Cooperation between Top-Down and Bottom-Up Theorem Provers

TL;DR

This paper explores integrating top-down and bottom-up theorem proving methods to leverage their respective strengths, using cooperation techniques and relevancy filtering, demonstrated through experiments with SETHEO and SPASS.

Contribution

It introduces novel cooperation strategies between top-down and bottom-up provers, enhancing proof efficiency and relevance filtering in theorem proving.

Findings

Cooperation techniques can shorten proofs.

Relevancy filtering improves proof relevance.

Experiments show high potential of combined approaches.

Abstract

Top-down and bottom-up theorem proving approaches each have specific advantages and disadvantages. Bottom-up provers profit from strong redundancy control but suffer from the lack of goal-orientation, whereas top-down provers are goal-oriented but often have weak calculi when their proof lengths are considered. In order to integrate both approaches, we try to achieve cooperation between a top-down and a bottom-up prover in two different ways: The first technique aims at supporting a bottom-up with a top-down prover. A top-down prover generates subgoal clauses, they are then processed by a bottom-up prover. The second technique deals with the use of bottom-up generated lemmas in a top-down prover. We apply our concept to the areas of model elimination and superposition. We discuss the ability of our techniques to shorten proofs as well as to reorder the search space in an appropriate manner. Furthermore, in order to identify subgoal clauses and lemmas which are actually relevant for the proof task, we develop methods for a relevancy-based filtering. Experiments with the provers SETHEO and SPASS performed in the problem library TPTP reveal the high potential of our cooperation approaches.