Interval Constraint Logic Programming

In this paper, we present a n o verview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate n-ary relations over R I with Cartesian products of intervals whose bounds are taken in a nite subset of R I. V ariables represent real values whose domains are intervals deened in the same manner. Narrowing operators are deened from approximations. These operators compute, from an interval and a relation, a s e t included in the initial interval. Sets of constraints are then processed thanks to a local consistency algorithm pruning at each s t e p v alues 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 link between approximations and local consistency notions and show that arc-consistency is an instance of the approximation framework. We nally describe recent w ork on diierent v ariants of the initial algorithm proposed by John Cleary and developed by W. Older and A. Vellino which h a ve been proposed in this context. These variants address four particular points: generalization of the constraint language, improvement of domain reductions, eeciency of the computation and nally, cooperation with other solvers. Some open questions are also identiied.