Solving Mixed Quantified Constraints over a Domain Based on $\mathcal{R}$eal Numbers and $\mathcal{H}$erbrand Terms

Abstract

Combining the logic of hereditary Harrop formulas HH with a constraint system, a logic programming language is obtained that extends Horn clauses in two different directions, thus enhancing substantially the expressivity of Prolog. The implementation of this new language requires the ability to test the satisfiability of constraints built up by means of terms and predicates belonging to the domain of the chosen constraint system, and by the connectives and quantifiers usual in first-order logic. In this paper we present a constraint system called $\mathcal{R}\mathcal{H}$ for a hybrid domain that mixes Herbrand terms and real numbers. It arises when joining the axiomatization of the arithmetic of real numbers and the axiomatization of the algebra of finite trees. We have defined an algorithm to solve certain constraints of this kind. The novelty relies on the combination of two different mechanisms, based on elimination of quantifiers, o ne used for solving unification and disunification problems, the other used to solve polynomials. This combination provides a procedure to solve $\mathcal{R}\mathcal{H}$-constraints in the context of HH with constraints.