Higher-order distributions for differential linear logic. - Inria - Institut national de recherche en sciences et technologies du numérique
Conference Papers Year : 2019

Higher-order distributions for differential linear logic.

Abstract

Linear Logic was introduced as the computational counterpart of the algebraic notion of linearity. Differential Linear Logic refines Linear Logic with a proof-theoretical interpretation of the geometrical process of differentiation. In this article, we construct a polarized model of Differential Linear Logic satisfying computational constraints such as an interpretation for higher-order functions, as well as constraints inherited from physics such as a continuous interpretation for spaces. This extends what was done previously by Kerjean for first order Differential Linear Logic without promotion. Concretely, we follow the previous idea of interpreting the exponential of Differential Linear Logic as a space of higher-order distributions with compact-support, and is constructed as an inductive limit of spaces of distributions on Euclidean spaces. We prove that this exponential is endowed with a co-monadic like structure, with the notable exception that it is functorial only on isomorphisms. Interestingly, as previously argued by Ehrhard, this still allows one to interpret differential linear logic without promotion.
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Dates and versions

hal-01969262 , version 1 (03-01-2019)
hal-01969262 , version 2 (16-01-2019)

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Marie Kerjean, Jean-Simon Lemay. Higher-order distributions for differential linear logic.. Foundations of Software Science and Computation Structures - 22nd International Conference, {FOSSACS} 2019, Held as Part of the European Joint Conferences on Theory and Practice of Software, {ETAPS} 2019, Prague, Czech Republic, April 6-11, 2019, Proceedings, Mikolaj Bojanczyk and Alex Simpson, 2019, Prague (Czech Republic), Czech Republic. ⟨10.1007/978-3-030-17127-8\_19⟩. ⟨hal-01969262v2⟩
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