Constraint Satisfaction - Constraint Satisfaction Problem

Constraint Satisfaction Problem

As originally defined in artificial intelligence, constraints enumerate the possible values a set of variables may take. Informally, a finite domain is a finite set of arbitrary elements. A constraint satisfaction problem on such domain contains a set of variables whose values can only be taken from the domain, and a set of constraints, each constraint specifying the allowed values for a group of variables. A solution to this problem is an evaluation of the variables that satisfies all constraints. In other words, a solution is a way for assigning a value to each variable in such a way that all constraints are satisfied by these values.

In some circumstances, there may exist additional requirements: one may be interested not only in the solution (and in the fastest or most computationally efficient way to reach it) but in how it was reached; e.g. one may want the "simplest" solution ("simplest" in a logical, non computational sense that has to be precisely defined). This is often the case in logic games such as Sudoku.

In practice, constraints are often expressed in compact form, rather than enumerating all values of the variables that would satisfy the constraint. One of the most used constraints is the one establishing that the values of the affected variables must be all different.

Problems that can be expressed as constraint satisfaction problems are the Eight queens puzzle, the Sudoku solving problem, the Boolean satisfiability problem, scheduling problems and various problems on graphs such as the graph coloring problem.

While usually not included in the above definition of a constraint satisfaction problem, arithmetic equations and inequalities bound the values of the variables they contain and can therefore be considered a form of constraints. Their domain is the set of numbers (either integer, rational, or real), which is infinite: therefore, the relations of these constraints may be infinite as well; for example, has an infinite number of pairs of satisfying values. Arithmetic equations and inequalities are often not considered within the definition of a "constraint satisfaction problem", which is limited to finite domains. They are however used often in constraint programming.