Superposition for Full Higher-order Logic

Abstract

We recently designed two calculi as stepping stones towards superposition for full higher-order logic: Boolean-free $$\lambda $$ λ -superposition and superposition for first-order logic with interpreted Booleans. Stepping on these stones, we finally reach a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics. In addition to the complexity of combining the calculus’s two predecessors, new challenges arise from the interplay between $$\lambda $$ λ -terms and Booleans. Our implementation in Zipperposition outperforms all other higher-order theorem provers and is on a par with an earlier, pragmatic prototype of Booleans in Zipperposition.

Extending a High-Performance Prover to Higher-Order Logic

“…The λ-superposition calculus is parameterized by a term order that is used to break symmetries in the search space. We implemented the versions of the Knuth-Bendix order (KBO) and lexicographic path order (LPO) for higher-order terms described by Bentkamp et al [4]. These orders encode λ-terms as first-order terms and then invoke the standard KBO or LPO.…”

“…The λ-superposition calculus for full higher-order logic [4] includes many rules that act on Boolean subterms, which are necessary for completeness. Other than Boolean simplification rules, which use simple tautologies such as p ∧ ∧ ∧ ↔ ↔ ↔ p to simplify terms, we have implemented none of the Boolean rules of this calculus in λE.…”

“…In 2019, we extended the state-of-the-art first-order prover E [32] with a λ-free superposition calculus [42], obtaining a version of E called Ehoh, as a stepping stone towards full higher-order logic. Together with Bentkamp, Tourret, and Waldmann, we have since developed calculi, called λ-superposition, corresponding to the other two milestones [5,4] and implemented them in the experimental superposition prover Zipperposition [14]. This OCaml prover is not nearly as efficient as E. Nevertheless, it has won the higher-order division of the CASC prover competition [39] in 2020, 2021, and 2022, ending nearly a decade of Satallax domination.…”

SAT-Inspired Higher-Order Eliminations

“…The more clausal structure it produces, the more effective the elimination techniques can be. Examples of provers compatible with the techniques are λE [VBS23], Leo-III [SB18], Vampire [BR20], and Zipperposition [BBTV21].…”

A Comprehensive Framework for Saturation Theorem Proving

“…When users of the framework instantiate these prover architectures with a concrete saturation calculus, they obtain the dynamic refutational completeness of the combination from the properties of the prover architecture and the static refutational completeness proof for the calculus. The framework is applicable to a wide range of calculi, including ordered resolution [6], unfailing completion [2], standard superposition [5], constraint superposition [29], theory superposition [44], and hierarchic superposition [8], It is already used in several published and ongoing works on combinatory superposition [15], λ-free superposition [12], λ-superposition [13,14], superposition with interpreted Booleans [33], AVATAR-style splitting [21], and superposition with SAT-inspired inprocessing [43].…”