alpha = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9}
         $$. 
  
         \item This expression can be plotted for $d\in\intcc{0,1}$
@@ -386,10 +485,10 @@
             \includegraphics[height=15em]{Pabc_alpha03.pdf}
         \end{center} 
  
-        \item If a data set $E$ entails \emph{e.g.} $P\at{abc \mid E} = 0.0015$ we can numerically solve
+        \item If a data set $E$ entails \emph{e.g.} $\pr{abc \mid E} = 0.0015$ we can numerically solve
         $$
         \begin{aligned}
-            P\at{abc \mid \alpha = 0.3} &= P\at{abc \mid E} \cr
+            \pr{abc \mid \alpha = 0.3} &= \pr{abc \mid E} \cr
             \iff\cr
             \frac{0.09 d \del{d - 1}}{0.09 d^{2} - 0.69 d - 7.9} &= 0.0015
         \end{aligned}
@@ -397,10 +496,9 @@
         which has two solutions, $d \approx 0.15861$ or $d \approx 0.83138$.
     \end{itemize}
 \end{frame}
-
+% ================================================================
 \subsection{Non-stratified programs}
-
-
+% ================================================================
 \begin{frame}
     The following LP is non-stratified, because has a cycle with negated arcs:
     $$
@@ -429,13 +527,13 @@
     }.
     $$
 \end{frame}
-
+% ================================================================
 \begin{frame}
  
-    Looking into probabilistic interpretations of the program and/or its models, we define $\alpha = P\at{\Theta = \theta_1}\in\intcc{0, 1}$ and $P\at{\Theta = \theta_2} = \co{\alpha}$.
+    Looking into probabilistic interpretations of the program and/or its models, we define $\alpha = \pr{\Theta = \theta_1}\in\intcc{0, 1}$ and $\pr{\Theta = \theta_2} = \co{\alpha}$.
  
-    Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $P\at{ s_1 } = P\at{\Theta = \theta_1}  = \alpha$. However, there is no clear way to assign $P\at{s_2}, P\at{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly, 
-    $$P\at{s_2 \mid \Theta} + P\at{s_3 \mid \Theta} =
+    Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $\pr{ s_1 } = \pr{\Theta = \theta_1}  = \alpha$. However, there is no clear way to assign $\pr{s_2}, \pr{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly, 
+    $$\pr{s_2 \mid \Theta} + \pr{s_3 \mid \Theta} =
     \begin{cases}
         0 & \text{if}~\Theta = \theta_1\cr
         1 & \text{if}~\Theta = \theta_2
@@ -444,16 +542,15 @@
     but further assumptions are not supported \emph{a priori}. So let's \textbf{parameterize} the equation above,
     $$
     \begin{cases}
-        P\at{s_2 \mid \Theta = \theta_2} = &\beta \in \intcc{0, 1} \cr
-        P\at{s_3 \mid \Theta = \theta_2} = &\co{\beta},
+        \pr{s_2 \mid \Theta = \theta_2} = &\beta \in \intcc{0, 1} \cr
+        \pr{s_3 \mid \Theta = \theta_2} = &\co{\beta},
     \end{cases}
     $$
     in order to explicit our knowledge, or lack of, with numeric values and relations.
 \end{frame}
-
-
+% ================================================================
 \begin{frame}
-    Now we are able to define the \textbf{joint distribution} of the boolean random variables $A,B,C$, :
+    Now we are able to define the \textbf{joint distribution} of the boolean random variables $A,B,C$:
  
     $$
     \begin{array}{cc|l}
@@ -467,10 +564,9 @@
     $$
     where $\alpha, \beta\in\intcc{0,1}$.
 \end{frame}
-
+% ================================================================
 \section{Conclusions}
-
-
+% ================================================================
 \begin{frame}
     \begin{itemize}
         \item We can use the basics of probability theory and logic programming to assign explicit \emph{parameterized} probabilities to the (stable) models of a program.
@@ -479,10 +575,9 @@
         \item However, it is non-restrictive since \emph{no unusual assumptions are made}.
     \end{itemize}
 \end{frame}
-
+% ================================================================
 \section*{ASP \& related definitions}
-
-
+% ================================================================
 \begin{frame}
  
