A Comprehensive Framework for Saturation Theorem Proving

Abstract

A crucial operation of saturation theorem provers is deletion of subsumed formulas. Designers of proof calculi, however, usually discuss this only informally, and the rare formal expositions tend to be clumsy. This is because the equivalence of dynamic and static refutational completeness holds only for derivations where all deleted formulas are redundant, but the standard notion of redundancy is too weak: A clause C does not make an instance $$C\sigma $$ C σ redundant. We present a framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution and superposition. The framework modularly extends redundancy criteria derived via a familiar ground-to-nonground lifting. It allows us to extend redundancy criteria so that they cover subsumption, and also to model entire prover architectures so that the static refutational completeness of a calculus immediately implies the dynamic refutational completeness of a prover implementing the calculus within, for instance, an Otter or DISCOUNT loop. Our framework is mechanized in Isabelle/HOL.

Superposition for Higher-Order Logic

“…The standard redundancy criterion for standard superposition cannot justify subsumption deletion. Following Waldmann et al [140], we incorporate subsumption into the redundancy criterion. A clause C subsumes D if there exists a substitution σ such that Cσ ⊆ D. A clause C strictly subsumes D if C subsumes D but D does not subsume C. Let stand for "is strictly subsumed by".…”

“…3. We use the saturation framework of Waldmann et al [140] to lift the static refutational completeness of GHInf to static and dynamic refutational completeness of HInf .…”

“…In the third step, we employ the saturation framework by Waldmann et al [140], which is largely based on Bachmair and Ganzinger's [10], to prove HInf refutationally complete. Like Bachmair and Ganzinger's, the saturation framework allows us to prove the static and dynamic refutational completeness of our calculus on the nonground level.…”

“…To lift the result to the nonground level, we employ the saturation framework of Waldmann et al [140]. It is easy to see that the entailment relation |= on GH is a consequence relation in the sense of the framework.…”

On Verified Automated Reasoning in Propositional Logic

“…For first-order logic there is leanTaP [1,6]: a small, unverified prover in Prolog. Larger and verified provers for first-order logic also exist [15,16,19,22,24].…”