Local consistency

In constraint satisfaction, local consistency conditions are properties of constraint satisfaction problems related to the consistency of subsets of variables or constraints. They can be used to reduce the search space and make the problem easier to solve. Various kinds of local consistency conditions are leveraged, including node consistency, arc consistency, and path consistency. In constraint satisfaction, local consistency conditions are properties of constraint satisfaction problems related to the consistency of subsets of variables or constraints. They can be used to reduce the search space and make the problem easier to solve. Various kinds of local consistency conditions are leveraged, including node consistency, arc consistency, and path consistency. Every local consistency condition can be enforced by a transformation that changes the problem without changing its solutions. Such a transformation is called constraint propagation. Constraint propagation works by reducing domains of variables, strengthening constraints, or creating new ones. This leads to a reduction of the search space, making the problem easier to solve by some algorithms. Constraint propagation can also be used as an unsatisfiability checker, incomplete in general but complete in some particular cases. Local consistency conditions can be grouped into various classes. The original local consistency conditions require that every consistent assignment can be consistently extended to another variable. Directional consistency only requires this condition to be satisfied when the other variable is higher than the ones in the assignment, according to a given order. Relational consistency includes extensions to more than one variable, but this extension is only required to satisfy a given constraint or set of constraints. In this article, a constraint satisfaction problem is defined as a set of variables, a set of domains, and a set of constraints. Variables and domains are associated: the domain of a variable contains all values the variable can take. A constraint is composed of a sequence of variables, called its scope, and a set of their evaluations, which are the evaluations satisfying the constraint. The constraint satisfaction problems referred to in this article are assumed to be in a special form. A problem is in normalized form, respectively regular form, if every sequence of variables is the scope of at most one constraint or exactly one constraint. The assumption of regularity done only for binary constraints leads to the standardized form. These conditions can always be enforced by combining all constraints over a sequence of variables into a single one and/or adding a constraint that is satisfied by all values of a sequence of variables. In the figures used in this article, the lack of links between two variables indicate that either no constraint or a constraint satisfied by all values exists between these two variables. The 'standard' local consistency conditions all require that all consistent partial evaluations can be extended to another variable in such a way the resulting assignment is consistent. A partial evaluation is consistent if it satisfies all constraints whose scope is a subset of the assigned variables.