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Journal ArticleDOI

Proof nets and the call-by-value λ-calculus

16 Nov 2015-Theoretical Computer Science (Elsevier Science Publishers Ltd.)-Vol. 606, pp 2-24
TL;DR: This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets, and identifies a subcalculus that is shown to be as expressive as the full calculus.
About: This article is published in Theoretical Computer Science.The article was published on 2015-11-16 and is currently open access. It has received 34 citations till now. The article focuses on the topics: Structural proof theory & Proof calculus.
Citations
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Proceedings ArticleDOI
05 Sep 2016
TL;DR: The Bang Calculus is introduced, an untyped functional calculus in which the promotion operation of Linear Logic is made explicit and where application is a bilinear operation, and an adequacy theorem is proved by means of a resourcebang Calculus whose design is based on Differential Linear Logic.
Abstract: We introduce and study the Bang Calculus, an untyped functional calculus in which the promotion operation of Linear Logic is made explicit and where application is a bilinear operation. This calculus, which can be understood as an untyped version of Call-By-Push-Value, subsumes both Call-By-Name and Call-By-Value lambda-calculi, factorizing the Girard's translations of these calculi in Linear Logic. We build a denotational model of the Bang Calculus based on the relational interpretation of Linear Logic and prove an adequacy theorem by means of a resource Bang Calculus whose design is based on Differential Linear Logic.

43 citations


Cites background from "Proof nets and the call-by-value λ-..."

  • ...Consider the Girard’s translations (·) of untyped call-byname [12, 28, 41] and (·) of untyped call-by-value [2] λ-calculi into LL proof-nets based on, respectively, the recursive types identity o = !o ( o and o = !(o ( o) (or, equivalently, o = !o ( !o): they differ essentially in handling boxes and derelictions....

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  • ...contraction and weakening) ingredients in the same way: since in CBN λ-calculus there is no restriction on firing a β-redex (its argument can be freely copied or erased), the translation (·) puts the argument of every application into a box (see [12, 28, 41]); on the other hand, the translation (·) puts only values into boxes (see [2])(2) since in CBV λ-calculus values are the only duplicable and discardable λ-terms....

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Book ChapterDOI
21 Nov 2016
TL;DR: A detailed comparative study of the operational semantics of four calculi, coming from different areas such as the study of abstract machines, denotational semantics, linear logic proof nets, and sequent calculus, showing that these calculi are all equivalent from a termination point of view.
Abstract: The elegant theory of the call-by-value lambda-calculus relies on weak evaluation and closed terms, that are natural hypotheses in the study of programming languages. To model proof assistants, however, strong evaluation and open terms are required, and it is well known that the operational semantics of call-by-value becomes problematic in this case. Here we study the intermediate setting—that we call Open Call-by-Value—of weak evaluation with open terms, on top of which Gregoire and Leroy designed the abstract machine of Coq. Various calculi for Open Call-by-Value already exist, each one with its pros and cons. This paper presents a detailed comparative study of the operational semantics of four of them, coming from different areas such as the study of abstract machines, denotational semantics, linear logic proof nets, and sequent calculus. We show that these calculi are all equivalent from a termination point of view, justifying the slogan Open Call-by-Value.

42 citations


Cites background or methods from "Proof nets and the call-by-value λ-..."

  • ...Open CBV Starting with the seminal work of Paolini and Ronchi Della Rocca [17, 16, 20], the dissonance between open terms and CBV has been repeatedly pointed out and studied per se via various calculi [8, 2, 1, 5, 10, 9, 3]....

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  • ...The Value Substitution Calculus λvsub, coming from the linear logic interpretation of CBV and using explicit substitutions and contextual rewriting rules to circumvent stuck β -redexes—it was introduced by Accattoli and Paolini [2] and it is a graph-free presentation of proof nets for CBV λ -calculus [1]; 3....

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  • ...In Naı̈ve Open CBV, t and u are premature βv-normal forms because they both have a stuck β -redex forbidding evaluation to keep going, while one would expect them to behave like the famous divergent term Ω := δδ (see [17, 20, 2, 1, 5, 10])....

