research-article

A Logical Account for Linear Partial Differential Equations

Author:

Marie KerjeanAuthors Info & Claims

LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Pages 589 - 598

https://doi.org/10.1145/3209108.3209192

Published: 09 July 2018 Publication History

Abstract

Differential Linear Logic (DiLL), introduced by Ehrhard and Regnier, extends linear logic with a notion of linear approximation of proofs. While DiLL is classical logic, i.e. has an involutive negation, classical denotational models of it in which this notion of differentiation corresponds to the usual one, defined on any smooth function, were missing. We solve this issue by constructing a model of it based on nuclear topological vector spaces and distributions with compact support.

This interpretation sheds a new light on the rules of DiLL, as we are able to understand them as the computational principles for the resolution of Linear Partial Differential Equations. We thus introduce D-DiLL, a deterministic refinement of DiLL with a D-exponential, for which we exhibit a cut-elimination procedure, and a categorical semantics. When D is a Linear Partial Differential Operator with constant coefficients, then the D-exponential is interpreted as the space of generalised functions ψ solutions to Dψ = φ. The logical inference rules represents the computational steps for the construction of the solution φ. We recover linear logic and its differential extension DiLL particular case.

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Cited By

Kerjean MPédrot PSobocinski PLago UEsparza J(2024)δ is for DialecticaProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662106(1-13)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662106
Kerjean MLemay J(2023)Taylor Expansion as a Monad in Models of DiLL2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175753(1-13)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175753
Wallbridge J(2020)Jets and differential linear logicMathematical Structures in Computer Science10.1017/S096012952000024930:8(865-891)Online publication date: 24-Nov-2020
https://doi.org/10.1017/S0960129520000249
Show More Cited By

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cover image ACM Conferences

LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

July 2018

960 pages

ISBN:9781450355834

DOI:10.1145/3209108

Copyright © 2018 ACM.

Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.

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SIGLOG: ACM Special Interest Group on Logic and Computation
EACSL: European Association for Computer Science Logic
IEEE-CS\DATC: IEEE Computer Society

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New York, NY, United States

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Published: 09 July 2018

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LICS '18: 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

July 9 - 12, 2018

Oxford, United Kingdom

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Cited By

Kerjean MPédrot PSobocinski PLago UEsparza J(2024)δ is for DialecticaProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662106(1-13)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662106
Kerjean MLemay J(2023)Taylor Expansion as a Monad in Models of DiLL2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175753(1-13)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175753
Wallbridge J(2020)Jets and differential linear logicMathematical Structures in Computer Science10.1017/S096012952000024930:8(865-891)Online publication date: 24-Nov-2020
https://doi.org/10.1017/S0960129520000249
Kerjean MPacaud Lemay J(2019)Higher-Order Distributions for Differential Linear LogicFoundations of Software Science and Computation Structures10.1007/978-3-030-17127-8_19(330-347)Online publication date: 5-Apr-2019
https://doi.org/10.1007/978-3-030-17127-8_19

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