Combining local consistency, symbolic rewriting and interval methods

Abstract

This paper is an attempt to address the processing of non-linear numerical constraints over the Reals by combining three different methods: local consistency techniques, symbolic rewriting and interval methods. To formalize this combination, we define a generic two-step constraint processing technique based on an extension of the Constraint Satisfaction Problem, called Extended Constraint Satisfaction Problem (ECSP). The first step is a rewriting step, in which the initial ECSP is symbolically transformed. The second step, called approximation step, is based on a local consistency notion, called weak arc-consistency, defined over ECSPs in terms of fixed point of contractant monotone operators. This notion is shown to generalize previous local consistency concepts defined over finite domains (arc-consistency) or infinite subsets of the Reals (arc B-consistency and interval, hull and box-consistency). A filtering algorithm, derived from AC-3, is given and is shown to be correct, confluent and to terminate. This framework is illustrated by the combination of Gröbner Bases computations and Interval Newton methods. The computation of Gröbner Bases for subsets of the initial set of constraints is used as a rewriting step and operators based on Interval Newton methods are used together with enumeration techniques to achieve weak arc-consistency on the modified ECSP. Experimental results from a prototype are presented, as well as comparisons with other systems.KeywordsConstraint Satisfaction Problemlocal consistencyarc-consistencyfiltering algorithmsnon-linear constraint solvingGröbner basesNewton methodsinterval arithmeticinterval constraints