An Abstract Tableau Calculus for the Description Logic SHOI Using UnrestrictedBlocking and Rewriting

Abstract This paper presents an abstract tableau calculus for the description logic SHOI. SHOI is the extension of ALC with singleton concepts, role inverse, transitive roles and role inclusion axioms. The presented tableau calculus is inspired by a recently introduced tableau synthesis framework. Termination is achieved by a variation of the unrestricted blocking mechanism that immediately rewrites terms with respect to the conjectured equalities. This approach leads to reduced search space for decision procedures based on the calculus. We also discuss restrictions of the application of the blocking rule by means of additional side conditions and/or additional premises.

Positive Unit Hyperresolution Tableaux and Their Application to Minimal Model Generation

Minimal Herbrand models of sets of first-order clauses are useful in several areas of computer science, e.g. automated theorem proving, program verification, logic programming, databases, and artificial intelligence. In most cases, the conventional model generation algorithms are inappropriate because they generate nonminimal Herbrand models and can be inefficient. This article describes an approach for generating the minimal Herbrand models of sets of first-order clauses. The approach builds upon positive unit hyperresolution (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 first one expands PUHR tableaux depth-first 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 given and its performance on benchmark suites is reported. The second minimal model generation procedure performs a breadth-first, 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-first minimal model generation procedure computes all minimal Herbrand models of the considered clauses provided these models are all finite. The breadth-first minimal model generation procedure computes all finite minimal Herbrand models of the set of clauses under consideration. The proposed procedures are compared with related work in terms of both principles and performance on benchmark problems