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  • ...The need arises, most notably, when trying to describe the implementation model of Coq [8], but also from other motivations, as denotational semantics [17, 20, 2, 5], monad and CPS translations and the associated equational theories [15, 21, 22, 7, 11], bisimulations [13], partial evaluation [12], linear logic proof nets [1], cost models [3]....

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Book ChapterDOI
02 Dec 2018
TL;DR: Operational and implementative studies of proposed call-by-value lambda-calculus extensions are provided, showing that they are equivalent with respect to termination, and also at the level of time cost models.
Abstract: The good properties of Plotkin’s call-by-value lambda-calculus crucially rely on the restriction to weak evaluation and closed terms. Open call-by-value is the more general setting where evaluation is weak but terms may be open. Such an extension is delicate and the literature contains a number of proposals. Recently, we provided operational and implementative studies of these proposals, showing that they are equivalent with respect to termination, and also at the level of time cost models.

29 citations


Cites background from "Proof nets and the call-by-value λ-..."

  • ...It can also be seen as the (open fragment of) Accattoli and Paolini’s value substitution calculus [6], where indeed inert terms are never substituted....

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  • ...For the issues of Plotkin’s setting with respect to open terms and for alternative presentations of Open CbV, see Accattoli and Guerrieri’s [3]....

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  • ...Among them, Ehrhard’s [23], Diaz-Caro, Manzonetto, and Pagani’s [22], Carraro and Guerrieri’s [13], Ehrhard and Guerrieri’s [24], and Guerrieri’s [31] deal with CbV, while de Carvalho’s [14,16], Bernadet and Lengrand’s [8], de Carvalho, Pagani, and Tortora de Falco’s [17], Accattoli, Graham-Lengrand, and Kesner’s [2] provide exact bounds....

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  • ...While property 0 is not problematic (type systems/denotational models are conceived to satisfy it), it turns out that none of the incarnations of Open CbV studied by Accattoli and Guerrieri in [3] (namely, Paolini and Ronchi della Rocca’s fireball calculus λfire [44,48,29,7], Accattoli and Paolini’s value substitution calculus λvsub [6,1], and Carraro and Guerrieri’s shuffling calculus λsh [13,32,30,33,31]) 5 satisfies all the properties 1–5 at the same time: λfire lacks property 1 (as shown here in Sect....

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  • ...Moreover, we provide a characterisation of types and type derivations that provide exact bounds, similarly to de Carvalho [14,16], Bernadet and Lengrand [8], and de Carvalho, Pagani, and Tortora de Falco [17], and along the lines of a very recent work by Accattoli, Graham-Lengrand, and Kesner [2], but using a slightly different approach....

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Book ChapterDOI
16 Oct 2018
TL;DR: It is shown that the linear substitution calculus, a simple refinement of the \(\lambda \)-calculus with sharing, is isomorphic to proof nets at the operational level.
Abstract: Since the very beginning of the theory of linear logic it is known how to represent the \(\lambda \)-calculus as linear logic proof nets. The two systems however have different granularities, in particular proof nets have an explicit notion of sharing—the exponentials—and a micro-step operational semantics, while the \(\lambda \)-calculus has no sharing and a small-step operational semantics. Here we show that the linear substitution calculus, a simple refinement of the \(\lambda \)-calculus with sharing, is isomorphic to proof nets at the operational level.

19 citations


Cites background from "Proof nets and the call-by-value λ-..."

  • ...Our presentation of proof nets, similar to the one in [5], is nonstandard in at least four points—we suggest to have a quick look to Fig....

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  • ...An ancestor of this paper is [5], that adopts essentially the same syntax for proof nets....

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  • ...Our presentation of proof nets also refines the one in [5] with a micro-step operational semantics....

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  • ...proof nets—already at work by the author [5]—that intuitively corresponds to interaction nets (to work modulo cut with axioms) with hyper-wires, that is, wires connecting more than two ports (to have smooth contractions)....

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Journal ArticleDOI
TL;DR: It is proved that this extended calculus is a conservative refinement of Plotkin's one, in particular, the notions of solvability and potential valuability for this calculus coincide with those for plotkin's call-by-value lambda-calculus.
Abstract: We study an extension of Plotkin's call-by-value lambda-calculus via two commutation rules (sigma-reductions). These commutation rules are sufficient to remove harmful call-by-value normal forms from the calculus, so that it enjoys elegant characterizations of many semantic properties. We prove that this extended calculus is a conservative refinement of Plotkin's one. In particular, the notions of solvability and potential valuability for this calculus coincide with those for Plotkin's call-by-value lambda-calculus. The proof rests on a standardization theorem proved by generalizing Takahashi's approach of parallel reductions to our set of reduction rules. The standardization is weak (i.e. redexes are not fully sequentialized) because of overlapping interferences between reductions.

