An introduction to differential linear logic: proof-nets, models and antiderivatives

Ehrhard, Thomas

doi:10.1017/s0960129516000372

Cited by 58 publications

(142 citation statements)

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“…The notation W was introduced by Ehrhard in [13]. Proof: Here we use the Leibniz rule One might expect the definition of a monoidal deriving transformation, that is a deriving transformation d for a monoidal coalgebra modality, to require that the differential monoidal rule should hold:…”

Section: Differential Categoriesmentioning

confidence: 99%

“…We next define the Taylor condition, which unlike compatibility and the fundamental theorems, is a property only of the deriving transformation. Definition 5.3 A deriving transformation d is said to be Taylor [13] if for every pair of maps f, g :…”

Section: Calculus Categoriesmentioning

confidence: 99%

“…This is the main result of this paper. We first show that Ehrhard's integral built in [13] is an integral transformation which furthermore satisfies Poincaré Condition. Proposition 6.7 In a differential category for which J is a natural isomorphism, the natural transformation s := d • (J −1 ⊗ 1) is an integral transformation which satisfies the Poincaré Condition.Proof: In string diagram notation, s is expressed as follows: J −1 === We must show that s satisfies [s.1] to [s.3].…”

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Integral categories and calculus categories

Cockett¹,

Lemay²

2018

Math. Struct. Comp. Sci.

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Differential categories are now an established abstract setting for differentiation. However not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, s A : !A − → !A ⊗ A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is called a calculus category.Modifying an approach to antiderivatives by T. Ehrhard, we define having antiderivatives as the demand that a certain natural transformation, K : !A − → !A, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category.When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence we show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure.The paper finishes by providing a suite of separating examples. Examples of differential categories, integral categories, and calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which are not integral categories are discussed and vice versa.The proof follows the form of (i): we show that J −1 n+1 := δJ n d • (ε ⊗ e) by first showing, with this definition, that J −1 n+1 J n+1 = 1 and then showing that J n+1 J −1 n+1 = J −1 n+1 J n+1 . First, by naturality of J n , [J.7], and [cd.7] we have:=== e δ ε = J −1 n ε === e δ J n = ε === e δ = Proof: These are mostly straightforward calculations by using the properties of J: [J −1 .1]: Here we use the property [J.1]: (n + 1) · 0 J −1 n = 0 J n J −1 n = 0 = J −1 n J n 0 = (n + 1) · J −1 n 0 [J −1 .2]: Here we use [J.2]: (n + 1) · e J −1 n = e J n J −1 n = e [J −1 .3]: Here we use [J.3]: (n + 2) · ε J −1 n = ε J n J −1 n = ε 42 [J −1 .4] Here we use [J.6]: J −1 n === = J −1 n === Jn+1 J −1 n+1 = J n === J −1 n+1 J −1 n = J −1 n+1 === [J −1 .5] Here we use [J.7]: J −1 n+1 === = J −1 n+1 === Jn J −1 n = J n+1 === J −1 n J −1 n+1 = J −1 n === [J −1 .6]: Here we use [J.12]: === === J −1 n = Jn === === J −1 n J −1 n = Jn === === J −1 n J −1 n = J −1 n === === [J −1 .7]: Here we use [J −1 .6]: === === J −1 n = === === J −1 n + J −1 n = J −1 n === === + J −1 n = === === J −1 n [J −1 .8]: Here we use [L.6] and that, as K = L + !(0), the differential of K and L are equal: === [J −1 .10]: Here we use [J −1 .4] and [d.5]: J −1 n === === = === === J −1 n+1 = === === J −1 n+1 = === === J −1 n [J −1 .11]: Here we use [J −1 .5] and [cd.6]:n ✷ Proposition 6.6 In a differential category with antiderivatives and a monoidal coalgebra modality, the followi...

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Section: Differential Categoriesmentioning

confidence: 99%

Section: Calculus Categoriesmentioning

confidence: 99%

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confidence: 99%

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Integral categories and calculus categories

Cockett¹,

Lemay²

2018

Math. Struct. Comp. Sci.

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“…In the early 2000's, Ehrhard and Regnier introduced the differential λ-calculus [10], an extension of the λ-calculus equipped with a differential combinator capable of taking the derivative of arbitrary higher-order functions. This development, based on models of linear logic equipped with a natural notion of "derivative" [11], sparked a wave of research into categorical models of differentiation.…”

Section: Introductionmentioning

confidence: 99%

Cartesian Difference Categories

Alvarez-Picallo

Lemay

2020

Lecture Notes in Computer Science

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Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential λ-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation such as the calculus of finite differences or the Boolean differential calculus. On the other hand, change action models have been shown to capture these examples as well as more "exotic" examples of differentiation. However, change action models are very general and do not share the nice properties of a Cartesian differential category. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.In this section we briefly review Cartesian differential categories, so that the reader may compare Cartesian differential categories with the new notion of Cartesian difference categories which we introduce in the next section. For a full detailed introduction on Cartesian differential categories, we refer the reader to the original paper [4]. Cartesian Left Additive CategoriesHere we provide the definition of Cartesian left additive categories [4] -where "additive" is meant being skew enriched over commutative monoids. In particular this means that we do not assume the existence of additive inverses, i.e., "negative elements". Definition 1. A left additive category [4] is a category X such that each hom-set X(A, B) is a commutative monoid with addition operation + : X(A, B)× X(A, B) → X(A, B) and zero element (called the zero map) 0 ∈ X(A, B), such that pre-composition preserves the additive structure: (f + g) • h = f • h + g • h and 0 • f = 0. A map k in a left additive category is additive if post-composition by k preserves the additive structure: k • (f + g) = k • f + k • g and k • 0 = 0.By a Cartesian category we mean a category X with chosen finite products where we denote the binary product of objects A and B by A×B with projection maps π 0 : A × B → A and π 1 : A × B → B and pairing operation −, − , and the chosen terminal object as ⊤ with unique terminal maps ! A : A → ⊤.Definition 2. A Cartesian left additive category [4] is a Cartesian category X which is also a left additive category such all projection maps π 0 : A × B → A and π 1 : A × B → B are additive 3 .By [4, Proposition 1.2.2], an equivalent axiomatization is of a Cartesian left additive category is that ...

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“…is an endofunctor on the category Refl iso of reflexive spaces and isomorphisms between them. We use a technique developed by Ehrhard in [6] to show that this still provides a model of finitary Differential linear logic (DiLL 0 ), that is, DiLL without the promotion rule. We also discuss how this construction also leads to a polarized model of DiLL 0 (Sect.…”

Section: Introductionmentioning

confidence: 99%

Higher-Order Distributions for Differential Linear Logic

Kerjean

Lemay

2019

Lecture Notes in Computer Science

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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, which 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 the interpretation of differential linear logic without promotion.

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An introduction to differential linear logic: proof-nets, models and antiderivatives

Cited by 58 publications

References 35 publications

Integral categories and calculus categories

Integral categories and calculus categories

Cartesian Difference Categories

Higher-Order Distributions for Differential Linear Logic

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