Rob Arthan

HOL with Definitions: Semantics, Soundness, and a Verified Implementation

We present a mechanised semantics and soundness proof for the HOL Light kernel including its definitional principles, extending Harrison’s verification of the kernel without definitions. Soundness of the logic extends to soundness of a theorem prover, because we also show that a synthesised implementation of the kernel in CakeML refines the inference system. Our semantics is the first for Wiedijk’s stateless HOL; our implementation, however, is stateful: we give semantics to the stateful inference system by translation to the stateless. We improve on Harrison’s approach by making our model of HOL parametric on the universe of sets. Finally, we prove soundness for an improved principle of constant specification, in the hope of encouraging its adoption. This paper represents the logical kernel aspect of our work on verified HOL implementations; the production of a verified machine-code implementation of the whole system with the kernel as a module will appear separately.

Self-Formalisation of Higher-Order Logic

We present a mechanised semantics for higher-order logic (HOL), and a proof of soundness for the inference system, including the rules for making definitions, implemented by the kernel of the HOL Light theorem prover. Our work extends Harrison’s verification of the inference system without definitions. Soundness of the logic extends to soundness of a theorem prover, because we also show that a synthesised implementation of the kernel in CakeML refines the inference system. Apart from adding support for definitions and synthesising an implementation, we improve on Harrison’s work by making our model of HOL parametric on the universe of sets, and we prove soundness for an improved principle of constant specification in the hope of encouraging its adoption. Our semantics supports defined constants directly via a context, and we find this approach cleaner than our previous work formalising Wiedijk’s Stateless HOL.

A general framework for sound and complete Floyd-Hoare logics

This article presents an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. Our abstraction is based on a traced monoidal functor from an arbitrary traced monoidal category into the category of preorders and monotone relations. We give several examples of how our theory generalizes usual Hoare logics (partial correctness of while programs, partial correctness of pointer programs), and provide some case studies on how it can be used to develop new Hoare logics (runtime analysis of while programs and stream circuits).

HOL Constant Definition Done Right

This note gives a proposal for a simpler and more powerful replacement for the mechanisms currently provided in the various HOL implementations for defining new constants.

A Hoare logic for linear systems

We consider reasoning about linear systems expressed as block diagrams that give a graphical representation of a system of differential equations or recurrence equations. We use the notion of additive relation borrowed from homological algebra to give a convenient framework in which all diagrams have a semantic value. We give a sound system of Hoare-style rules for the block diagram constructors that singles out a tractable subset of the block diagram language in which all diagrams represent total functions. We show these rules in action on some simple examples from a variety of applications domains.

On Definitions of Constants and Types in HOL

This paper reports on a simpler and more powerful replacement for the principles for defining new constants that were previously provided in the various HOL implementations. We discuss the problems that the new principle is intended to solve and sketch the proofs that it is conservative and that it subsumes the earlier definitional principles. The new definitional principle for constants has been implemented in HOL4 and in ProofPower and has been adopted in OpenTheory and in the work of Kumar, Myreen and Owens on a fully verified implementation of HOL. Kumar et al. have formally verified that the new definitional principle is conservative with respect to the standard set theoretic semantics of HOL. We continue this line of thought with a look at the mechanisms for defining new types and consider potential improvements, one of which has now been adopted in OpenTheory.

Dual) hoops have unique halving

Continuous logic extends the multi-valued Łukasiewicz logic by adding a halving operator on propositions. This extension is designed to give a more satisfactory model theory for continuous structures. The semantics of these logics can be given using specialisations of algebraic structures known as hoops and coops. As part of an investigation into the metatheory of propositional continuous logic, we were indebted to Prover9 for finding proofs of important algebraic laws.