Add FAIKR2 probabilistic logic programming

src/fundamentals-of-ai-and-kr/module2/img/_probabilistic_logic_example.drawio

@@ -0,0 +1,34 @@
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src/fundamentals-of-ai-and-kr/module2/main.tex

@@ -12,5 +12,6 @@
     \input{sections/_descriptive_logic.tex}
     \input{sections/_semantic_web.tex}
     \input{sections/_time_reasoning.tex}
+    \input{sections/_probabilistic_reasoning.tex}
 
 \end{document}

src/fundamentals-of-ai-and-kr/module2/sections/_probabilistic_reasoning.tex

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+\chapter{Probabilistic logic reasoning}
+
+\begin{description}
+    \item[Probabilistic logic programming] \marginnote{Probabilistic logic programming}
+        Adds probability distributions over logic programs allowing to define different worlds.
+        Joint distributions can also be defined over worlds and allows to answer to queries.
+\end{description}
+
+
+
+\section{Logic programs with annotated disjunctions (LPAD)}
+\marginnote{LPAD}
+
+\subsection{Syntax}
+
+\begin{description}
+    \item[\texttt{null}] 
+        Atom that can only appear in the head of a clause and 
+        cancels the clause (i.e. equivalent of not having the clause).
+\end{description}
+
+The head of each clause is defined as a disjunction of atoms, each with a probability.
+More specifically, each clause has a probability distribution over its head.
+
+\begin{example}
+    \phantom{}
+    \begin{lstlisting}
+    sneezing(X):0.7 ; null:0.3 :- flu(X).
+    sneezing(X):0.8 ; null:0.2 :- hay_fever(X).
+    \end{lstlisting}
+\end{example}
+
+
+\subsection{Distribution semantics}
+
+\begin{description}
+    \item[Worlds] \marginnote{World}
+        Given a clause $C$ and a substitution $\theta$ such that $C\theta$ is ground,
+        the following operations are defined for LPAD:
+        \begin{descriptionlist}
+            \item[Atomic choice] \marginnote{Atomic choice}
+                An atomic choice $(C, \theta, i)$ is the selection of the $i$-th atom in the head of $C$ for grounding.
+
+            \item[Composite choice] \marginnote{Composite choice}
+                A composite choice $\kappa$ is a set of atomic choices.
+                The probability of a composite choice is the following:
+                \[ \prob{\kappa} = \prod_{(C, \theta, i) \in \kappa} \prob{C, i} \]
+                where $\prob{C, i}$ is the probability of choosing the $i$-th atom in the head of $C$.
+
+            \item[Selection] \marginnote{Selection}
+                A selection $\sigma$ is a composite choice where an atom from the head of each clause for each grounding has been chosen.
+                In other words, a selection can be defined only when the program is ground.
+
+                A selection $\sigma$ identifies a world $w_\sigma$ and has probability:
+                \[ \prob{w_\sigma} = \prob{\sigma} = \prod_{(C, \theta, i) \in \sigma} \prob{C, i} \]
+        \end{descriptionlist}
+
+        \begin{example}
+            Given the program:
+            \begin{lstlisting}
+    sneezing(X):0.7 ; null:0.3 :- flu(X).
+    sneezing(X):0.8 ; null:0.2 :- hay_fever(X).
+            \end{lstlisting}
+
+            The possible worlds are:
+            \begin{center}
+                \includegraphics[width=0.7\textwidth]{img/_probabilistic_logic_example.pdf}
+            \end{center}
+        \end{example}
+
+    \item[Queries] \marginnote{Queries}
+        Given a ground query $Q$ and a world $w$, the probability of $Q$ being true in $w$ is trivially:
+        \[ 
+            \prob{Q \mid w}
+            \begin{cases}
+                1 & \text{ if $Q$ is true in $w$}\\
+                0 & \text{ otherwise}
+            \end{cases} 
+        \]
+
+        The overall probability of $Q$ is:
+        \[ \prob{Q} = \sum_w \prob{Q, w} = \sum_w \prob{Q \mid w}\prob{w} = \sum_{w \models Q} \prob{w} \]
+
+        \begin{example}
+            Given the program:
+            \begin{lstlisting}
+    sneezing(X):0.7 ; null:0.3 :- flu(X).
+    sneezing(X):0.8 ; null:0.2 :- hay_fever(X).
+            \end{lstlisting}
+
+            The probability of \texttt{sneezing(bob)} is:
+            \[ \prob{\texttt{sneezing(bob)}} = \prob{w_1} + \prob{w_2} + \prob{w_3} \]
+        \end{example}
+\end{description}