Periodic Reporting for period 1 - Nekoka (Realizing the Promise of Higher-Order SMT and Superposition for Interactive Verification)
Reporting period: 2023-07-01 to 2025-12-31
Proof assistants (also called interactive theorem provers) have a long history of being very tedious to use. The situation has improved markedly in the past decade with the integration of first-order automatic theorem provers as backends. And recently, there have been exciting developments for more expressive logics, with the emergence of automatic provers based on optimized higher-order calculi. The Nekoka project's aim is to make two theorem proving technologies, higher-order SMT and lambda-superposition, a good fit for logical problems emerging from the verification of software and mathematics.
The Nekoka project pursues three main scientific objectives. In the first two years of the project, work has commenced toward achieving all three objectives. The first objective is to extend higher-order SMT and lambda-superposition to make them more suitable as push-button backends to interactive verification platforms. The second objective is about integration in interactive verification platforms. The third objective is to carry out case studies about quantum information and a big data framework.
In terms of scientific impact, the improved higher-order SMT and lambda-superposition are expected to advance the art of higher-order automation and help reorient research in automated reasoning towards the needs of end users, whether computer scientists or mathematicians. Our tools will outlive the project, serving end users and continuing to be useful for future research. At the societal level, the project will herald a future in which automatic provers and proof assistants are routinely deployed in tandem to verify critical computing infrastructure and to formalize research in computer science and mathematics, thereby leading to more trustworthy software and science.
The Nekoka project pursues three main scientific objectives. In the first two years of the project, work has commenced toward achieving all three objectives. The first objective is to extend higher-order SMT and lambda-superposition to make them more suitable as push-button backends to interactive verification platforms. The second objective is about integration in interactive verification platforms. The third objective is to carry out case studies about quantum information and a big data framework.
In terms of scientific impact, the improved higher-order SMT and lambda-superposition are expected to advance the art of higher-order automation and help reorient research in automated reasoning towards the needs of end users, whether computer scientists or mathematicians. Our tools will outlive the project, serving end users and continuing to be useful for future research. At the societal level, the project will herald a future in which automatic provers and proof assistants are routinely deployed in tandem to verify critical computing infrastructure and to formalize research in computer science and mathematics, thereby leading to more trustworthy software and science.
In the first two years of the project, we made progress on all three objectives.
The first objective is concerned with improving higher-order SMT and lambda-superposition.
First, for higher-order SMT, we designed a new quantifier instantiation strategy, MBQI-Enum, based on a combination of models and grammars, which is now the best strategy on a vast benchmark suite, thereby improving on the state of the art. This work was published at the highly competitive TACAS 2025 conference. In ongoing follow-up work that is submitted for review, we extended the strategy to come up with Hilbert choice instantiations, further increasing the success rate.
Second, for lambda-superposition, we continued work started in 2021 as part of my ERC starting grant project Matryoshka and currently have an almost finished 120 page draft of the design and correctness proof of a new “optimistic” version of the lambda-superposition calculus, which addresses the three weaknesses of old, “pessimistic” lambda-superposition head on: imprecise term order, explosive function extensionality, and explosive higher-order unification. We expect this work to be submitted for publication by the end of 2025.
Third, for both higher-order SMT and lambda-superposition, we designed a technique called iterative monomorphization that transforms problems expressed in an expressive, so-called polymorphic logical language to problems in a less expressive, so-called monomorphic language.
Objective 2 is about integration in interactive verification platforms. We have developed a new built-in lambda-superposition prover, called slam, for the Isabelle/HOL proof assistant, increasing both the success rate of proof automation and its trustworthiness. This work is under review. We have started work on integrating external proofs by induction, as developed by other teams. We have developed a technique to extract lemma instantiations from superposition proofs (published at CADE 2025). And we have developed a new architecture for “hammer” tools that does not rely on automated theorem provers (published at ITP 2025).
Objective 3 is about case studies. This part will become more important later in the project. For the moment, we can point to a smaller (but nonetheless sizable) case study: the formal verification of the superposition calculus (published at ITP 2024).
The first objective is concerned with improving higher-order SMT and lambda-superposition.
First, for higher-order SMT, we designed a new quantifier instantiation strategy, MBQI-Enum, based on a combination of models and grammars, which is now the best strategy on a vast benchmark suite, thereby improving on the state of the art. This work was published at the highly competitive TACAS 2025 conference. In ongoing follow-up work that is submitted for review, we extended the strategy to come up with Hilbert choice instantiations, further increasing the success rate.
Second, for lambda-superposition, we continued work started in 2021 as part of my ERC starting grant project Matryoshka and currently have an almost finished 120 page draft of the design and correctness proof of a new “optimistic” version of the lambda-superposition calculus, which addresses the three weaknesses of old, “pessimistic” lambda-superposition head on: imprecise term order, explosive function extensionality, and explosive higher-order unification. We expect this work to be submitted for publication by the end of 2025.
Third, for both higher-order SMT and lambda-superposition, we designed a technique called iterative monomorphization that transforms problems expressed in an expressive, so-called polymorphic logical language to problems in a less expressive, so-called monomorphic language.
Objective 2 is about integration in interactive verification platforms. We have developed a new built-in lambda-superposition prover, called slam, for the Isabelle/HOL proof assistant, increasing both the success rate of proof automation and its trustworthiness. This work is under review. We have started work on integrating external proofs by induction, as developed by other teams. We have developed a technique to extract lemma instantiations from superposition proofs (published at CADE 2025). And we have developed a new architecture for “hammer” tools that does not rely on automated theorem provers (published at ITP 2025).
Objective 3 is about case studies. This part will become more important later in the project. For the moment, we can point to a smaller (but nonetheless sizable) case study: the formal verification of the superposition calculus (published at ITP 2024).
The project is still at an early stage, but much of our work is already integrated in existing theorem proving tools and benefits users. This is the case on MBQI-Enum, which is part of the standard version of the state-of-the-art cvc5 SMT solver, as well as the new "hammer" architecture, which is part of standard Isabelle. Our Isabelle formal proofs are part of the online Archive of Formal Proofs.