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Book ChapterDOI

Proof Nets and Explicit Substitutions

25 Mar 2000-pp 63-81
TL;DR: The simulation technique introduced in [10] is refined to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets and a version of typed λl with named variables is proposed which helps to better understand the complex mechanism of the explicit weakening notation.
Abstract: We refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the λl-calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed λl to proof nets. Finally, we propose a version of typed λl with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the λl-calculus with de Bruijn indices [8].

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Citations
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Proceedings ArticleDOI
19 Aug 2014
TL;DR: The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic, and shows that the LSC is a complexity-preserving abstraction of abstract machines.
Abstract: It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between small-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and a sketch of the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.

77 citations


Cites background from "Proof Nets and Explicit Substitutio..."

  • ...Explicit substitutions (ES) have been connected to linear logic by Kesner and co-authors in a sequence of works [26, 32, 33], culminating in the linear substitution calculus (LSC), a new formalism with ES behaviorally isomorphic to proof nets (introduced in [6], developed in [1, 3, 4, 7, 10], and bearing similarities with calculi by De Bruijn [25], Nederpelt [42], and Milner [41])....

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Book ChapterDOI
Delia Kesner1
11 Sep 2007
TL;DR: Very simple technology is used to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition.
Abstract: Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with meta-level substitutions) they were implementing. In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic's proof-nets.

53 citations


Cites background from "Proof Nets and Explicit Substitutio..."

  • ...But the λws-calculus has a complicated syntax and its named version [13] is even less intelligible....

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  • ...However, the strong normalisation proof for λws given in [13] reveals a natural semantics for composition of ES via Linear Logic’s proof-nets [19], suggesting that weakening (explicit erasure) and contraction (explicit duplication) can be added to the calculus without losing strong normalisation....

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Journal ArticleDOI
TL;DR: The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening.
Abstract: Since Mellies showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening. A typed version is also presented.

49 citations

Journal ArticleDOI
TL;DR: Normalization of the exponential reduction and confluence of the full one is proved and a translation of Boudol?s untyped ?-calculus with resources extended with a linear?nonlinear reduction a la Ehrhard and Regnier?s differential ?
Abstract: We define pure intuitionistic differential proof nets, extending Ehrhard and Regnier?s differential interaction nets with the exponential box of Linear Logic. Normalization of the exponential reduction and confluence of the full one is proved. These results are directed and adjusted to give a translation of Boudol?s untyped ?-calculus with resources extended with a linear?nonlinear reduction a la Ehrhard and Regnier?s differential ?-calculus. Such reduction comes in two flavours: baby-step and giant-step s-reduction. The translation, based on Girard?s encoding A?B~!A?B and as such extending the usual one for ?-calculus into proof nets, enjoys bisimulation for giant-step s-reduction. From this result we also derive confluence of both reductions.

44 citations


Cites background from "Proof Nets and Explicit Substitutio..."

  • ...Solutions proposed in LL include – adopting a syntax which identifies contractions made at several exponential depths, as in [23] – for now it seems hard to apply it in differential nets with boxes, we will see how the rule of codereliction against box introduces many difficulties; – using such an identification as an equivalence relation, as hinted in [12] for DINs and investigated in [3,8] for LL proof nets – an elegant solution, though it is less so with respect to freely moving around weakenings, as it may generate infinite trees with weakened leaves; – using it as a set of reductions, as in [4] – which is is the way we are adopting here....

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Proceedings Article
23 Aug 2010
TL;DR: In this article, an untyped structural λ-calculus, called λj, was introduced, which combines action at a distance with exponential rules decomposing the substitution by means of weakening, contraction and dereliction.
Abstract: Inspired by a recent graphical formalism for λ-calculus based on Linear Logic technology, we introduce an untyped structural λ-calculus, called λj, which combines action at a distance with exponential rules decomposing the substitution by means of weakening, contraction and dereliction. Firstly, we prove fundamental properties such as confluence and preservation of β-strong normalisation. Secondly, we use λj to describe known notions of developments and superdevelopments, and introduce a more general one called XL-development. Then we show how to reformulate Regnier's s-equivalence in λj so that it becomes a strong bisimulation. Finally, we prove that explicit composition or de-composition of substitutions can be added to λj while still preserving β-strong normalisation.

37 citations

References
More filters
Book ChapterDOI
08 Jul 1992
TL;DR: It is demonstrated how this may be used to describe standard binding constructions (let and letrec)—directly using substitution and fixed point induction as well as using ‘small-step’ rewriting semantics where substitution is interleaved with the mechanics of the following β-reductions.
Abstract: In this paper we consider rewrite systems that describe the λ-calculus enriched with recursive and non-recursive local definitions by generalizing the lsexplicit substitutions’ used by Abadi, Cardelli, Curien, and Levy [1] to describe sharing in λ-terms. This leads to ‘explicit cyclic substitutions’ that can describe the mutual sharing of local recursive definitions. We demonstrate how this may be used to describe standard binding constructions (let and letrec)—directly using substitution and fixed point induction as well as using ‘small-step’ rewriting semantics where substitution is interleaved with the mechanics of the following β-reductions.

