Curry-Howard-Lambek Correspondence for Intuitionistic Belief

This paper introduces a natural deduction calculus for intuitionistic logic of belief $\mathsf{IEL}^{-}$ which is easily turned into a modal $\lambda$-calculus giving a computational semantics for deductions in $\mathsf{IEL}^{-}$. By using that interpretation, it is also proved that $\mathsf{IEL}^{-}$ has good proof-theoretic properties. The correspondence between deductions and typed terms is then extended to a categorical semantics for identity of proofs in $\mathsf{IEL}^{-}$ showing the general structure of such a modality for belief in an intuitionistic framework.Comment: Submitted to Studia Logica on January 31st, 202

A formal proof of modal completeness for provability logic

This work presents a formalized proof of modal completeness for G\"odel-L\"ob provability logic (GL) in the HOL Light theorem prover. We describe the code we developed, and discuss some details of our implementation, focusing on our choices in structuring proofs which make essential use of the tools of HOL Light and which differ in part from the standard strategies found in main textbooks covering the topic in an informal setting. Moreover, we propose a reflection on our own experience in using this specific theorem prover for this formalization task, with an analysis of pros and cons of reasoning within and about the formal system for GL we implemented in our code

Investigations of proof theory and automated reasoning for non-classical logics

This thesis presents some new results in structural proof theory for modal, intuitionistic, and intuitionistic modal logics. The first part introduces three original Gentzen-style natural deduction calculi for, respectively, intuitionistic verification-based epistemic states -- namely, belief and knowledge operators -- and intuitionistic strong L\"ob logic for arithmetical provability. For each of these calculi strong normalisation results are proven w.r.t. several systems of proof rewritings, which are considered on the basis of their structural relevance, e.g.\ for establishing the related subformula principles, or for providing a categorical semantics of normal deductions. The presentation of new and original sequent calculi for a wide family of interpretability logics closes this first part of the thesis. These sequent systems are modularly designed by recurring to internalisation techniques which make possible their fine grained structural analysis, this way establishing both their semantic and structural completeness. The second part has a more applicative nature. It presents first an implementation in the HOL Light proof assistant of an internal theorem prover and countermodel constructor for G\"odel-L\"ob logic, relying on a previous computerised proof of modal completeness for that logic within the same formal environment. The design of that proof search algorithm is surveyed, and examples of both its interactive and automated use are shown. An overview of an ongoing automation-oriented implementation in UniMath of the basics of univalent universal algebra closes this second part of the thesis. The coding style and methodology used are discussed besides some concrete formalisation examples of algebraic structures. Finally, two appendices describe the logical engine underlying each of the proof assistants that are used for the results presented in the second part, namely classical higher order logic for HOL Light, and univalent type theory for UniMath