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Interval Constraint Logic Programming Frederic Benhamou Laboratoire d'Informatique Fondamentale d'Orleans I.I.I.A., Universite d'Orleans, Rue Leonard de Vinci B.P. 6759 45067 ORLEANS Cedex 2 France
Abstract. 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 nite subset of IR. Variables represent real values whose domains are intervals de ned in the same manner. Narrowing operators are de ned 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 link between approximations and local consistency notions and show that arc-consistency is an instance of the approximation framework. We nally describe recent work on di erent variants of the initial algorithm proposed by John Cleary and developed 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, eciency of the computation and nally, cooperation with other solvers. Some open questions are also identi ed.
1 Introduction The use of intervals in Constraint Logic Programming has been investigated by several authors in the last few years. The eld was pioneered by J. Cleary [12] and the main ideas were extended and implemented by W. Older and A. Vellino [42, 43] in their system BNR-Prolog. A number of systems followed, among which one can cite CLP(BNR)[5, 44], Interlog [28], Newton[6], ICHIP [36], Echidna [46], CIAL [11], ICE [8], PRINCE [9]. All these systems have in common a constraint language based on a relational form of interval arithmetic and the use of a xpoint algorithm to process networks of constraints. As we establish in this paper, this last method is closely related to arc-consistency, a well-known concept in Arti cial Intelligence [39, 38]. This is not the only connection with Arti cial Intelligence, since interval constraint solving has been studied by several authors
to address temporal reasoning ([2]) or Constraint Satisfaction Problems ([29, 25, 34]). As it is developed in this paper, this common theoretical and algorithmic platform addresses a number of issues previously pointed out in Prolog and in CLP. The rst issue concerns numerical computations in Logic programming. The original Prolog way of tackling numbers (pre-de ned relations tested after evaluation of ground numerical terms using oating point operations) was lacking both correctness and relationality. The introduction of the CLP paradigm [30, 13, 32, 3] and the design and implementation of CLP languages such as CLP(
