Superposition with Lambdas

Abstract

We designed a superposition calculus for a clausal fragment of extensional polymorphic higher-order logic that includes anonymous functions but excludes Booleans. The inference rules work on $$\beta \eta $$ β η -equivalence classes of $$\lambda $$ λ -terms and rely on higher-order unification to achieve refutational completeness. We implemented the calculus in the Zipperposition prover and evaluated it on TPTP and Isabelle benchmarks. The results suggest that superposition is a suitable basis for higher-order reasoning.

A Comprehensive Framework for Saturation Theorem Proving

“…We start with a technical lemma: F ) and that the definitions of a saturated set and of static refutational completeness do not depend on the second component of a redundancy criterion. The following lemmas are immediate consequences of these observations: Example 47 For higher-order calculi such as higher-order resolution [25] and λ-superposition [14], the instantiation ordering is not well founded, as witnessed by the chain…”

“…Example 82 Higher-order unification can give rise to infinitely many incomparable unifiers. As a result, in λ-superposition [14], performing all inferences between two clauses can lead to infinitely many conclusions, which need to be enumerated fairly. The Zipperposition prover [14], which implements the calculus, performs this enumeration in an extended DISCOUNT loop.…”

“…A formula is an orphan when it has lost one of its parents to the redundancy criterion [24,39]; removing such formulas reduces the search space. Dovetailing makes it possible to support infinitary inference rules, such as λ-superposition and its possibly infinite sequences of higher-order unifiers [14].…”

“…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].…”

Extending a High-Performance Prover to Higher-Order Logic

“…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.…”