Completion of first-order clauses with equality by strict superposition

Abstract

We have previously shown that strict superposition together with merging paramodulation is refutationally complete for first-order clauses with equality. This paper improves these results by considering a more powerful framework for simplification and elimination of clauses. The framework gives general criteria under which simplification and elimination do not destroy the refutation completeness of the superposition calculus. One application is a proof of the refutation completeness for alternative superposition strategies with arbitrary selection functions for negative literals. With these powerful simplification mechanisms it is often possible to compute the closure of nontrivial sets of clauses under superposition in a finite number of steps. Refutation or solving of goals for such closed or complete sets of clauses is simpler than for arbitrary sets of clauses. The results in this paper contain as special cases or generalize many known results about about ordered Knuth-Bendix-like completion of equations, of Horn clauses, of Horn clauses over built-in Booleans, about completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses.