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DOI: 10.4230/LIPIcs.FSCD.2024.9

Laplace Distributors and Laplace Transformations for Differential Categories

Authors: Marie Kerjean and Jean-Simon Pacaud Lemay

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

In a differential category and in Differential Linear Logic, the exponential conjunction ! admits structural maps, characterizing quantitative operations and symmetric co-structural maps, characterizing differentiation. In this paper, we introduce the notion of a Laplace distributor, which is an extra structural map which distributes the linear negation operation (_)^∗ over ! and transforms the co-structural rules into the structural rules. Laplace distributors are directly inspired by the well-known Laplace transform, which is all-important in numerical analysis. In the star-autonomous setting, a Laplace distributor induces a natural transformation from ! to the exponential disjunction ?, which we then call a Laplace transformation. According to its semantics, we show that Laplace distributors correspond precisely to the notion of a generalized exponential function e^x on the monoidal unit. We also show that many well-known and important examples have a Laplace distributor/transformation, including (weighted) relations, finiteness spaces, Köthe spaces, and convenient vector spaces.

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Marie Kerjean and Jean-Simon Pacaud Lemay. Laplace Distributors and Laplace Transformations for Differential Categories. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kerjean_et_al:LIPIcs.FSCD.2024.9,
  author =	{Kerjean, Marie and Lemay, Jean-Simon Pacaud},
  title =	{{Laplace Distributors and Laplace Transformations for Differential Categories}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{9:1--9:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.9},
  URN =		{urn:nbn:de:0030-drops-203382},
  doi =		{10.4230/LIPIcs.FSCD.2024.9},
  annote =	{Keywords: Differential Categories, Differential Linear Logic, Laplace Distributor, Laplace Transformation, Exponential Function}
}

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DOI: 10.4230/LIPIcs.FSCD.2023.21

Unifying Graded Linear Logic and Differential Operators

Authors: Flavien Breuvart, Marie Kerjean, and Simon Mirwasser

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

Abstract

Linear Logic refines Classical Logic by taking into account the resources used during the proof and program computation. In the past decades, it has been extended to various frameworks. The most famous are indexed linear logics which can describe the resource management or the complexity analysis of a program. From another perspective, Differential Linear Logic is an extension which allows the linearization of proofs. In this article, we merge these two directions by first defining a differential version of Graded linear logic: this is made by indexing exponential connectives with a monoid of differential operators. We prove that it is equivalent to a graded version of previously defined extension of finitary differential linear logic. We give a denotational model of our logic, based on distribution theory and linear partial differential operators with constant coefficients.

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Flavien Breuvart, Marie Kerjean, and Simon Mirwasser. Unifying Graded Linear Logic and Differential Operators. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{breuvart_et_al:LIPIcs.FSCD.2023.21,
  author =	{Breuvart, Flavien and Kerjean, Marie and Mirwasser, Simon},
  title =	{{Unifying Graded Linear Logic and Differential Operators}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{21:1--21:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.21},
  URN =		{urn:nbn:de:0030-drops-180052},
  doi =		{10.4230/LIPIcs.FSCD.2023.21},
  annote =	{Keywords: Linear Logic, Denotational Semantics, Functional Analysis, Graded Logic}
}

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Documents authored by Kerjean, Marie

Laplace Distributors and Laplace Transformations for Differential Categories

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Unifying Graded Linear Logic and Differential Operators

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