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Modal proof theory through a focused telescope

Abstract : In this thesis, we use in two ways the concept of synthetic inference rules that can be obtained from a focused proof system; from one side of the “telescope”, focusing allows us to analyse the internal machinery of inference rules; on the other side, it allows us to consider more global behaviours. In the first part, we review existing proof systems for modal logic, concentrating our efforts around the sequent calculus and its extensions. We underline the issues that drive the modal proof theory community, such as the usual distinction between labelled and unlabelled systems that we aim at deconstructing. We present these questions and concepts in parallel for classical and intuitionistic modal logic in chapter 2 and 3 respectively. We in particular go through Fitting’s indexed nested sequents, for which we demonstrated a new completeness result via cut-elimination. The second part recalls first the notion of focusing and of synthetic inference rules in chapter 4, then presents two of our contributions in chapter 5 and 6. Firstly, we show how to emulate Simpson’s labelled sequent calculus for intuitionistic modal logic with Liang and Miller’s focused sequent calculus for first-order logic, therefore extending the result of Miller and Volpe. Secondly, we propose a similar encoding though for ordinary (unlabelled) sequent calculus via an intermediate focused framework based on Negri’s labelled sequent calculus. The third part reports on two other contributions, namely the completeness proofs of two nested sequent calculi for both classical and intuitionistic modal logic, first a focused version in chapter 7, and then a system merely based on synthetic inference rules in chapter 8. These rules only retain the transitions between big steps of reasoning forgetting most of the focused rules, which renders the system presentation clear and elegant while also simplifying the cut-elimination and completeness proofs.
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Submitted on : Tuesday, December 11, 2018 - 12:29:45 PM
Last modification on : Friday, August 5, 2022 - 12:35:03 PM
Long-term archiving on: : Tuesday, March 12, 2019 - 2:39:59 PM


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  • HAL Id : tel-01951291, version 1


Sonia Marin. Modal proof theory through a focused telescope. Logic in Computer Science [cs.LO]. Université Paris Saclay, 2018. English. ⟨NNT : ⟩. ⟨tel-01951291⟩



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