Ground Joinability and Connectedness in the Superposition Calculus

Abstract

Problems in many theories axiomatised by unit equalities (UEQ), such as groups, loops, lattices, and other algebraic structures, are notoriously difficult for automated theorem provers to solve. Consequently, there has been considerable effort over decades in developing techniques to handle these theories, notably in the context of Knuth-Bendix completion and derivatives. The superposition calculus is a generalisation of completion to full first-order logic; however it does not carry over all the refinements that were developed for it, and is therefore not a strict generalisation. This means that (i) as of today, even state of the art provers for first-order logic based on the superposition calculus, while more general, are outperformed in UEQ by provers based on completion, and (ii) the sophisticated techniques developed for completion are not available in any problem which is not in UEQ. %In particular, this includes key simplifications such as ground joinability, which have been known for more than 30 years. In fact, all previous completeness proofs for ground joinability rely on proof orderings and proof reductions, which are not easily extensible to general clauses together with redundancy elimination. In this paper we address this limitation and extend superposition with ground joinability, and show that under an adapted notion of redundancy, simplifications based on ground joinability preserve completeness. Another recently explored simplification in completion is connectedness. We extend this notion to ``ground connectedness'' and show superposition is complete with both connectedness and ground connectedness. %We implemented ground joinability and connectedness in a theorem prover, iProver, the former using a novel algorithm which we also present in this paper, and evaluated over the TPTP library with encouraging results.

Rewriting and Inductive Reasoning

“…Bold entries show the maximum number of solved benchmarks among each subgroup. demodulation [15]; and -thsq on to control pure theory derivations [19]. Experiments were run on computers with AMD Epyc 7502 2.5GHz processors and 1TB RAM, with each individual benchmark run given a single core.…”