17 citations


Cites background from "Proof nets and the call-by-value λ-..."

  • ...cut-elimination via the call-by-value “boring” translation (·)v of λ-terms into proof-nets [Gir87, pp. 81-82], which decomposes the intuitionistic implication as follows: (A ⇒B)v = !(A ⊸B)v (see also [Acc15]). It turns out that the images under (·)v of a σ-redex and its contractum are equal modulo some non-structural cut-elimination steps. Note that Regnier’s σ-rules are contained in β-equivalence, while...

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References
More filters
Journal ArticleDOI
30 Jan 1987

3,947 citations


Additional excerpts

  • ...[19] Jean-Yves Girard (1987): Linear Logic....

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Journal ArticleDOI
TL;DR: This contribution was made possible only by the miraculous fact that the first members of the Editorial Board were sharing the same conviction about the necessity of Theoretical Computer Science.

1,480 citations

Journal ArticleDOI
TL;DR: This paper examines the old question of the relationship between ISWIM and the λ-calculus, using the distinction between call-by-value and call- by-name, and finds that operational equality is not preserved by either of the simulations.

1,240 citations


"Proof nets and the call-by-value λ-..." refers background in this paper

  • ...Usually, it is said to represent Plotkin’s call-by-value (CBV) λβv-calculus [34]....

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  • ...[34] Gordon D. Plotkin (1975): Call-by-Name, Call-by-Value and the lambdaCalculus....

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Dissertation
01 Jan 1990
TL;DR: In this paper, a representation of the normalisation of the lambda-calcul, a notion of calcul sous-jacente aux preuves-programmes, comme d'un parcours de graphe subordonne a des calculs dans une algebre equationnelle, is presented.
Abstract: Certaines preuves formelles peuvent aussi bien etre vues comme des programmes. D'ou vient qu'il y ait un recoupement entre theorie de la demonstration et informatique. Un evenement d'importance dans cette intersection recemment decouverte fut la construction en 86 par girard de la logique lineaire (11), decomposition des logiques classiques et intuitionnistes, pourvue de bonnes proprietes comme sa posterite en temoigne. Cette these presente un groupe de resultats sur 11: diverses caracterisations des reseaux, nouvelle synthaxe utilisee par girard pour noter le fragment multiplicatif de 11; extension d'une condition faible a 11 tout entiere; definition des connecteurs multiplicatifs generaux. Elle propose ensuite une representation de la normalisation du lambda-calcul, notion de calcul sous-jacente aux preuves-programmes, comme d'un parcours de graphe subordonne a des calculs dans une algebre equationnelle. Pour definir cette representation et prouver qu'elle preserve la finitude des calculs, on a recours a une representation intermediaire: les reseaux purs, dont l'etude est egalement poursuivie pour leur interet propre

151 citations


"Proof nets and the call-by-value λ-..." refers background in this paper

  • ...Such a calculus can be seen as an algebraic reformulation of proof nets for λ-calculus [14, 36], and turned out to have a simpler meta-theory than previous calculi with explicit substitutions....

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  • ...[15] Vincent Danos & Laurent Regnier (1999): Reversible, Irreversible and Optimal lambda-Machines....

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  • ...[14] Vincent Danos (1990): La Logique Linéaire appliqué à l’étude de divers processus de normalisation (principalement du λ-calcul)....

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  • ...The first such works were the PhD thesis of Vincent Danos [14] and Laurent Regnier [36], that focused on the call-by-name (CBN) translation....

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  • ...[21] Dimitri Hendriks & Vincent van Oostrom (2003): adbmal....

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Book ChapterDOI
01 Jun 1995

129 citations


"Proof nets and the call-by-value λ-..." refers methods in this paper

  • ...Regnier avoid explicit substitutions, use n-ary contractions, explicit axioms, and small-step exponential rules, see also [13]....

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