Cofree coalgebras and differential linear logic

Clift, James; Murfet, Daniel

doi:10.1017/s0960129520000134

“…and where the codereliction V η − →!V is defined as the injection of V into the 0 ∈ V component of !V . For full details on this example, see [8]. Similarly to the previous example, !…”

Section: Define the Natural Transformation !(A × B)mentioning

confidence: 93%

Exponential Functions in Cartesian Differential Categories

Lemay

¹

2020

Appl Categor Struct

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In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $$e^x$$ e x from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of $$e^0 = 1$$ e 0 = 1 , $$e^{x+y} = e^x e^y$$ e x + y = e x e y , and $$\frac{\partial e^x}{\partial x} = e^x$$ ∂ e x ∂ x = e x all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. In particular, differential exponential maps can be defined without the need of limits, converging power series, or unique solutions of certain differential equations—which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, the split complex exponential function, and the dual numbers exponential function. As another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category.

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“…The bridge between the ordinary derivative of f (x) and the Ehrhard-Regnier derivative of π is the concept of a primitive element in the theory of coalgebras, which is an equivalent but dual point of view on the concept of tangent vectors; see Clift and Murfet (2017), Section 2.3. A vector z ∈ !…”

Section: !Bool Boolmentioning

confidence: 99%

“…The semantics of intuitionistic linear logic in vector spaces using the cofree coalgebra to interpret the exponential, which we call the Sweedler semantics since Sweedler was the first to give a detailed study of the cofree coalgebra (Sweedler 1969), was introduced as an example by Hyland and Schalk (2003) and revisited in Murfet (2014) and Clift and Murfet (2017) with a focus on explicit formulas for the involved structures based on the results of Murfet (2015). Denotations of formulas and proofs will throughout be denoted by − .…”

Section: Linear Logic and The Sweedler Semanticsmentioning

confidence: 99%

“…The Sweedler semantics is a denotational semantics of first-order intuitionistic linear logic in the category of vector spaces over an algebraically closed field k of characteristic zero (Murfet 2014;Clift and Murfet 2017). The ⊗ and & connectives of linear logic are interpreted by the usual tensor product and direct sum of vector spaces, while the exponential !…”

Section: Introductionmentioning

confidence: 99%

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Encodings of Turing machines in linear logic

Clift

¹

,

Murfet

²

2020

Math. Struct. Comp. Sci.

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The Sweedler semantics of intuitionistic differential linear logic takes values in the category of vector spaces, using the cofree cocommutative coalgebra to interpret the exponential and primitive elements to interpret the differential structure. In this paper, we explicitly compute the denotations under this semantics of an interesting class of proofs in linear logic, introduced by Girard: the encodings of step functions of Turing machines. Along the way we prove some useful technical results about linear independence of denotations of Church numerals and binary integers.

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“…The first is concerned with a certain categorical axiomatics for differential structure; it originates in the work of Ehrhard and Regnier on the differential λ-calculus [21], with the definitive notions of tensor differential category and cartesian differential category being identified by Blute, Cockett and Seely in [7,8], and further studied by the Canadian school of category theorists [4][5][6]14,16,17,33]. This has led to novel applications in computer science [11,13,15,20,23,36] and in other areas such as abelian functor calculus [2].…”

Section: Introductionmentioning

confidence: 99%

Cartesian Differential Categories as Skew Enriched Categories

Garner

¹

,

Lemay

²

2021

Appl Categor Struct

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We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

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Cofree coalgebras and differential linear logic

Abstract: We prove that the semantics of intuitionistic linear logic in vector spaces which uses cofree coalgebras is also a model of differential linear logic, and that the Cartesian closed category of cofree coalgebras is a model of the simply typed differential λ-calculus.

Cited by 6 publications

References 23 publications

Exponential Functions in Cartesian Differential Categories

Exponential Functions in Cartesian Differential Categories

Encodings of Turing machines in linear logic

Cartesian Differential Categories as Skew Enriched Categories

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