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We propose a "modal linear logic" to reformulate intuitionistic modal logicS4 (IS4) in terms of linear logic, establishing an S4-version of Girardtranslation from IS4 to it. While the Girard translation from intuitionisticlogic to linear logic is well-known, its extension to modal logic isnon-trivial since a naive combination of the S4 modality and the exponentialmodality causes an undesirable interaction between the two modalities. To solvethe problem, we introduce an extension of intuitionistic multiplicativeexponential linear logic with a modality combining the S4 modality and theexponential modality, and show that it admits a sound translation from IS4.Through the Curry-Howard correspondence we further obtain a Geometry ofInteraction Machine semantics of the modal lambda-calculus by Pfenning andDavies for staged computation.

A Linear-logical Reconstruction of Intuitionistic Modal Logic S4