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Communication Dans Un Congrès Année : 2024

Laplace Distributors and Laplace Transformations for Differential Categories

Résumé

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 ex on the monoidal unit. We also show that many well-known and important examples have a Laplace distributor/transformation, including (weighted) relations, finiteness spaces, K¨othe spaces, and convenient vector spaces.
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Dates et versions

hal-04568289 , version 1 (04-05-2024)

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  • HAL Id : hal-04568289 , version 1

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Marie Kerjean, Jean-Simon Pacaud Lemay. Laplace Distributors and Laplace Transformations for Differential Categories. 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024), Jul 2024, Tallinn, France. ⟨hal-04568289⟩
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