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We present a formalisation of Aristotle’s Assertoric Syllogistic in the proof assistant (interactive theorem prover) Isabelle/HOL and discuss the insights gained by the formalisation, as we demonstrate that the formal proofs and especially Isabelle’s automation can enhance the user’s understanding and facilitate metatheoretical explorations.

SAT solvers decide the satisfiability of Boolean formulas in conjunctive normal form. They are commonly used for software and hardware verification. Modern SAT solvers are highly complex and optimized programs. As a single bug in the solver may invalidate the verification of many systems, SAT solvers output certificates for their answer, which are then checked independently. However, even certificate checking requires highly optimized non-trivial programs. This paper presents the first SAT solver certificate checker that is formally verified down to the integer sequence representing the formula. Our tool supports the full DRAT standard, and is even faster than the unverified state-of-the-art tool drat-trim, on a realistic set of benchmarks drawn from the 2016 and 2017 SAT competitions. An optional multi-threaded mode further reduces the runtime, in particular for big certificates.

We introduce a single-set axiomatisation of cubical ω-categories, including connections and inverses. We justify these axioms by establishing a series of equivalences between the category of single-set cubical ω-categories, and their variants with connections and inverses, and the corresponding cubical ω-categories. We also report on the formalisation of cubical ω-categories with the Isabelle/HOL proof assistant, which has been instrumental in developing the single-set axiomatisation.

In this article we present an ongoing effort to formalise quantum algorithms and results in quantum information theory using the proof assistant Isabelle/HOL. Formal methods being critical for the safety and security of algorithms and protocols, we foresee their widespread use for quantum computing in the future. We have developed a large library for quantum computing in Isabelle based on a matrix representation for quantum circuits, successfully formalising the no-cloning theorem, quantum teleportation, Deutsch’s algorithm, the Deutsch–Jozsa algorithm and the quantum Prisoner’s Dilemma. We discuss the design choices made and report on an outcome of our work in the field of quantum game theory.

Brzozowski introduced the notion of derivatives for regular expressions. They can be used for a very simple regular expression matching algorithm. Sulzmann and Lu cleverly extended this algorithm in order to deal with POSIX matching, which is the underlying disambiguation strategy for regular expressions needed in lexers. Their algorithm generates POSIX values which encode the information of how a regular expression matches a string—that is, which part of the string is matched by which part of the regular expression. In this paper we give our inductive definition of what a POSIX value is and show that Sulzmann and Lu’s algorithm always generates such a value. We also show that our inductive definition of a POSIX value is equivalent to an alternative definition by Okui and Suzuki which identifies POSIX values as least elements according to an ordering of values.

Weak memory presents a new challenge for program verification and has resulted in the development of a variety of specialised logics. For C11-style memory models, our previous work has shown that it is possible to extend Hoare logic and Owicki–Gries reasoning to verify correctness of weak memory programs. The technique introduces a set of high-level assertions over C11 states together with a set of basic Hoare-style axioms over atomic weak memory statements (e.g. reads/writes), but retains all other standard proof obligations for compound statements. This paper takes this line of work further by introducing the first deductive verification environment in Isabelle/HOL for C11-like weak memory programs. This verification environment is built on the Nipkow and Nieto’s encoding of Owicki–Gries in the Isabelle theorem prover. We exemplify our techniques over several litmus tests from the literature and two non-trivial examples: Peterson’s algorithm and a read–copy–update algorithm adapted for C11. For the examples we consider, the proof outlines can be automatically discharged using the existing Isabelle tactics developed by Nipkow and Nieto. The benefit here is that programs can be written using a familiar pseudocode syntax with assertions embedded directly into the program.

We present a novel approach to modelling and verifying ant colony pathfinding behaviour in an idealised scenario mirroring the double bridge experiment from biological research. We implement our analysis framework in the HOL Light proof assistant to provide rigorous verification of emergent collective dynamics. Unlike most of the existing approaches, which are limited by state explosion or fixed-size constraints, we formally prove that an ant colony of any size converges on the optimal path, given specified preconditions. This establishes that the selection of a shortest path is a stable, emergent property independent of the colony’s population . To enhance the computational performance of the analysis on colonies of a given size, we implement a faithful translation between HOL Light and SMT-LIB2. This bridge allows proof obligations to be discharged efficiently by modern SAT solvers, thereby integrating the expressive power of higher-order logic with the speed of automated reasoning tools, and creating a division of labour between formal verification and dynamic simulation. Our work advances the application of computerised mathematics to collective adaptive systems, providing a unified framework for modelling, simulation, and formal verification.

We describe the design, implementation and verification of an automated theorem prover for first-order logic with functions. The proof search procedure is based on sequent calculus and we formally verify its soundness and completeness in Isabelle/HOL using an existing abstract framework for coinductive proof trees. Our analytic completeness proof covers both open and closed formulas. Since our deterministic prover considers only the subset of terms relevant to proving a given sequent, we do the same when building a countermodel from a failed proof. Finally, we formally connect our prover with the proof system and semantics of the existing SeCaV system. In particular, the prover can generate human-readable SeCaV proofs which are also machine-verifiable proof certificates. The abstract framework we rely on requires us to fix a stream of proof rules in advance, independently of the formula we are trying to prove. We discuss the efficiency implications of this and the difficulties in mitigating them.

Machine-checked proofs of security are important to increase the rigour of provable security. In this work we present a formalised theory of two fundamental two party cryptographic primitives: Σ-protocols and Commitment Schemes. Σ-protocols allow a prover to convince a verifier that they possess some knowledge without leaking information about the knowledge. Commitment schemes allow a committer to commit to a message and keep it secret until revealing it at a later time. We use CryptHOL (Lochbihler in Archive of formal proofs, 2017) to formalise both primitives and prove secure multiple examples namely; the Schnorr, Chaum-Pedersen and Okamoto Σ-protocols as well as a construction that allows for compound (AND and OR) Σ-protocols and the Pedersen and Rivest commitment schemes. A highlight of the work is a formalisation of the construction of commitment schemes from Σ-protocols (Damgard in Lecture notes, 2002). We formalise this proof at an abstract level using the modularity available in Isabelle/HOL and CryptHOL. This way, the proofs of the instantiations come for free.

We describe the library of theorems about N-dimensional Euclidean space that has been formalized in the HOL Light prover. This formalization was started in 2005 and has been extensively developed since then, partly in direct support of the Flyspeck project, partly out of a general desire to develop a well-rounded and comprehensive theory of basic analytical, geometrical and topological machinery. The library includes various ‘big name’ theorems (Brouwer’s fixed point theorem, the Stone-Weierstrass theorem, the Tietze extension theorem), numerous non-trivial results that are useful in applications (second mean value theorem for integrals, power series for real and complex transcendental functions) and a host of supporting definitions and lemmas. It also includes some specialized automated proof tools. The library has as planned been applied to the Flyspeck project and has become the basis of a significant development of results in complex analysis, among others.