Analytic constraint solving and interval arithmetic

Abstract

In this paper we describe the syntax, semantics, and implementation of the constraint logic programming language CLP(F) and we prove that the implementation is sound. A CLP(F) constraint is a conjunction of equations and inequations in a first order theory of analytic univariate functions over the reals. The theory allows vector-valued functions over closed intervals to be constrained in several ways, including specifying functional equations (possibly involving the differentiation operator) that must hold at each point in the domain, arithmetic constraints on the value of a function at a particular point in its domain, and bounds on the range of a function over its domain. After describing the syntax and semantics of the constraint language for CLP(F) and giving several examples, we show how to convert these analytic constraints into a subclass of simpler functional constraints which involve neither differentiation nor evaluation of functions. We then present an algorithm for solving these latter constraints and prove that it is sound. This implies the soundness of the CLP(F) interpreter. We also provide some timing results from an implementation of CLP(F) based on GNU Prolog. The current implementation is able to solve a wide variety of analytic constraints, but on particular classes of constraints (such as initial value problems for autonomous ODEs), it is not competitive with other non-constraint based, interval solvers such as Lohner's AWA system. CLP(F) should be viewed as a first step toward the long term goal of developing a practical, declarative, logic-based approach to numerical analysis.