Partial Redundancy in Saturation

TL;DR

Partial redundancy generalizes standard redundancy by attaching redundancy formulas to clauses, enabling pruning of ground instances that satisfy $R$ via partial clauses like $C\\setminus R$. The PaRC calculus formalizes this idea and proves soundness and refutational completeness, while also generalizing demodulation. It is implemented in the Vampire prover and demonstrates practical impact by solving 24 TPTP problems previously unsolved. This work provides a new framework for redundancy criteria in first-order reasoning with equality, with potential extensions to theories and higher-order logic.

Abstract

Redundancy elimination is one of the crucial ingredients of efficient saturation-based proof search. We improve redundancy elimination by introducing a new notion of redundancy, based on partial clauses and redundancy formulas, which is more powerful than the standard notion: there are both clauses and inferences that are redundant when we use our notions and not redundant when we use standard notions. In a way, our notion blurs the distinction between redundancy at the level of inferences and redundancy at the level of clauses. We present a superposition calculus PaRC on partial clauses. Our calculus is refutationally complete and is strong enough to capture some standard restrictions of the superposition calculus. We discuss the implementation of the calculus in the theorem prover Vampire. Our experiments show the power of the new approach: we were able to solve 24 TPTP problems not previously solved by any prover, including previous versions of Vampire.