Partial Regularization of First-Order Resolution Proofs

TL;DR

This paper extends a proof compression technique called partial regularization from propositional to first-order logic, improving the conciseness of automated theorem prover proofs through a generalized algorithm and empirical evaluation.

Contribution

It introduces a generalized partial regularization algorithm for first-order resolution proofs, adapting propositional techniques to first-order logic and evaluating its effectiveness.

Findings

The generalized algorithm effectively compresses first-order resolution proofs.

Combining partial regularization with existing methods yields better proof compression.

Empirical results demonstrate improved proof conciseness in theorem proving.

Abstract

Resolution and superposition are common techniques which have seen widespread use with propositional and first-order logic in modern theorem provers. In these cases, resolution proof production is a key feature of such tools; however, the proofs that they produce are not necessarily as concise as possible. For propositional resolution proofs, there are a wide variety of proof compression techniques. There are fewer techniques for compressing first-order resolution proofs generated by automated theorem provers. This paper describes an approach to compressing first-order logic proofs based on lifting proof compression ideas used in propositional logic to first-order logic. One method for propositional proof compression is partial regularization, which removes an inference $\eta$ when it is redundant in the sense that its pivot literal already occurs as the pivot of another inference in every path from $\eta$ to the root of the proof. This paper describes the generalization of the partial-regularization algorithm RecyclePivotsWithIntersection [10] from propositional logic to first-order logic. The generalized algorithm performs partial regularization of resolution proofs containing resolution and factoring inferences with unification. An empirical evaluation of the generalized algorithm and its combinations with the previously lifted GreedyLinearFirstOrderLowerUnits algorithm [12] is also presented