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A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4

Authors Yosuke Fukuda, Akira Yoshimizu

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Yosuke Fukuda
  • Graduate School of Informatics, Kyoto University, Japan
Akira Yoshimizu
  • French Institute for Research in Computer Science and Automation (INRIA), France


The authors thank Kazushige Terui for his helpful comments on our work and pointing out related work on multicolored linear logic, and Atsushi Igarashi for his helpful comments on earlier drafts. Special thanks are also due to anonymous reviewers for their fruitful comments. This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

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Yosuke Fukuda and Akira Yoshimizu. A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 20:1-20:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear logic
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Proof theory
  • Theory of computation → Type theory
  • linear logic
  • modal logic
  • Girard translation
  • Curry-Howard correspondence
  • geometry of interaction
  • staged computation


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