CHAD for expressive total languages
Nunes, Fernando Lucatelli; Vákár, Matthijs
(2023) Mathematical Structures in Computer Science, volume 33, issue 4-5, pp.
(Article)
Abstract
We show how to apply forward and reverse mode Combinatory Homomorphic Automatic Differentiation (CHAD) (Vákár (2021). ESOP, 607–634; Vákár and Smeding (2022). ACM Transactions on Programming Languages and Systems 44 (3) 20:1–20:49.) to total functional programming languages with expressive type systems featuring the combination of • tuple types; • sum
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types; • inductive types; • coinductive types; • function types. We achieve this by analyzing the categorical semantics of such types in Σ-types (Grothendieck constructions) of suitable categories. Using a novel categorical logical relations technique for such expressive type systems, we give a correctness proof of CHAD in this setting by showing that it computes the usual mathematical derivative of the function that the original program implements. The result is a principled, purely functional and provably correct method for performing forward- and reverse-mode automatic differentiation (AD) on total functional programming languages with expressive type systems.
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Keywords: (Co)monadicity, Artin gluing, Automatic differentiation, Cartesian closed categories, Coalgebras, Coinductive types, Comma categories, Creation of initial algebras, Denotational semantics, Dependently typed languages, Exponentiability, Extensive categories, Extensive indexed categories, Fibered categories, Free cocompletion under coproducts, Grothendieck construction, Inductive types, Initial algebra semantics, Linear types, Logical relations, Polynomial functors, Program transformations, Programming languages, Scientific computing, Software correctness, Type systems, Variant types, Mathematics (miscellaneous), Computer Science Applications
ISSN: 0960-1295
Publisher: Cambridge University Press
Note: Funding Information: This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 895827 and from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek under NWO Veni grant number VI.Veni.202.124. This research was also supported through the program “Oberwolfach Leibniz Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2022 and partially supported by the CMUC, Centre for Mathematics of the University of Coimbra – UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Publisher Copyright: © The Author(s), 2023. Published by Cambridge University Press.
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