Differentiable Causal Computations via Delayed Trace

Sprunger, David; Katsumata, Shin-ya

doi:10.1109/lics.2019.8785670

“…For this we will introduce an idempotent infinitesimal extension on streams that plays a similar role in this setting as Rutten's stream derivative operator [Rut05], which is given by discarding the head of the stream. On the other hand, our work is more closely related to Sprunger et al's work [SJ19,SK19] as it focuses on the differentiation of functions between streams, rather than the description of single streams in terms of differential equations.…”

Section: Stream Calculusmentioning

confidence: 98%

“…1 A similar approach to the one in [SK19] 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: :28mentioning

confidence: 99%

Cartesian Difference Categories

Alvarez-Picallo¹,

Lemay²

2021

Logical Methods in Computer Science

View full text Add to dashboard Cite

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.

show abstract

“…Moreover, enforcing that all feedback is delay-guarded would also prevent the construction of a freely generated traced monoidal category. There do exist weakenings of traced categories in which this 'delay-guardedness' principle holds, in the form of categories with feedback [24] or delayed trace [41]. However, these are not suitable in this context as the unfolding rule would not hold.…”

Section: (C )mentioning

confidence: 99%

A compositional theory of digital circuits

Ghica¹,

Kaye²,

Sprunger³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

This paper refines the existing axiomatic semantics of digital circuits with delay and feedback, in which circuits are constructed as morphisms in a freely generated cartesian traced (dataflow) category. First, we give a cleaner presentation, making a clearer distinction between syntax and semantics, including a full formalisation of the semantics as stream functions. As part of this effort, we refocus the categorical framework through the lens of string diagrams, which not only makes reading equations more intuitive but removes bureaucracy such as associativity from proofs. We also extend the existing framework with a new axiom, inspired by the Kleene fixed-point theorem, which allows circuits with non-delay-guarded feedback, typically handled poorly by traditional methodologies, to be replaced with a series of finitely iterated circuits. This eliminates the possibility of infinitely unfolding a circuit; instead, one can always reduce a circuit to some (possibly undefined) value. To fully characterise the stream functions that correspond to digital circuits, we examine how the behaviour of the latter can be modelled using Mealy machines. By establishing that the translation between sequential circuits and Mealy machines preserves their behaviour, one can observe that circuits always implement monotone stream functions with finite stream derivatives.

show abstract

“…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%

“…For example, in [22], a notion of stream derivative operator is introduced, and streams are characterized as the solutions of stream differential equations involving stream derivatives. More recent work in the setting of causal functions between streams of real numbers [23,24] has focused on extending the "classical" notion of the derivative of a real-valued function to stream-valued functions.…”

Section: Stream Calculusmentioning

confidence: 99%

Cartesian Difference Categories

Alvarez-Picallo

¹

,

Lemay

²

2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

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 ...

show abstract

Differentiable Causal Computations via Delayed Trace

Cited by 15 publications

References 28 publications

Cartesian Difference Categories

Cartesian Difference Categories

A compositional theory of digital circuits

Cartesian Difference Categories

Contact Info

Product

Resources

About