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Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2019.111

Differential Logical Relations, Part I: The Simply-Typed Case (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors: Ugo Dal Lago, Francesco Gavazzo, and Akira Yoshimizu

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract

We introduce a new form of logical relation which, in the spirit of metric relations, allows us to assign each pair of programs a quantity measuring their distance, rather than a boolean value standing for their being equivalent. The novelty of differential logical relations consists in measuring the distance between terms not (necessarily) by a numerical value, but by a mathematical object which somehow reflects the interactive complexity, i.e. the type, of the compared terms. We exemplify this concept in the simply-typed lambda-calculus, and show a form of soundness theorem. We also see how ordinary logical relations and metric relations can be seen as instances of differential logical relations. Finally, we show that differential logical relations can be organised in a cartesian closed category, contrarily to metric relations, which are well-known not to have such a structure, but only that of a monoidal closed category.

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Ugo Dal Lago, Francesco Gavazzo, and Akira Yoshimizu. Differential Logical Relations, Part I: The Simply-Typed Case (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 111:1-111:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dallago_et_al:LIPIcs.ICALP.2019.111,
  author =	{Dal Lago, Ugo and Gavazzo, Francesco and Yoshimizu, Akira},
  title =	{{Differential Logical Relations, Part I: The Simply-Typed Case}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{111:1--111:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.111},
  URN =		{urn:nbn:de:0030-drops-106879},
  doi =		{10.4230/LIPIcs.ICALP.2019.111},
  annote =	{Keywords: Logical Relations, lambda-Calculus, Program Equivalence, Semantics}
}

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DOI: 10.4230/LIPIcs.FSCD.2019.20

A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4

Authors: Yosuke Fukuda and Akira Yoshimizu

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Abstract

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.

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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)

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@InProceedings{fukuda_et_al:LIPIcs.FSCD.2019.20,
  author =	{Fukuda, Yosuke and Yoshimizu, Akira},
  title =	{{A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{20:1--20:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.20},
  URN =		{urn:nbn:de:0030-drops-105271},
  doi =		{10.4230/LIPIcs.FSCD.2019.20},
  annote =	{Keywords: linear logic, modal logic, Girard translation, Curry-Howard correspondence, geometry of interaction, staged computation}
}

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Documents authored by Yoshimizu, Akira

Differential Logical Relations, Part I: The Simply-Typed Case (Track B: Automata, Logic, Semantics, and Theory of Programming)

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A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4

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