Interval Constraint Logic Programming

In this paper, we present an overview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate n-ary relations over IR with Cartesian products of intervals whose bounds are taken in a finite subset of IR. Variables represent real values whose domains are intervals defined in the same manner. Narrowing operators are defined from approximations. These operators compute, from an interval and a relation, a set included in the initial interval. Sets of constraints are then processed thanks to a local consistency algorithm pruning at each step values from initial intervals. This algorithm is shown to be correct and to terminate, on the basis of a certain number of properties of narrowing operators. We focus here on the description of the general framework based on approximations, on its application to interval constraint solving over continuous and discrete quantities, we establish a strong ling between approximations and local consistency notions and show that arc-consistency is an instance of the approximation framework. We finally describe recent work on different variants of the initial algorithm proposed by John Cleary and developped by W. Older and A. Vellino which have been proposed in this context. These variants address four particular points: generalization of the constraint language, improvement of domain reductions, efficiency of the computation and finally, cooperation with other solvers. Some open questions are also identified.