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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
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Proceedings ArticleDOI
John O. Lamping1
01 Dec 1989
TL;DR: An algorithm for lambda expression reduction that avoids any copying that could later cause duplication of work and is optimal in the sense defined by Lévy is presented.
Abstract: We present an algorithm for lambda expression reduction that avoids any copying that could later cause duplication of work. It is optimal in the sense defined by Levy. The basis of the algorithm is a graphical representation of the kinds of commonality that can arise from substitutions; the idea can be adapted to represent other kinds of expressions besides lambda expressions. The algorithm is also well suited to parallel implementations, consisting of a fixed set of local graph rewrite rules.

275 citations


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

  • ...1 Using various translations of the λ-calculus into proof nets, new abstract machines have been proposed, exploiting the Geometry of Interaction and the Dynamic Algebras [14, 2, 5], leading to the works on optimal reduction [15, 17]....

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Proceedings ArticleDOI
01 Feb 1992
TL;DR: This paper connects and explains the geometry of interaction and Lamping's graphs, which offer a new understanding of computation, as well as ideas for efficient and correct implementations.
Abstract: Lamping discovered an optimal graph-reduction implementation of the l-calculus. Simultaneously, Girard invented the geometry of interaction, a mathematical foundation for operational semantics. In this paper, we connect and explain the geometry of interaction and Lamping's graphs. The geometry of interaction provides a suitable semantic basis for explaining and improving Lamping's system. On the other hand, graphs similar to Lamping's provide a concrete representation of the geometry of interaction. Together, they offer a new understanding of computation, as well as ideas for efficient and correct implementations.

257 citations


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

  • ...1 Using various translations of the λ-calculus into proof nets, new abstract machines have been proposed, exploiting the Geometry of Interaction and the Dynamic Algebras [14, 2, 5], leading to the works on optimal reduction [15, 17]....

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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

Book ChapterDOI
10 Apr 1995
TL;DR: A simply typed λ-term whose computation in the λσ-calculus does not always terminate is presented.
Abstract: We present a simply typed λ-term whose computation in the λσ-calculus does not always terminate.

142 citations


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

  • ...But λσ does not preserve β-strong normalization as shown by Mellies, who exhibited a well-typed term which, due to the substitution composition rules in λσ, is not λσ-strongly normalizing [18]....

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Book ChapterDOI
01 Jun 1995

129 citations


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

  • ...Meanwhile, in the linear logic community, many studies focused of the connection between λ-calculus (without explicit substitutions) and proof nets, trying to find the proper variant or extension of proof nets that could be used to cleanly simulate β-reduction, like in [7]....

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