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Pré-Publication, Document De Travail Année : 2024

Cut-elimination for the circular modal mu-calculus: linear logic and super exponentials to the rescue

Résumé

The aim of this paper is twofold: contributing to the non-wellfounded and circular proof-theory of the modal mu-calculus and to that of extensions of linear logic with fixed points. Contrarily to Girard's linear logic which is equipped with a rich proof theory, Kozen's modal mu-calculus has an underdeveloped one: for instance the modal mu-calculus is lacking a proper syntactic cut-elimination theorem. A strategy to prove the cut-elimination theorem for the modal mu-calculus is to prove cut-elimination for a "linear translation" of the modal mu-calculus (that is define a translation of the statements and proofs of the modal mu-calculus into a linear sequent calculus) and to deduce the desired cut-elimination results as corollaries. While designing this linear translation, we come up with a sequent calculus exhibiting several modalities (or exponentials). It happens that the literature of linear logic offers tools to manage such calculi, for instance subexponentials by Nigam and Miller and super\linebreak exponentials by the first author and Laurent. We actually extend super exponentials with fixed-points and non-wellfounded proofs (of which the linear decomposition of the modal mu-calculus is an instance) and prove a syntactic cut-elimination theorem for these sequent calculi in the framework of non-wellfounded proof theory. Back to the classical modal mu-calculus, we deduce its cut-elimination theorem from the above results, investigating both non-wellfounded and regular proofs.
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hal-04496648 , version 1 (08-03-2024)

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  • HAL Id : hal-04496648 , version 1

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Esaïe Bauer, Alexis Saurin. Cut-elimination for the circular modal mu-calculus: linear logic and super exponentials to the rescue. 2024. ⟨hal-04496648⟩
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