Quantitative semantics of the lambda calculus: Some generalisations of the relational model

Ong, C.-H. Luke

doi:10.1109/lics.2017.8005064

Cited by 16 publications

(20 citation statements)

References 63 publications

(82 reference statements)

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“…It would be interesting to have a categorical semantics of CbNeed, as well as a categorical way of discriminating our quantitative precise model from the quantitatively lax one given by CbN multi types. It would also be interesting to obtain game semantics of CbNeed, hopefully satisfying a strong correspondence with our multi types in the style of what happens in CbN [30,31,51,56].…”

Section: Discussionmentioning

confidence: 91%

Types by Need

Accattoli

Guerrieri

Leberle

2019

Programming Languages and Systems

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A cornerstone of the theory of λ-calculus is that intersection types characterise termination properties. They are a flexible tool that can be adapted to various notions of termination, and that also induces adequate denotational models. Since the seminal work of de Carvalho in 2007, it is known that multi types (i.e. non-idempotent intersection types) refine intersection types with quantitative information and a strong connection to linear logic. Typically, type derivations provide bounds for evaluation lengths, and minimal type derivations provide exact bounds. De Carvalho studied call-by-name evaluation, and Kesner used his system to show the termination equivalence of call-by-need and call-byname. De Carvalho's system, however, cannot provide exact bounds on call-by-need evaluation lengths. In this paper we develop a new multi type system for call-by-need. Our system produces exact bounds and induces a denotational model of callby-need, providing the first tight quantitative semantics of call-by-need.

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Section: Discussionmentioning

confidence: 91%

Types by Need

Accattoli

Guerrieri

Leberle

2019

Programming Languages and Systems

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“…In fact, REL was one of the original examples of a differential category (Blute et al, 2006) and of a differential category with antiderivatives (Cockett and Lemay, 2018b;Ehrhard, 2017). For more details on generalizations of the relational model, we invite the reader to see Laird et al (2013), Lamarche (1992) and Ong (2017).…”

Section: Biproduct Completion Of Complete Semiringsmentioning

confidence: 99%

“…This gives a coalgebra modality which satisfies the Seely isomorphisms, and therefore provides a monoidal coalgebra modality on R (for a full description of this monoidal coalgebra modality see Laird et al (2013), Lamarche (1992) and Ong (2017)). This monoidal coalgebra modality is in fact a free exponential modality (Melliès et al, 2017), making R a Lafont category (Melliès, 2009).…”

Section: Biproduct Completion Of Complete Semiringsmentioning

confidence: 99%

Convenient antiderivatives for differential linear categories

Lemay

2020

Math. Struct. Comp. Sci.

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Abstract Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation K, which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson’s differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

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“…In fact, REL was one of the original examples of a differential category [4] and of a differential category with antiderivatives [8,9]. For more details on generalizations of the relational model, we invite the reader to see [19,25].…”

Section: Biproduct Completion Of Complete Semiringsmentioning

confidence: 99%

Convenient Antiderivatives For Differential Linear Categories

Lemay¹

2018

Preprint

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Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation K, which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category -which is a differential category with a monoidal coalgebra modality -has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. As well, we show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential categories with antiderivatives.Acknowledgements. The author would like to thank Kellogg College, the Clarendon Fund, and the Oxford-Google DeepMind Graduate Scholarship for financial support.

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Quantitative semantics of the lambda calculus: Some generalisations of the relational model

Cited by 16 publications

References 63 publications

Types by Need

Types by Need

Convenient antiderivatives for differential linear categories

Convenient Antiderivatives For Differential Linear Categories

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