Rewrite-based Equational Theorem Proving with Selection and Simplification

Abstract

We present various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses. Refutational completeness is established via the use of well-founded orderings on clauses for defining a Herbrand model for a consistent set of clauses. We also formulate an abstract notion of redundancy and show that the deletion of redundant clauses during the theorem proving process preserves refutational completeness. It is often possible to compute the closure of non-trivial sets of clauses under application of non-redundant inferences. The refutation of goals for such complete sets of clauses is simpler than for arbitrary sets of clauses, in particular it suffices to consider only those proofs that have support from the goals, without compromising refutational completeness. For certain classes of formulas the search space can be restricted even further, as we demonstrate for so-called quasi-Horn clauses. The results in this paper contain as special cases or generalize many known results about Knuth–Bendix-like completion procedures (for equations, Horn clauses, and Horn clauses with Boolean conditions), completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses.