In this and the next chapter, algorithms for reasoning in constraint systems, i.e. for solving, simplification, and propagation of constraints, will be presented as logical inference rules that are directly executable in CHR. Unless otherwise noted, these constraint solvers are well-behaved (terminating and confluent). For most solvers, we will contrast the worst-case time complexity given by the meta-complexity Theorem 5.35 with empirical results from a preliminary set of test runs. Typically, the observed time complexity has lower exponents than those predicted by the meta-complexity theorem.

There are two main approaches for constraint-solving algorithms, variable elimination and local consistency (local propagation) techniques. A clear distinction between the two approaches is not always possible. Variable elimination substitutes terms for variables. This typically results in solvers defined by rules with recursive calls to the given constraints that contain fewer variables. In the local propagation approach, we derive simple constraints from the given constraints at hand and let the given constraints react to them. Local consistency techniques typically have to be interleaved with search to achieve global consistency, i.e. satisfaction-completeness. (Satisfaction-) completeness means that the solver can always detect unsatisfiability of allowed constraints.

In this chapter, we deal with constraint solvers for problems where the variables can only take values from a finite domain. Here the solvers try to reduce the set of possible values for a variable, i.e. to remove values from its domain that do not occur in any solution. So finite domain constraint solving proceeds by making domains smaller and smaller. The smallest useful finite domain contains the two truth values of Boolean algebra.