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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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Journal ArticleDOI
TL;DR: The origins of the λx family of calculi of explicit substitution with proper variable names are recounted, including the original result of preservation of strong β-normalization based on the use of synthetic reductions for garbage collection.
Abstract: This paper recounts the origins of the ?x family of calculi of explicit substitution with proper variable names, including the original result of preservation of strong β-normalization based on the use of synthetic reductions for garbage collection. We then discuss the properties of a variant of the calculus which is also confluent for "open" terms (with meta-variables), and verify that a version with garbage collection preserves strong β-normalization (as is the state of the art), and we summarize the relationship with other efforts on using names and garbage collection rules in explicit substitution.

14 citations


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

  • ...It is not known whether the weakening used by λws carries over to using names [19]....

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Proceedings ArticleDOI
01 Jan 2013
TL;DR: This paper gives a new presentation of MELL proof nets, without any commutative cut-elimination rule, which is the first proof of strong normalization for MELL which does not rely on any form of confluence, and so it smoothly scales up to full linear logic.
Abstract: Strong normalization for linear logic requires elaborated rewriting techniques. In this paper we give a new presentation of MELL proof nets, without any commutative cut-elimination rule. We show how this feature induces a compact and simple proof of strong normalization, via reducibility candidates. It is the first proof of strong normalization for MELL which does not rely on any form of confluence, and so it smoothly scales up to full linear logic. Moreover, it is an axiomatic proof, as more generally it holds for every set of rewriting rules satisfying three very natural requirements with respect to substitution, commutation with promotion, full composition, and Kesner's IE property. The insight indeed comes from the theory of explicit substitutions, and from looking at the exponentials as a substitution device.

13 citations

Journal ArticleDOI
TL;DR: A framework called the prismoid of resources where each vertex is a language which refines the @l-calculus by using a different choice to make explicit or implicit the definition of the contraction, weakening, and substitution operations.

13 citations

Dissertation
08 Dec 2006
TL;DR: This dissertation contributes to extend its framework in the directions of proof-theoretic formalisms that are appealing for logical purposes like proof-search, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning.
Abstract: At the heart of the connections between Proof Theory and Type Theory, the Curry-Howard correspondence provides proof-terms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the directions of proof-theoretic formalisms (such as sequent calculus) that are appealing for logical purposes like proof-search, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning. Part I is entitled Proof-terms for Intuitionistic Implicational Logic. Its contributions use rewriting techniques on proof-terms for natural deduction (lambda-calculus) and sequent calculus, and investigate normalisation and cut-elimination, with call-by-name and call-by-value semantics. In particular, it introduces proof-term calculi for multiplicative natural deduction and for the depth-bounded sequent calculus G4. The former gives rise to a calculus with explicit substitutions, weakenings and contractions that refines the lambda-calculus and beta-reduction, and preserves strong normalisation with a full notion of composition of substitutions. The latter gives a new insight to cut-elimination in G4. Part II, entitled Type Theory in Sequent Calculus develops a theory of Pure Type Sequent Calculi (PTSC), which are sequent calculi that are equivalent (with respect to provability and normalisation) to Pure Type Systems but better suited for proof-search, in connection with proof-assistant tactics and proof-term enumeration algorithms. Part III, entitled Towards Classical Logic, presents some approaches to classical type theory. In particular it develops a sequent calculus for a classical version of System Fomega. Beyond such a type theory, the notion of equivalence of classical proofs becomes crucial and, with such a notion based on parallel rewriting in the Calculus of Structures, we compute canonical representatives of equivalent proofs.

13 citations

Book ChapterDOI
07 Jul 2008
TL;DR: The strong normalisation proof is based on implicit substitution rather than explicit substitution, so that it turns out to be modular w.r.t. the well-known proofs for typed lambda-calculus.
Abstract: We study perpetuality in calculi with explicit substitutions having full composition A simple perpetual strategy is used to define strongly normalising terms inductively This gives a simple argument to show preservation of β-strong normalisation as well as strong normalisation for typed terms Particularly, the strong normalisation proof is based on implicit substitution rather than explicit substitution, so that it turns out to be modular wrt the well-known proofs for typed lambda-calculus All the proofs we develop are constructive

10 citations


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

  • ...Proofs using the first technique are for example those for λws [6] and λlxr [14], based on the wellfoundedness of the reduction relation for multiplicative exponential linear logic...

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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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