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arXiv · 2304.12094

Intuitionistic S4 is decidable

Marianna Girlando·Roman Kuznets·Sonia Marin·Marianela Morales·Lutz Straßburger
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Abstract

In this paper we demonstrate decidability for the intuitionistic modal logic S4 first formulated by Fischer Servi. This solves a problem that has been open for almost thirty years since it had been posed in Simpson's PhD thesis in 1994. We obtain this result by performing proof search in a labelled deductive system that, instead of using only one binary relation on the labels, employs two: one corresponding to the accessibility relation of modal logic and the other corresponding to the order relation of intuitionistic Kripke frames. Our search algorithm outputs either a proof or a finite counter-model, thus, additionally establishing the finite model property for intuitionistic S4, which has been another long-standing open problem in the area.

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Marianna Girlando, Roman Kuznets, Sonia Marin, Marianela Morales, Lutz Straßburger. 2023-04-24. Intuitionistic S4 is decidable. https://doi.org/10.1109/lics56636.2023.10175684

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