Positive Unit Hyper-Resolution Tableaux for Minimal Model Generation

Abstract

Minimal Herbrand models for clausal theories are useful in several areas of computer science, e.g. automated theorem proving, program veri cation, logic programming, databases, and arti cial intelligence. In most cases, the conventional model generation algorithms are inappropriate because they generate nonminimal Herbrand models and can be ine cient. This article describes a novel approach for generating the minimal Herbrand models of sets of clauses. The approach builds upon positive unit hyper-resolution (PUHR) tableaux, that are in general smaller than conventional tableaux. PUHR tableaux formalize the approach initially introduced with the theorem prover SATCHMO. Two minimal model generation procedures are described. The rst one expands PUHR tableaux depth- rst relying on a complement splitting expansion rule and on a form of backtracking involving constraints. A Prolog implementation, named MM-SATCHMO, of this procedure is described. The second minimal model generation procedure performs a breadth- rst, constrained expansion of PUHR (complement) tableaux. Both procedures are optimal in the sense that each minimal model is constructed only once, and the construction of nonminimal models is interrupted as soon as possible. They are complete in the following sense: The depth- rst minimal model generation procedure computes all minimal Herbrand models of the considered clauses provided these models are all nite. The breadth- rst minimal model generation procedure computes all nite minimal Herbrand models of the set of clauses under consideration.