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Laplace Distributors and Laplace Transformations for Differential Categories

Authors Marie Kerjean , Jean-Simon Pacaud Lemay

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Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Theory of computation → Denotational semantics
  • Theory of computation → Linear logic
Keywords
  • Differential Categories
  • Differential Linear Logic
  • Laplace Distributor
  • Laplace Transformation
  • Exponential Function

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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) https://doi.org/10.4230/LIPIcs.FSCD.2024.9

Author Details

Marie Kerjean
  • CNRS, Université Sorbonne Paris Nord, France
Jean-Simon Pacaud Lemay
  • School of Mathematical and Physical Sciences, Macquarie University, Sydney, Australia

Funding

  • Kerjean, Marie: Part of the ANR DIFFERENCE #{A}NR-20-CE48-0002 and the ANR NuSCAP #{A}NR-20-CE48-0014.
  • Lemay, Jean-Simon Pacaud: ARC DECRA (DE230100303) & AFOSR (FA9550-24-1-0008)

Acknowledgements

We are grateful to Yoann Dabrowski for enriching discussions on Laplace and Fourier transformations.

References

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