In a CSP, what is an extensional constraint?

Question

When talking about Constraint Satisfaction Problems (CSP), what does the term extensional constraint mean?

Answer 1

Let $X$ be a finite set of variables, where each variable $x \in X$ has a domain of values. Let $D(x)$ denote the domain of the variable $x$. Let $C$ be a finite set of constraints on the variables.

Each constraint $c \in C$ has a scope, which is a subset of the Cartesian product $D(x_i) \times \cdots \times D(x_j)$. In other words, the scope of the constraints gives the set of legal combinations of values.

An extensional constraint is the relational view of the constraint. It is an explicit listing of the subset of the Cartesian product. You could also have something called an intensional constraint, where the idea is that the values are instead computed by an algorithm.