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

##### Citations

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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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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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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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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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

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3,947 citations

### Additional excerpts

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

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1,480 citations

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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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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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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