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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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Book ChapterDOI
07 Sep 2009
TL;DR: It is proved that pure differential nets are Church-Rosser modulo such equivalences, which generalizes to linear logic regular proof nets, and uses a result of finiteness of developments, given by strong normalization when blocking a suitable notion of "new" cuts.
Abstract: We study the confluence of Ehrhard and Regnier's differential nets with exponential promotion, in a pure setting. Confluence fails with promotion and codereliction in absence of associativity of (co)contractions. We thus introduce it as a necessary equivalence, together with other optional ones. We then prove that pure differential nets are Church-Rosser modulo such equivalences. This result generalizes to linear logic regular proof nets, where the same notion of equivalence was already studied in the literature, but only with respect to the problem of normalization in a typed setting. Our proof uses a result of finiteness of developments, which in this setting is given by strong normalization when blocking a suitable notion of "new" cuts.

9 citations


Cites background or methods or result from "Proof Nets and Explicit Substitutio..."

  • ...Furtherly, as can be deducted from [9], this can be the ground for new work on calculi with explicit substitutions: whether by extending some results to untyped calculi; or by considering explicit substitutions for nondeterministic calculi akin to Boudol’s λ-calculus with resources (see [6])....

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  • ...Our system of reductions and equivalences bears close resemblance to the one developed for MELL in [9]....

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  • ...We thus generalize the equivalences and reductions of [9], providing as a byproduct the first proof of confluence1 for such LL proof nets with equivalences in the completely pure case, as previous works concentrated on normalization in the typed one....

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  • ...The push equivalence9 has already been studied in the literature on proof nets and explicit substitutions [8,9]....

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  • ...In fact, stripping DiLL of all its differential features, the only difference is the absence in [9] of anything related to the p-reduction....

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Dissertation
21 Dec 2007
TL;DR: This Ph.D. thesis addresses the problem of giving computational interpretation to proofs in classical logic by presenting three calculi reflecting different approaches in the study of this area, and presents the dX calculus, the diagrammatic calculus for classical logic, whose diagrams originate from *X-terms.
Abstract: This PhD thesis addresses the problem of giving computational interpretation to proofs in classical logic As such, it presents three calculi reflecting different approaches in the study of this area The thesis consists of three parts The first part introduces the *X calculus, whose terms represent proofs in the classical sequent calculus, and whose reduction rules capture most of the features of cut-elimination in sequent calculus This calculus introduces terms which enable explicit implementation of erasure and duplication and to the best of our knowledge it is the first such calculus for classical logic The second part studies the possibility to represent classical computation diagrammatically We present the dX calculus, the diagrammatic calculus for classical logic, whose diagrams originate from *X-terms The principal difference lies in the fact that dX has a higher level of abstraction, capturing the essence of sequent calculus proofs, as well as the essence of classical cut-elimination The third part relates the first two It presents the ©X calculus, a one-dimensional counterpart of the diagrammatic calculus We start from *X, where we explicitly identify terms which should be considered the same These are the terms that code sequent proofs which are equivalent up to permutations of independent inference rules They also have the same diagrammatic representation Such identification induces the congruence relation on terms The reduction relation is defined modulo congruence rules, and reduction rules correspond to those of dX calculus

7 citations

Journal ArticleDOI
TL;DR: The conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL) is proved, which turns the quest for strong normalisation into one for non-erasing weak normalisation (WN), and indeed this result is used to prove SN of simply typed DiLL.
Abstract: We prove the conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL). The conservation theorem states that the property of having infinite reductions (here infinite chains of cut elimination steps) is preserved by non-erasing steps. This turns the quest for strong normalisation (SN) into one for non-erasing weak normalisation (WN), and indeed we use this result to prove SN of simply typed DiLL (with promotion). Along the way to the theorem we achieve a number of additional results having their own interest, such as a standardisation theorem and a slightly modified system of nets, DiLL ∂ϱ.

6 citations


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

  • ...After the first results stated in Ehrhard and Regnier (2006) but restricted to the promotion-free fragment, other works have strengthened this claim: as already mentioned, the first author showed WN of simply typed DiLL in Pagani (2009) (see also Gimenez (2011) for a different proof, based on reducibility candidates), the second showed confluence (or rather Church–Rosser modulo, as we will have occasion to explain) of the untyped case in Tranquilli (2009a), and we here add yet another brick to the solidity of the system....

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  • ...After the first results stated in Ehrhard and Regnier (2006) but restricted to the promotion-free fragment, other works have strengthened this claim: as already mentioned, the first author showed WN of simply typed DiLL in Pagani (2009) (see also Gimenez (2011) for a different proof, based on reducibility candidates), the second showed confluence (or rather Church–Rosser modulo, as we will have occasion to explain) of the untyped case in Tranquilli (2009a), and we here add yet another brick to the solidity of the system. Confluence in presence of a formal sum modelling nondeterminism means that such a nondeterministic choice is completely internal and does not depend on the reduction strategy. Finally, an enticing point of interest is the link with concurrency. Differentiation, as modelled by the codereliction rule, can be in fact seen as the introduction of a one-use entity that goes to only one among the possibly many who are asking for it. This has been employed by Ehrhard and Laurent (2007) to model concurrent communication, where only one among many may answer to a given signal....

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  • ...The push equivalence7 has already been studied in the literature on proof nets and explicit substitutions (Di Cosmo and Guerrini 1999; Di Cosmo et al. 2003)....

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  • ...Unlike LL, where such semantic equivalences can be added and studied a posteriori (as, for example, in Di Cosmo et al. (2003)), in DiLL they must be included in order to assure confluence (Tranquilli 2009a)....

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Journal ArticleDOI
TL;DR: This paper defines two labelled calculi with explicit substitutions and resource management, and observes a tight relationship between labels and the dynamics of substitutions, which will guide the design of a third calculus that combines the advantages of the previous two.
Abstract: Levy’s labelled λ-calculus has played an important role in the understanding of the Geometry of Interaction and its applications to the implementation of λ-evaluators: labels relate to the multiplicative information of paths In this paper, we generalise the structure of labels, and the underlying term structure, in order to keep track of exponential information too We first define two labelled calculi with explicit substitutions and resource management, where labels are in close correspondence with paths in call-by-value and call-by-name translations of the λ-calculus into linear logic proof nets, respectively We observe a tight relationship between labels and the dynamics of substitutions; this will then guide us through the design of a third calculus that combines the advantages of the previous two, where labels fully reflect the dynamics of substitutions

6 citations

01 Jan 2010
TL;DR: It is proved that the stratification condition on regions, already used in type and effect systems to assure termination, is equivalent to completely avoid the use of recursion in the types used in the translation, obtaining a logical characterization of stratification.
Abstract: We study a lambda-calculus with references and a types and effects system. In the first part of the paper, we translate it into the ordinary lambda-calculus with products, implementing an interacting family of state monads localized at sets of regions. In general the target language must be endowed with recursive types. However we prove that the stratification condition on regions, already used in type and effect systems to assure termination, is equivalent to completely avoid the use of recursion in the types used in the translation. We thus obtain a logical characterization of stratification, and by simulation we also provide a new proof that it yields termination. In the second part of the paper we extend the call-by-value translation of ordinary lambda-terms in linear logic proof nets to the calculus with references. This allows for a parallel evaluation of the calculus that preserves its sequential semantics.

5 citations


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

  • ...Such equivalences where already studied in literature about explicit substitutions [13], or for differential nets [14](3)....

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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]....

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  • ...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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