Algorithmic Aspects of Herbrand Models Represented by Ground Atoms with Ground Equations

Abstract

Automated model building has evolved as an important sub-discipline of automated deduction over the past decade. One crucial issue in automated model building is the selection of an appropriate (finite) representation of (in general infinite) models. Quite a few such formalisms have been proposed in the literature. In this paper, we concentrate on the representation of Herbrand models by ground atoms with ground equations (GAE-models), introduced in [9]. For the actual work with any model representation, efficient algorithms for two decision problems are required, namely: The clause evaluation problem (i.e.: Given a clause C and a representation $\mathcal{M}$ of a model, does C evaluate to “true” in this model?) and the model equivalence problem (i.e.: Given two representations $\mathcal{M}_1$ and $\mathcal{M}_2$, do they represent the same model?). Previously published algorithms for these two problems in case of GAE-models require exponential time. We prove that the clause evaluation problem is indeed intractable (that is, coNP-complete), whereas the model equivalence problem can be solved in polynomial time. Moreover, we show how our new algorithm for the model equivalence problem can be used to transform an arbitrary GAE-model into an equivalent one with better computational properties.