Fixing Incremental Computation

Alvarez-Picallo, Mario; Eyers-Taylor, Alex; Jones, Michael Peyton; Ong, C.-H. Luke

doi:10.1007/978-3-030-17184-1_19

“…On the one hand, the derivatives arising from differential logical relations (which essentially coincide with the derivatives from Q) have been compared [48] with those found in some recent literature on discrete differentiation (e.g. finite difference operators, Boolean derivatives), and approaches based on the so-called incremental λ-calculus [2], [3]. On the other hand, the derivatives from Section VI can be compared with the literature on Cartesian Differential Categories, originating in Ehrhard and Regnier's work on differential Linear Logic and the differential λ-calculus [30].…”

Section: Related Workmentioning

confidence: 99%

On Generalized Metric Spaces for the Simply Typed Lambda-Calculus

Pistone

¹

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Generalized metrics, arising from Lawvere's view of metric spaces as enriched categories, have been widely applied in denotational semantics as a way to measure to which extent two programs behave in a similar, although non equivalent, way. However, the application of generalized metrics to higher-order languages like the simply typed lambda calculus has so far proved unsatisfactory. In this paper we investigate a new approach to the construction of cartesian closed categories of generalized metric spaces. Our starting point is a quantitative semantics based on a generalization of usual logical relations. Within this setting, we show that several families of generalized metrics provide ways to extend the Euclidean metric to all higher-order types.

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“…Change actions [1,2] have recently been proposed as a setting for reasoning about higher-order incremental computation, based on a discrete notion of differentiation. Together with Cartesian differential categories, they provide the core ideas behind Cartesian difference categories.…”

Section: Change Action Modelsmentioning

confidence: 99%

“…We will restrict ourselves to considering streams over abelian groups 1 , so let Ab ω be the category whose objects are all the abelian groups and whose morphisms are causal maps 1 A similar approach to the one in [24] is possible where we consider streams on arbitrary difference categories, and lift the difference operator of the underlying category to its category of streams, although it would complicate the presentation of this section without gaining clarity.…”

Section: Stream Calculusmentioning

confidence: 99%

“…This led to the development of change structures [6] and change actions [2]. Change action models have been successfully used to provide a model for incrementalizing Datalog programs [1], but have also been shown to model the calculus of finite differences as well as the Kleisli category of the tangent bundle monad of a Cartesian differential category. Change action models, however, are very general, lacking many of the nice properties of Cartesian differential categories (for example, addition in a change action model is not required to be commutative), even though they are verified in most change action models.…”

Section: Introductionmentioning

confidence: 99%

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Cartesian Difference Categories

Alvarez-Picallo

¹

,

Lemay

²

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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“…This led to the development of change structures [CGRO14] and change actions [APO19]. Change action models have been successfully used to provide a model for incrementalizing Datalog programs [APETJO19], and have also been shown to model the calculus of finite differences. Change action models, however, are very general, lacking many of the nice properties of Cartesian differential categories (for example, addition in a change action model is not required to be commutative), even though they are verified in most change action models.…”

Section: Introductionmentioning

confidence: 99%

Cartesian Difference Categories

Alvarez-Picallo¹,

Lemay²

2021

Logical Methods 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 $\lambda$-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation of a more discrete nature such as the calculus of finite differences. On the other hand, change action models have been shown to capture these examples as well as more "exotic" examples of differentiation. But change action models are very general and do not share the nice properties of Cartesian differential categories. 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.

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Fixing Incremental Computation

Cited by 16 publications

References 31 publications

On Generalized Metric Spaces for the Simply Typed Lambda-Calculus

On Generalized Metric Spaces for the Simply Typed Lambda-Calculus

Cartesian Difference Categories

Cartesian Difference Categories

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