The logic of satisfaction constraint

Mackworth, A.K., The logic of constraint satisfaction, Artificial Intelligence 58 (1992) 3-20. The constraint satisfaction problem (CSP) formalization has been a productive tool within Artificial Intelligence and related areas. The finite CSP (FCSP) framework is presented here as a restricted logical calculus within a space of logical representation and reasoning systems. FCSP is formulated in a variety of logical settings: theorem proving in first order predicate calculus, propositional theorem proving (and hence SAT), the Prolog and Datalog approaches, constraint network algorithms, a logical interpreter for networks of constraints, the constraint logic programming (CLP) paradigm and propositional model finding (and hence SAT, again). Several standard, and some not-so-standard, logical methods can therefore be used to solve these problems. By doing this we obtain a specification of the semantics of the common approaches. This synthetic treatment also allows algorithms and results from these disparate areas to be imported, and specialized, to FCSP; the special properties of FCSP are exploited to achieve, for example, completeness and to improve efficiency. It also allows export to the related areas. By casting CSP both as a generalization of FCSP and as a specialization of CLP it is observed that some, but not all, FCSP techniques lift to CSP and thereby to CLP. Various new connections are uncovered, in particular between the proof-finding approaches and the alternative model-finding approaches that have arisen in depiction and diagnosis applications. 1. Logical frameworks for constraint satisfaction Informally, a constraint satisfaction problem (CSP) is posed as follows. Given a set of variables and a set of constraints, each specifying a relation on a particular subset of the variables, find the relation on the set of all the variables which satisfies all the given constraints. Typically, the given unary relation for each variable specifies its domain as a set of possible values; the required solution relation is a subset of the Cartesian product of the variable domains. If Correspondence to: A.K. Mackworth, Department of Computer Science, University of British Columbia, Vancouver, BC, V6T 1W5 * Shell Canada Fellow, Canadian Institute for Advanced Research. 0004-3702/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved