Polyadic approximations, fibrations and intersection types

Mazza, Damiano; Pellissier, Luc; Vial, Pierre

doi:10.1145/3158094

Cited by 31 publications

(34 citation statements)

References 31 publications

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“…Apart from the papers already cited, let us mention some other related works. A recent, general categorical framework to define intersection and multi type systems is in [Mazza et al 2018].…”

Section: Other Related Workmentioning

confidence: 99%

Tight typings and split bounds

Accattoli

Graham-Lengrand

Kesner

2018

Proc. ACM Program. Lang.

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Multi types-aka non-idempotent intersection types-have been used to obtain quantitative bounds on higherorder programs, as pioneered by de Carvalho. Notably, they bound at the same time the number of evaluation steps and the size of the result. Recent results show that the number of steps can be taken as a reasonable time complexity measure. At the same time, however, these results suggest that multi types provide quite lax complexity bounds, because the size of the result can be exponentially bigger than the number of steps.Starting from this observation, we refine and generalise a technique introduced by Bernadet & Graham-Lengrand to provide exact bounds for the maximal strategy. Our typing judgements carry two counters, one measuring evaluation lengths and the other measuring result sizes. In order to emphasise the modularity of the approach, we provide exact bounds for four evaluation strategies, both in the λ-calculus (head, leftmostoutermost, and maximal evaluation) and in the linear substitution calculus (linear head evaluation).Our work aims at both capturing the results in the literature and extending them with new outcomes. Concerning the literature, it unifies de Carvalho and Bernadet & Graham-Lengrand via a uniform technique and a complexity-based perspective. The two main novelties are exact split bounds for the leftmost strategythe only known strategy that evaluates terms to full normal forms and provides a reasonable complexity measure-and the observation that the computing device hidden behind multi types is the notion of substitution at a distance, as implemented by the linear substitution calculus.

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Section: Other Related Workmentioning

confidence: 99%

Tight typings and split bounds

Accattoli

Graham-Lengrand

Kesner

2018

Proc. ACM Program. Lang.

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“…3.4, the parameters were integers, but one may imagine more complex structures. This leads us towards a type discipline in the style of bounded linear logic [Girard et al 1992], which is known to be related to intersection types (via the notion of approximation considered in [Mazza et al 2018]). More precisely, it seems that a similar application of (linear) dependent types as that given by [Dal Lago and Gaboardi 2011] is possible here, which would yield more practical type systems, necessarily sacrificing completeness but hopefully still of remarkable expressiveness.…”

Section: Toy Operatingmentioning

confidence: 99%

Intersection types and runtime errors in the pi-calculus

Lago

Visme

Mazza

et al. 2019

Proc. ACM Program. Lang.

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We introduce a type system for the π-calculus which is designed to guarantee that typable processes are well-behaved, namely they never produce a run-time error and, even if they may diverge, there is always a chance for them to łfinish their workž, i.e., to reduce to an idle process. The introduced type system is based on non-idempotent intersections, and is thus very powerful as for the class of processes it can capture. Indeed, despite the fact that the underlying property is Π 0 2-complete, there is a way to show that the system is complete, i.e., that any well-behaved process is typable, although for obvious reasons infinitely many derivations need to be considered.

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“…Proof p. 31 Let t be a sh -normalizable term and t 0 be its sh -normal form. For every reduction sequence d : t → * sh t 0 and every π t and π 0 t 0 such that |π| = inf{|π | | π t} and |π 0 | = inf{|π 0 | | π 0 t 0 }, one has…”

Section: If T Is a Value Then |T|mentioning

confidence: 99%

“…Starting from [11], research on relational semantics/non-idempotent intersection types has proliferated: various works in the literature explore their power in bounding the execution time or in characterizing normalization [12,8,6,27,5,13,34,28,9,31]. All these works study relational semantics/non-idempotent intersection types either in LL proof-nets (the graphical representation of proofs in LL), or in some variant of ordinary (i.e.…”

Section: Introductionmentioning

confidence: 99%

Towards a Semantic Measure of the Execution Time in Call-by-Value lambda-Calculus

Guerrieri¹

2018

EasyChair Preprints

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We investigate the possibility of a semantic account of the execution time (i.e. the number of β v -steps leading to the normal form, if any) for the shuffling calculus, an extension of Plotkin's call-by-value λ -calculus. For this purpose, we use a linear logic based denotational model that can be seen as a nonidempotent intersection type system: relational semantics. Our investigation is inspired by similar ones for linear logic proof-nets and untyped call-by-name λ -calculus. We first prove a qualitative result: a (possibly open) term is normalizable for weak reduction (which does not reduce under abstractions) if and only if its interpretation is not empty. We then show that the size of type derivations can be used to measure the execution time. Finally, we show that, differently from the case of linear logic and call-by-name λ -calculus, the quantitative information enclosed in type derivations does not lift to types (i.e. to the interpretation of terms). To get a truly semantic measure of execution time in a call-by-value setting, we conjecture that a refinement of its syntax and operational semantics is needed.

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Polyadic approximations, fibrations and intersection types

Cited by 31 publications

References 31 publications

Tight typings and split bounds

Tight typings and split bounds

Intersection types and runtime errors in the pi-calculus

Towards a Semantic Measure of the Execution Time in Call-by-Value lambda-Calculus

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