Constructing Differential Categories and Deconstructing Categories of Games

Laird, Jim; Manzonetto, Giulio; McCusker, Guy

doi:10.1007/978-3-642-22012-8_14

Cited by 9 publications

(15 citation statements)

References 26 publications

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“…This homotopy-theoretic assumption is thus reminiscent of the idea advocated in [2], [4] that one should only consider the strategies which are saturated modulo the action of the symmetric group Σ n on the tensorial powers A ⊗n of n copies of the game A. As additional precursor to this work, let us mention the game semantics of the differential λ-calculus, the intuitionistic fragment of DiLL, designed in [16] and based on the nondeterministic pointer game semantics formulated by Harmer and McCusker [13].…”

Section: Related Workmentioning

confidence: 99%

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Template games and differential linear logic

Melliès

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We extend our template game model of multiplicative additive linear logic (MALL) with an exponential modality of linear logic (LL) derived from the standard categorical construction Sym of the free symmetric monoidal category. We obtain in this way the first game semantics of differential linear logic (DiLL). Its formulation relies on a careful and healthy comparison with the model of distributors and generalised species designed ten years ago by Fiore, Gambino, Hyland and Winskel. Besides the resolution of an old open problem of game semantics, the study reveals an unexpected and promising convergence between linear logic and homotopy theory.

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Section: Related Workmentioning

confidence: 99%

“…Property A. The two spaces [0] and [1] in (15) of the internal category are fibrant objects in S, and its structural S-morphisms s, t, m, e in (16) and (18) are fibrations in S.…”

Section: A a Quillen Model Structurementioning

confidence: 99%

Template games and differential linear logic

Melliès

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…The category CPM is "almost" capable of handling this case, but not quite, because it cannot express infinite tuples of matrices. The model we propose in this paper is essentially an extension of CPM to infinite biproducts, using methods developed in [5,15,11,12].…”

Section: Limitations Of Cpm As a Modelmentioning

confidence: 99%

Applying quantitative semantics to higher-order quantum computing

Pagani

Selinger

Valiron

2014

Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

101

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Finding a denotational semantics for higher order quantum computation is a long-standing problem in the semantics of quantum programming languages. Most past approaches to this problem fell short in one way or another, either limiting the language to an unusably small finitary fragment, or giving up important features of quantum physics such as entanglement. In this paper, we propose a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic.

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“…The theory of differential categories now has a rich literature of its own and has led to other abstract formulations of several notions of differentiation such as the directional derivative [3] and smooth manifolds [6]. As differentiation is an important tool throughout quantum mechanics and quantum information, it makes sense to study applications of the theory of differential categories to categorical quantum foundations (as suggested briefly in the conclusion of [20]).…”

Section: Introductionmentioning

confidence: 99%

Why FHilb is Not an Interesting (Co)Differential Category

Lemay

2020

Electron. Proc. Theor. Comput. Sci.

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Differential categories provide an axiomatization of the basics of differentiation and categorical models of differential linear logic. As differentiation is an important tool throughout quantum mechanics and quantum information, it makes sense to study applications of the theory of differential categories to categorical quantum foundations. In categorical quantum foundations, compact closed categories (and therefore traced symmetric monoidal categories) are one of the main objects of study, in particular the category of finite-dimensional Hilbert spaces FHilb. In this paper, we will explain why the only differential category structure on FHilb is the trivial one. This follows from a sort of incompatibility between the trace of FHilb and possible differential category structure. That said, there are interesting non-trivial examples of traced/compact closed differential categories, which we also discuss.The goal of this paper is to introduce differential categories to the broader categorical quantum foundation community and hopefully open the door to further work in combining these two fields. While the main result of this paper may seem somewhat "negative" in achieving this goal, we discuss interesting potential applications of differential categories to categorical quantum foundations.

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Constructing Differential Categories and Deconstructing Categories of Games

Cited by 9 publications

References 26 publications

Template games and differential linear logic

Template games and differential linear logic

Applying quantitative semantics to higher-order quantum computing

Why FHilb is Not an Interesting (Co)Differential Category

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