A Comprehensive Framework for Saturation Theorem Proving

Abstract

We present a framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution or superposition. The framework relies on modular extensions of lifted redundancy criteria. It allows us to extend redundancy criteria so that they cover subsumption, and also to model entire prover architectures in such a way that the static refutational completeness of a calculus immediately implies the dynamic refutational completeness of a prover implementing the calculus, for instance within an Otter or DISCOUNT loop. Our framework is mechanized in Isabelle/HOL.

Superposition with Lambdas

“…We can view the above loop as an instance of the abstract Zipperposition loop prover ZL of Waldmann et al [69,Example 34]. Their Theorem 32 allows us to obtain dynamic completeness for this prover architecture from our static completeness result (Theorem 54).…”

“…To lift the result to the nonground level, we employ the saturation framework of Waldmann et al [69]. It is easy to see that the entailment relation |= on GH is a consequence relation in the sense of the framework.…”

“…4). Our proof employs the new saturation framework by Waldmann et al [69] to derive dynamic completeness of a given clause prover from ground static completeness.…”

A Unifying Splitting Framework

“…Our framework is parameterized by abstract notions of formulas, consequence relations, inferences, and redundancy. We largely follow the conventions of Waldmann et al [23]. A-formulas generalize Voronkov's A-clauses [22].…”

“…Before we can answer this open question, we must mathematize splitting. Our starting point is the saturation framework by Waldmann, Tourret, Robillard, and Blanchette [23], based on Bachmair and Ganzinger [2]. It covers a wide array of techniques, but "the main missing piece of the framework is a generic treatment of clause splitting" [23, p. 332].…”