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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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BookDOI
01 Jan 2014
TL;DR: Strong Bridges and Strong Articulation Points are shown as well as weak bridges and strong articulation points in the second of a two-part series on how human interaction with concrete objects affects their ability to communicate.
Abstract: Strong Bridges and Strong Articulation Points

22 citations

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 Explicit Substitutio..."

  • ...Kesner and co-authors then explored the connection in various directions [20,28,29]....

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Journal ArticleDOI
TL;DR: An untyped structural lambda-calculus, called lambda j, is introduced, which combines actions at a distance with exponential rules decomposing the substitution by means of weakening, contraction and derelicition and some fundamental properties of lambda j such as confluence and preservation of beta-strong normalisation are proved.
Abstract: Inspired by a recent graphical formalism for lambda-calculus based on linear logic technology, we introduce an untyped structural lambda-calculus, called lambda j, which combines actions at a distance with exponential rules decomposing the substitution by means of weakening, contraction and derelicition. First, we prove some fundamental properties of lambda j such as confluence and preservation of beta-strong normalisation. Second, we add a strong bisimulation to lambda j by means of an equational theory which captures in particular Regnier's sigma-equivalence. We then complete this bisimulation with two more equations for (de)composition of substitutions and we prove that the resulting calculus still preserves beta-strong normalization. Finally, we discuss some consequences of our results.

18 citations

Dissertation
01 Jan 2009
TL;DR: In this paper, the authors present le terrain commun sur lequel s'appuient les parties suivantes, i.e., les reseaux de preuve de MALL, and present a version du lambda-calcul avec ressources de Boudol, ainsi qu'une traduction de celle-ci dans les intuitionnistes de DiLL.
Abstract: Cette these de theorie de la demonstration etudie les proprietes de la syntaxe parallele de la logique lineaire (LL) de Girard, les reseaux de preuve. La premiere partie presente le terrain commun sur lequel s'appuient les parties suivantes. En particulier, le paradigme des reseaux d'interaction de Lafont est utilise pour presenter les principaux objets de recherche de la these : d'un cote, les reseaux de preuve de Hughes et van Glabbeek pour la logique lineaire multiplicative additive (MAIL) ; de l'autre, les reseaux differentiels d'Ehrhard et Regnier pour la logique lineaire differentielle (DiLL) obtenue en ajoutant des operateurs differentiels a LL multiplicative exponentielle. Dans la deuxieme partie, nous nous concentrons sur MALL, en etudiant ses relations avec la semantique denotationnelle des espace hypercoherents d'Ehrhard. Un critere est etabli sur les reseaux de preuve de MALL caracterisant ceux interpretes (par la notion d'experience de Girard) comme des hypercliques, c'est-a-dire des objets des espaces hypercoherents. On montre ensuite la stabilite de ce critere par reduction des coupures. Dans la troisieme partie, nous passons a DiLL. Nous prouvons la confluence des reseaux purs de DiLL en utilisant un resultat de developpements finis. Ensuite, nous montrons un theoreme correspondant a la standardisation de LL (recemment prouve par Pagani et Tortora de Falco), a partir duquel la normalisation forte du cas simplement type peut etre deduite. Enfin, nous presentons une version du lambda-calcul avec ressources de Boudol, ainsi qu'une traduction de celle-ci dans les reseaux intuitionnistes de DiLL. Cette traduction permet de prouver la confluence de ce calcul.

17 citations

Book ChapterDOI
Catalin Dima1
06 Sep 2003
TL;DR: It is shown that discretizations of timed automata are, in general, context- sensitive languages over context-sensitive languages over Σ ∪ {1,δ + ,δ −, and give a class of automata that equals the class of languages that are discretization of timed Automata, and show that their emptiness problem is decidable.
Abstract: We give a new discretization of behaviors of timed automata In this discretization, timed languages are represented as sets of words containing action symbols, a clock tick symbol 1, and two delay symbols δ − (negative delay) and δ + (positive delay) Unlike the region construction, our discretization commutes with intersection We show that discretizations of timed automata are, in general, context-sensitive languages over Σ ∪ {1,δ + ,δ −, and give a class of automata that equals the class of languages that are discretizations of timed automata, and show that their emptiness problem is decidable

14 citations

References
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Journal ArticleDOI
30 Jan 1987

3,947 citations

Journal ArticleDOI
TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.
Abstract: Linear logic was introduced by Girard in 1987 [11] . Since then many results have supported Girard' s statement, \"Linear logic is a resource conscious logic,\" and related slogans . Increasingly, computer scientists have come to recognize linear logic as an expressive and powerful logic with connection s to a variety of topics in computer science . This column presents a.n intuitive overview of linear logic, some recent theoretical results, an d summarizes several applications of linear logic to computer science . Other introductions to linear logic may be found in [12, 361 .

2,304 citations


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

  • ...In this paper 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]....

    [...]

  • ...While we refer the interested reader to [13] for more details on linear logic in general, we give here a one-sided presentation of the sequent calculus for MELL:...

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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

Proceedings ArticleDOI
01 Dec 1989
TL;DR: The λ&sgr;-calculus is a refinement of the λ-Calculus where substitutions are manipulated explicitly, and provides a setting for studying the theory of substitutions, with pleasant mathematical properties.
Abstract: The ls-calculus is a refinement of the l-calculus where substitutions are manipulated explicitly. The ls-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical l-calculus and concrete implementations.

577 citations


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

  • ...The pioneer calculus with explicit substitutions, λσ, was introduced in [1] as a bridge between the classical λ-calculus and concrete implementations of functional programming languages....

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Book ChapterDOI
TL;DR: The chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism and subjectivism and that a representative class of algorithms can be modelized by means of standard mathematics.
Abstract: Publisher Summary This chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism (that leads to static modification) and subjectivism (that leads to syntactical abuses, in other terms, bureaucracy). The new approach initiated in this chapter rests on the use of a specific C*-algebra Λ* that has the distinguished property of bearing a (non associative) inner tensor product. The chapter describes that a representative class of algorithms can be modelized by means of standard mathematics.

321 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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