Extending SMT Solvers to Higher-Order Logic

Abstract

SMT solvers have throughout the years been able to cope with increasingly expressive formulas, from ground logics to full first-order logic (FOL). In contrast, the extension of SMT solvers to higher-order logic (HOL) is mostly unexplored. We propose a pragmatic extension for SMT solvers to support HOL reasoning natively without compromising performance on FOL reasoning, thus leveraging the extensive research and implementation efforts dedicated to efficient SMT solving. We show how to generalize data structures and the ground decision procedure to support partial applications and extensionality, as well as how to reconcile quantifier instantiation techniques with higher-order variables. We also discuss a separate approach for redesigning an HOL SMT solver from the ground up via new data structures and algorithms. We apply our pragmatic extension to the CVC4 SMT solver and discuss a redesign of the veriT SMT solver. Our evaluation shows they are competitive with state-of-the-art HOL provers and often outperform the traditional encoding into FOL.

Extending a brainiac prover to lambda-free higher-order logic

“…Zipperposition now includes Ehoh as a backend in a cooperative architecture. Finally, there is recent work by the developers of Vampire [14] and of the SMT (satisfiability modulo theories) solvers CVC4 and veriT [6] to extend their provers to higher-order logic. Native higher-order reasoning was pioneered by Robinson [39], Andrews [1], and Huet [24].…”

“…To ascertain the effectiveness of our approach, we evaluated Ehoh against E used on applicative encodings of problems (denoted by @+E). For reference, we also evaluated the latest versions of higher-order provers that competed 2 https://doi.org/10.5281/zenodo.4045452 in the THF division of the 2019 edition of CASC [52]: CVC4 1.8 prerelease [6], Leo-III 1.4 [46], Satallax 3.4 [18], Vampire 4.4 [14], and Zipperposition 1.6 [9]. Like at CASC, we used different versions of Vampire for first-order and higher-order problems.…”

Reliable Reconstruction of Fine-grained Proofs in a Proof Assistant

“…We hope to be able to reuse the current reconstruction code by only adding support for CVC4-specific rules. Generating and reconstructing proofs from the veriT version with higher-order logic [9] could also improve the usefulness of veriT on Isabelle problems. The current proof rules [40] should accommodate the more expressive logic.…”

Superposition with Lambdas

“…We selected all contenders in the THF division of CASC 2019 as representatives of the state of the art: CVC4 1.8 prerelease [9], Leo-III 1.4 [62], Satallax 3.4 [24], and Vampire 4.4 [18]. We also included Ehoh [67], the λ-free clausal higher-order mode of E 2.4.…”

“…AgsyHOL [50] is based on a focused sequent calculus guided by narrowing. The SMT solvers CVC4 and veriT have recently been extended to higher-order logic [9]. Vampire now implements both combinatory superposition and a version of stan-dard superposition with first-order unification replaced by restricted combinatory unification [18].…”