Abstract. We study here constraint satisfaction problems that are based on predefined, explicitly given finite constraints. To solve them we propose a notion of rule consistency that can be expressed in terms of rules derived from the explicit representation of the initial constraints. This notion of local consistency is weaker than arc consistency for constraints of arbitrary arity but coincides with it when all domains are unary or binary. For Boolean constraints rule consistency coincides with the closure under the wellknown propagation rules for Boolean constraints. By generalizing the format of the rules we obtain a characterization of arc consistency in terms of so-called inclusion rules. The advantage of rule consistency and this rule based characterization of the arc consistency is that the algorithms that enforce both notions can be automatically generated, as CHR rules. So these algorithms could be integrated into constraint logic programming systems such as ECL i PS e . We illustrate the usefulness of this approach to constraint propagation by discussing the implementations of both algorithms and their use on various examples, including Boolean constraints, three valued logic of Kleene, constraints dealing with Waltz's language for describing polyhedreal scenes, and Allen's qualitative approach to temporal logic.
We study here a natural situation when constraint programming can be entirely reduced to rule-based programming. To this end we explain first how one can compute on constraint satisfaction problems using rules represented by simple first-order formulas. Then we consider constraint satisfaction problems that are based on predefined, explicitly given constraints. To solve them we first derive rules from these explicitly given constraints and limit the computation process to a repeated application of these rules, combined with labeling.We consider here two types of rules. The first type, that we call equality rules, leads to a new notion of local consistency, called rule consistency that turns out to be weaker than arc consistency for constraints of arbitrary arity (called hyper-arc consistency in (Marriott & Stuckey, 1998)). For Boolean constraints rule consistency coincides with the closure under the well-known propagation rules for Boolean constraints. The second type of rules, that we call membership rules, yields a rule-based characterization of arc consistency.To show feasibility of this rule-based approach to constraint programming we show how both types of rules can be automatically generated, as CHR rules of (Frühwirth, 1995). This yields an implementation of this approach to programming by means of constraint logic programming.We illustrate the usefulness of this approach to constraint programming by discussing various examples, including Boolean constraints, two typical examples of many valued logics, constraints dealing with Waltz's language for describing polyhedral scenes, and Allen's qualitative approach to temporal logic.Note. A preliminary version of this article appeared as (Apt & Monfroy, 1999). In this version we also present a framework for computing with rules on constraint satisfaction problems and discuss in detail the results of various experiments.
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