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. Refutation 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 refutation completeness. It is often possible to compute the closure of nontrivial 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 one can restrict attention to proofs that have support from the goals without compromising refutation completeness. Additional syntactic properties allow to restrict the search space 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 over built-in Booleans), completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses.

E 2.0 User Manual

“…As such, it implements an instance of the superposition calculus described in [BG94]. We have added some stronger contraction rules and a more general approach to literal selection, and have also extended the calculus to simple, monomorphic many-sorted logic (in the sense of the TPTP-3 TFF format [SSCB12]).…”

“…However, the proof of completeness for the general case seems to be rather involved, as it requires a very different clause ordering than the one introduced [BG94], and we are not aware of any existing proof in the literature. The variant rule allows rewriting of maximal terms of maximal literals under certain circumstances:…”

Embedding Deduction Modulo into a Prover

“…The usual proof of completeness of superposition relies on saturation up to redundancies w.r.t. [27,28]. If we want to adapt this proof directly, we have to require that the one-way clauses corresponding to the rewrite rules are saturated for .…”

Polymorphic+Typeclass Superposition

“…Inferences We take the superposition calculus of [3] and sketch that it still refutationally complete in our typed setting.…”

“…The completeness of typed ground superposition now follows from the completeness proof for untyped superposition given in [3]. To show the lifting, we need to show for each inference that if σ is a grounding substitution, then every inference from Cσ 5 is an ground instance of an inference of C. After that the model construction lemma and the remaining completeness proof can be reused without further changes.…”