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Laplace Distributors and Laplace Transformations for Differential Categories

Authors Marie Kerjean , Jean-Simon Pacaud Lemay

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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.

Cite As Get BibTex

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

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.

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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References

  1. R. Blute, T. Ehrhard, and C. Tasson. A convenient differential category. Cahiers de topologie et géométrie différentielle catégoriques, 2012. Google Scholar
  2. R. F. Blute, J. R. B. Cockett, J.-S. P. Lemay, and R. A. G. Seely. Differential categories revisited. Applied Categorical Structures, 2020. Google Scholar
  3. R. F. Blute, J. R. B. Cockett, and R. A. G. Seely. Differential categories. Math. Struct. Comput. Sci., 2006. Google Scholar
  4. F. Breuvart, M. Kerjean, and S. Mirwasser. Unifying graded linear logic and differential operators. In 8th International Conference on Formal Structures for Computation and Deduction, FSCD, 2023. Google Scholar
  5. J. R. B. Cockett and R. A. G. Seely. Weakly distributive categories. Journal of Pure and Applied Algebra, 114(2), 1997. Google Scholar
  6. J. R. B. Cockett and R.A.G. Seely. Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories. Theory and Applications of categories, 3(5), 1997. Google Scholar
  7. J. R. B. Cockett and R.A.G. Seely. Linearly distributive functors. Journal of Pure and Applied Algebra, 143(1-3), 1999. Google Scholar
  8. J.R.B. Cockett and P. V. Srinivasan. Exponential modalities and complementarity (extended abstract). In Kohei Kishida, editor, Proceedings of the Fourth International Conference on Applied Category Theory, Cambridge, United Kingdom, 12-16th July 2021, volume 372 of Electronic Proceedings in Theoretical Computer Science. Open Publishing Association, 2022. Google Scholar
  9. T. Ehrhard. On Köthe Sequence Spaces and Linear Logic. Mathematical Structures in Computer Science, 12(5), 2002. Google Scholar
  10. T. Ehrhard. Finiteness spaces. Mathematical Structures in Computer Science, 15(4), 2005. Google Scholar
  11. T. Ehrhard. An introduction to differential linear logic: proof-nets, models and antiderivatives. Mathematical Structures in Computer Science, 2017. Google Scholar
  12. T. Ehrhard and L. Regnier. Differential interaction nets. Theoretical Computer Science, 364(2), 2006. Google Scholar
  13. A. Fleury and C. Retoré. The mix rule. Mathematical Structures in Computer Science, 4(2), 1994. Google Scholar
  14. R. Gannoun, R. Hachaichi, H. Ouerdiane, and A. Rezgui. Un théorème de dualité entre espaces de fonctions holomorphes à croissance exponentielle. Journal of Functional Analysis, 171(1), 2000. Google Scholar
  15. J.-Y. Girard. Linear Logic. Theoretical Computer Science, 50(1), 1987. Google Scholar
  16. A. Grothendieck. Produits tensoriels topologiques et espaces nucléaires. Memoirs of the AMS, 16, 1966. Publisher: American Mathematical Society. Google Scholar
  17. M. Kerjean. A Logical Account for Linear Partial Differential Equations. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, 2018. Google Scholar
  18. M. Kerjean and J.-S. P. Lemay. Taylor Expansion as a Monad in Models of DiLL. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2023. Google Scholar
  19. O. Laurent. Etude de la polarisation en logique. Thèse de Doctorat, Université Aix-Marseille II, March 2002. Google Scholar
  20. J.-S. P. Lemay. Exponential functions in cartesian differential categories. Applied Categorical Structures, 29, 2021. Google Scholar
  21. J.-S. P. Lemay and J.-B. Vienney. Graded differential categories and graded differential linear logic. In Mathematical Foundations of Programming Semantics, 2023. Google Scholar
  22. P.-A. Melliès. Categorical semantics of linear logic. Société Mathématique de France, 2008. Google Scholar
  23. C.-H. L. Ong. Quantitative semantics of the lambda calculus: Some generalisations of the relational model. In 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2017. Google Scholar
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