Essential and relational models

Paolini, Luca; Piccolo, Mauro; Rocca, Simonetta Ronchi Della

doi:10.1017/s0960129515000316

Cited by 17 publications

(30 citation statements)

References 24 publications

(43 reference statements)

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“…reducibility candidates). Several papers explore this approach under the call-by-name [Bucciarelli et al 2012;Kesner and Ventura 2015;Kesner and Vial 2017;Ong 2017;Paolini et al 2017] or the call-by-value [Carraro and Guerrieri 2014;Díaz-Caro et al 2013;Ehrhard 2012] operational semantics, or both [Ehrhard and Guerrieri 2016]. Sometimes, precise quantitative bounds are provided instead, as in [Bernadet and Graham-Lengrand 2013b;de Carvalho 2007].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Tight typings and split bounds

Accattoli

Graham-Lengrand

Kesner

2018

Proc. ACM Program. Lang.

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“…As discussed thoroughly in [21], the choice of presenting a relational model as a reflexive object or as a non-idempotent intersection type system is more a matter of taste rather than a technical decision. Here we provide the latter presentation.…”

Section: Non-idempotent Intersection Type Systemsmentioning

confidence: 99%

“…First, we introduce the class of relational graph models (rgms) of λ-calculus, which are the relational analogous of graph models [3], and describe them as nonidempotent intersection type systems [21]. This class is general enough to encompass all relational models individually introduced in the literature [6,14], including D ω (while Scott's D ∞ cannot be a graph model since it is extensional).…”

Section: Introductionmentioning

confidence: 99%

Relational Graph Models, Taylor Expansion and Extensionality

Manzonetto¹,

Ruoppolo²

2014

Electronic Notes in Theoretical Computer Science

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International audienceWe define the class of relational graph models and study the induced order-and equational-theories. Using the Taylor expansion, we show that all λ-terms with the same Böhm tree are equated in any relational graph model. If the model is moreover extensional and satisfies a technical condition, then its order-theory coincides with Morris's observational pre-order. Finally, we introduce an extensional version of the Taylor expansion, then prove that two λ-terms have the same extensional Taylor expansion exactly when they are equivalent in Morris's sense

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“…Starting from [15,16], 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 [17,12,10,33,9,18,40,34,13,37,3]. All these works study relational semantics/nonidempotent 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

Guerrieri

2019

Electron. Proc. Theor. Comput. Sci.

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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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Essential and relational models

Cited by 17 publications

References 24 publications

Tight typings and split bounds

Tight typings and split bounds

Relational Graph Models, Taylor Expansion and Extensionality

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

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