Add FAIRK1 constraints satisfaction intro

src/fundamentals-of-ai-and-kr/module1/main.tex

@@ -13,5 +13,6 @@
     \input{sections/_swarm_intelligence.tex}
     \input{sections/_games.tex}
     \input{sections/_planning.tex}
+    \input{sections/_constraints.tex}
 
 \end{document}

src/fundamentals-of-ai-and-kr/module1/sections/_constraints.tex

@@ -0,0 +1,65 @@
+\chapter{Constraints satisfaction}
+
+\begin{description}
+    \item[Constraints satisfaction problem (CSP)] \marginnote{Constraints satisfaction problem (CSP)}
+        Problem defined on a finite set of variables\\$(X_1, \dots, X_n)$,
+        each belonging to a domain $(D_1, \dots, D_n)$ 
+        (notation: $X_i :: [d_{i,1}, \dots, d_{i,m}]$ or $X_i \in [d_{i,1}, \dots, d_{i,m}]$).
+
+        A constraint is a relationship between variables (e.g. $X < Y$). 
+        Formally, a constraint $c(X_{k_1}, \dots, X_{k_m})$ is a subset of the cartesian product
+        $D_{k_1} \times \dots \times D_{k_m}$ of the admissible values of the variables.
+
+        A solution to a CSP is an admissible assignment of all the variables.
+
+        CSP can be solved as a search problem.
+        \begin{descriptionlist}
+            \item[A posteriori algorithms] \marginnote{A posteriori algorithms}
+                Constraints are checked after the construction of the search tree.
+        
+            \item[Propagation algorithms] \marginnote{Propagation algorithms}
+                Once a variable has been assigned, 
+                constraints are propagated to the unassigned variables to prune part of the search space.
+        
+            \item[Consistency techniques] \marginnote{Consistency techniques}
+                Constraints are propagated to derive a simpler problem.
+        \end{descriptionlist}
+\end{description}
+
+
+
+\section{A posteriori algorithms}
+
+\subsection{Generate and test}
+\marginnote{Generate and test}
+
+The search tree is constructed depth-first. 
+At level $i$, a value is assigned to the variable $X_i$.
+Constraints are only checked when a leaf has been reached.
+
+In other words, it computes all the permutations of the values of the variables.
+
+
+\subsection{Standard backtracking}
+\marginnote{Standard backtracking}
+
+The search tree is constructed depth-first. 
+At level $i$, a value is assigned to the variable $X_i$ and 
+constraints involving $X_1, \dots, X_i$ are checked.
+In case of failure, the path is not further explored.
+
+A problem of this approach is that it requires to backtrack in case of failure
+and reassign all the variables in the worst case.
+
+
+
+\section{Propagation algorithms}
+
+\subsection{Forward checking}
+\marginnote{Forward checking}
+
+The search tree is constructed depth-first. 
+At level $i$, a value is assigned to the variable $X_i$ and 
+constraints involving $X_i$ are propagated
+to restrict the domain of the remaining unassigned variables.
+If the domain of a variable becomes empty, the path is considered a failure and the algorithm backtracks.