44 citations


"Proof Nets and Explicit Substitutio..." refers background in this paper

  • ...Finally, in [10], the first two authors of this work showed for the first time that explicit substitutions could be tightly related to linear logic’s proof nets, by providing a translation into a variant of proof nets from λx [19, 4], a simple calculus with explicit substitutions and named variables, but no composition....

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Book ChapterDOI
02 Jul 1999
TL;DR: A notion of reduction for the proof nets of Linear Logic modulo an equivalence relation on the contraction links is proposed, that essentially amounts to consider the contraction as an associative commutative binary operator that can float freely in and out of proof net boxes.
Abstract: This paper proposes a notion of reduction for the proof nets of Linear Logic modulo an equivalence relation on the contraction links, that essentially amounts to consider the contraction as an associative commutative binary operator that can float freely in and out of proof net boxes. The need for such a system comes, on one side, from the desire to make proof nets an even more parallel syntax for Linear Logic, and on the other side from the application of proof nets to l-calculus with or without explicit substitutions, which needs a notion of reduction more flexible than those present in the literature. The main result of the paper is that this relaxed notion of rewriting is still strongly normalizing.

15 citations


"Proof Nets and Explicit Substitutio..." refers background or methods in this paper

  • ...We first remind the following result from [9]: Lemma 1 (Termination of R E )....

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  • ...Extended reduction modulo an equivalence relation Unfortunately, the original notion of reduction on PN is not well adapted to simulate neither the β rule of λ-calculus, nor the rules dealing with propagation of substitution in explicit substitution calculi: too many inessential details on the order of application of the rules are still present, and to make abstraction from them, one is naturally led to define an equivalence relation on PN , as is done in [9], where the following two equivalences are introduced:...

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  • ...We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating....

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  • ...3 Termination of RE We know from [9] that R E is terminating, and we can show easily that wc is terminating too, so if we could show that the wc-rule can be postponed with respect to all the other rules of R E , we would be easily done using a well-known abstract lemma....

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  • ...Finally, besides the equivalence relation defined in [9], for the sake of simulating explicit substitutions, we will also need an extra reduction rule allowing to remove unneeded weakening links:...

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Dissertation
01 Jan 1999
TL;DR: Kamareddine et rios as mentioned in this paper propose le lambda-se-calculus, which is a lambda-sigma-calculus with substitutions explicite.
Abstract: Le lambda-calcul est un formalisme simple et abstrait qui joue un role crucial dans l'etude des langages de programmation fonctionnelle et dans les assistants a la demonstration L'operation de base du lambda-calcul (la beta-reduction) pose de nombreux problemes d'implementation, c'est pourquoi abadi, cardelli, curien et levy ont propose le lambda-sigma-calcul, dans lequel l'operation de beta-reduction est decomposee en operations plus elementaires en introduisant des substitutions explicites De nombreux autres calculs avec substitutions explicites ont ete proposes Le principal probleme est de trouver un tel calcul qui soit a la fois confluent sur les termes avec metavariables (mc) et qui preserve la forte normalisation du lambda-calcul (psn) Dans cette these, on montre que le lambda-se-calcul (un calcul propose par kamareddine et rios) n'a pas la psn L'etude de la non psn dans lambda-sigma et dans lambda-se a permis d'obtenir le resultat principal de cette these, c'est-a-dire un nouveau calcul (nomme lambda-l) dont on montre qu'il possede les deux proprietes souhaitees Il s'agit de l'un des premiers calculs qui possedent ces deux proprietes et qui simulent la beta-reduction pas a pas Les techniques de preuve (pour la psn) sont nouvelles et ne ressemblent pas du tout aux preuves existantes pour des calculs sans composition de substitutions Une partie de la preuve de psn, ainsi que les preuves de confluence locale ont ete verifiees par un programme caml

6 citations


"Proof Nets and Explicit Substitutio..." refers background in this paper

  • ...As expected the λl-calculus enjoys the subject reduction property (see [16] for a detailed proof)....

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Journal ArticleDOI
TL;DR: In this article, the preservation of β-strong normalization by two different confluent λ-calculi with explicit substitutions defined in [96] was studied. But the particularity of these calculi, called λ d and λ n respectively, is that both have a (weak) composition operator for substitutions.
Abstract: This paper studies preservation of β-strong normalization by two different confluent λ-calculi with explicit substitutions defined in [96]; the particularity of these calculi, called λ d and λ dn respectively, is that both have a (weak) composition operator for substitutions. We apply an abstract simulation technique to reduce preservation of β-strong normalization of λ d and λ dn to that of another calculus, called λ f having no composition operator. Then, preservation of β-strong normalization of λ f is shown using the same technique as in [2]. As a consequence, λ d and λ dn become the first λ-calculi with explicit substitutions having (weak) composition and preserving β-strong normalization. As an aside, we also show how to apply our technique to reduce preservation of β-strong normalization of the calculus λ v in [20] to that of λ f .

5 citations


"Proof Nets and Explicit Substitutio..." refers methods in this paper

  • ...To achieve this result, we use the following abstract theorem (see for example [12]) : Theorem 4....

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