     \begin{itemize}
@@ -536,4 +631,5 @@
         \end{itemize}
     \end{itemize}
 \end{frame}
+% ================================================================
 \end{document}
 \ No newline at end of file
 \beamer@sectionintoc {1}{Introduction}{2}{0}{1}
-\beamer@sectionintoc {2}{Cases \& Examples}{10}{0}{2}
-\beamer@subsectionintoc {2}{1}{Programs with disjunctive heads}{10}{0}{2}
-\beamer@subsectionintoc {2}{2}{Non-stratified programs}{14}{0}{2}
-\beamer@sectionintoc {3}{Conclusions}{18}{0}{3}
+\beamer@sectionintoc {2}{Motivation}{8}{0}{2}
+\beamer@sectionintoc {3}{Resolution}{10}{0}{3}
+\beamer@sectionintoc {4}{Cases \& Examples}{14}{0}{4}
+\beamer@subsectionintoc {4}{1}{Programs with disjunctive heads}{15}{0}{4}
+\beamer@subsectionintoc {4}{2}{Non-stratified programs}{20}{0}{4}
+\beamer@sectionintoc {5}{Conclusions}{25}{0}{5}
1	1	# Probabilistic ILP
2	2
	3	+Check Conformal prediction.
	4	+
3	5	> Fonte: [Turning 30: New Ideas in Inductive Logic Programming](https://arxiv.org/abs/2002.11002)
4	6
5	7	## Introduction
...	...	@@ -62,4 +64,4 @@ Recursion; Predicate Invention; Higher order, ASP Hypotheses; Optimality; Prolog
62	64
63	65	## Applications
64	66
65		-### ELearning
66	67	\ No newline at end of file
	68	+### ELearning
...	...
...	...	@@ -6,7 +6,7 @@
6	6
7	7	Answer Set Programming (ASP) is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. 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.
8	8
9		-The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts labelled with a probability $p$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 − p$.
	9	+The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts labeled with a probability $p$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 − p$.
10	10
11	11	Crucially, a joint distribution of atoms derived from an ASP specification can be used to _quantitatively measure the performance of that specification_ given data observed from the system it is intended to describe. Then, given competing specifications to describe a certain system, these performance measures can be applied in various optimization techniques in order to obtain one that best describes the target system.
12	12
...	...	@@ -36,7 +36,7 @@ A team of two (or three?) researchers and a graduate student, working over s
36	36	- The formalization of the methods outlined above including the parameter estimation from observations and the joint distribution extending the probabilities of the stable models.
37	37	- Application and evaluation of this approach to well-known problems, using available software tools, such as
38	38	- Problems: Assim, de momento, não me lembro de nenhum!
39		- - Software tools: [`s(casp)`](https://ciao-lang.org/playground/scasp.html), Potassco suit, _etc._
	39	+ - Software tools: [`s(casp)`](https://ciao-lang.org/playground/scasp.html), [Potassco suit](https://potassco.org/), _etc._
40	40
41	41	## References
42	42
...	...	@@ -44,4 +44,4 @@ A team of two (or three?) researchers and a graduate student, working over s
44	44	2. Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021)
45	45	3. Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020)
46	46	4. Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018)
47		-6. Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan & Claypool Publishers (2013)
48	47	\ No newline at end of file
	48	+6. Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan & Claypool Publishers (2013)
...	...
...	...	@@ -29,14 +29,6 @@
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...	...
1	1	# Fdb version 3
2		-["xdvipdfmx"] 1663582707 "00_PASP.xdv" "00_PASP.pdf" "00_PASP" 1663582708
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	2	+["xdvipdfmx"] 1667928801 "00_PASP.xdv" "00_PASP.pdf" "00_PASP" 1667928802
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4	4	(generated)
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7		- "/home/fc/sci/projetos/zugzwang/text/00_PASP.tex" 1663582702 24211 60f6eac8176f95783413aef5958f7e49 ""
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