Finite Herbrand Models for Restricted First-Order Clauses

Abstract

We call a Herbrand model of a set of first-order clauses finite, if each of the predicates in the clauses is interpreted by a finite set of ground terms. We consider first-order clauses with the signature restricted to unary predicate and function symbols and one variable. Deciding the existence of a finite Herbrand model for a set of such clauses is known to be ExpTime-hard even when clauses are restricted to an anti-Horn form. Here we present an ExpTime algorithm to decide if a finite Herbrand model exists in the more general case of arbitrary clauses. Moreover, we describe a way to generate finite Herbrand models, if they exist. Since there can be infinitely many minimal finite Herbrand models, we propose a new notion of acyclic Herbrand models. If there is a finite Herbrand model for a set of restricted clauses, then there are finitely many (at most triple-exponentially many) acyclic Herbrand models. We show how to generate all of them.