Interval Constraints

Interval constraint processing is an alternative technology designed to process sets of (generally non-linear) continuous or mixed constraints over the real numbers. It associates propagation and search techniques developed in artiicial intelligence and methods from interval analysis. Interval constraints are used in the design of the constraint solving and optimization engines of most modern constraint programming languages and have been used to solve industrial applications in areas like mechanical design, chemistry, aeronautics, medical diagnosis or image synthesis. The term interval constraint is a generic term denoting a constraint (that is a rst order atomic formula such as an equation, inequation or more generally a relation) in which variables are associated with intervals. These intervals denote domains of possible values for these variables. In general, intervals are deened over the real numbers but the concept is general enough to address other constraint domains (e.g. non negative integers, Booleans, lists, sets, etc.). When deened over the real numbers, interval constraint sets are often called continuous constraint satisfaction problem (CSP) or numerical constraint satisfaction problems. The main idea underlying interval constraint processing | also called interval propagation | is, given a set of constraints S involving variables fv 1 ; : : : ; v n g and a set of oating-point intervals fI 1 ; : : : ; I n g representing the variables' domains of possible values, to isolate a set of fng-ary canonical boxes (Carte-sian products of I i s sub-intervals whose bounds are either equal or consecutive oating-point numbers) approximating the constraint system solution space. To compute such a set, a search procedure navigates through the Cartesian product I 1 : : : I n alternating pruning and branching steps. The pruning step uses a relational form of interval arithmetic Moo66], AH83]. Given a set of constraints over the reals, interval arithmetic is used to compute local approximations of the solution space for a given constraint. This approximation results in the elimination of values from the variables' domains 1