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Proceedings ArticleDOI

Collapsing non-idempotent intersection types

01 Jan 2012-pp 273
TL;DR: This work construction of a new model, which features a new duality, is presented, and how to use it for reducing normalization results in idempotent intersection types to purely combinatorial methods is explained.
Abstract: We proved recently that the extensional collapse of the relational model of linear logic coincides with its Scott model, whose objects are preorders and morphisms are downwards closed relations. This result is obtained by the construction of a new model whose objects can be understood as preorders equipped with a realizability predicate. We present this model, which features a new duality, and explain how to use it for reducing normalization results in idempotent intersection types (usually proved by reducibility) to purely combinatorial methods. We illustrate this approach in the case of the call-by-value lambda-calculus, for which we introduce a new resource calculus, but it can be applied in the same way to many different calculi.
Citations
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Journal ArticleDOI
TL;DR: This article explores the use of non-idempotent intersection types in the framework of the λ-calculus by replacing the reducibility technique with trivial combinatorial arguments.
Abstract: This article explores the use of non-idempotent intersection types in the framework of the λ-calculus. Different topics are presented in a uniform framework: head normalization, weak normalization, weak head normalization, strong normalization, inhabitation, exact bounds and principal typings. The reducibility technique, traditionally used when working with idempotent types, is replaced in this framework by trivial combinatorial arguments.

53 citations


Cites background or methods from "Collapsing non-idempotent intersect..."

  • ...In [30], strong normalization for a call-by-value λ-calculus is characterized by means of intersection types, both idempotent and non-idempotent....

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  • ...Following the same approach, and exploiting in particular the validity of the Taylor formula in the relational model of the λ-calculus and a resource call-by-value λ-calculus presented in [30], [13] provides a characterization of solvability for a call-by-value λ-calculus in terms of non-idempotent IT1....

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Book ChapterDOI
05 Apr 2014
TL;DR: Preserving confluence, Plotkin’s original reduction is extended without adding extra syntactical constructors, and a call-by-value operational characterization of solvable terms is given in a relational model, based on Linear Logic, satisfying the Taylor expansion formula.
Abstract: In Plotkin’s call-by-value lambda-calculus, solvable terms are characterized syntactically by means of call-by-name reductions and there is no neat semantical characterization of such terms. Preserving confluence, we extend Plotkin’s original reduction without adding extra syntactical constructors, and we get a call-by-value operational characterization of solvable terms. Moreover, we give a semantical characterization of solvable terms in a relational model, based on Linear Logic, satisfying the Taylor expansion formula. As a technical tool, we also use a resource-sensitive calculus (with tests) in which the elements of the model are definable.

50 citations


Cites background or methods from "Collapsing non-idempotent intersect..."

  • ...Its syntax is defined by the following grammar (the same as in [15]):...

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  • ...We use the relational model of [15], which is also a model of ordinary λ-calculus, unlike the model V of [8]....

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  • ...For more details we refer the reader to [20,15]....

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  • ...The starting points of our work are [7,6,15]....

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  • ...The resource λv -calculus consists of the language rΛ t and the reduction →v: it is the resource CBV λ-calculus of [15] plus the σ1- and σ3-rules....

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Proceedings ArticleDOI
05 Sep 2016
TL;DR: The Bang Calculus is introduced, an untyped functional calculus in which the promotion operation of Linear Logic is made explicit and where application is a bilinear operation, and an adequacy theorem is proved by means of a resourcebang Calculus whose design is based on Differential Linear Logic.
Abstract: We introduce and study the Bang Calculus, an untyped functional calculus in which the promotion operation of Linear Logic is made explicit and where application is a bilinear operation. This calculus, which can be understood as an untyped version of Call-By-Push-Value, subsumes both Call-By-Name and Call-By-Value lambda-calculi, factorizing the Girard's translations of these calculi in Linear Logic. We build a denotational model of the Bang Calculus based on the relational interpretation of Linear Logic and prove an adequacy theorem by means of a resource Bang Calculus whose design is based on Differential Linear Logic.

43 citations


Cites background or methods from "Collapsing non-idempotent intersect..."

  • ...Actually, it can be shown that Lemma 7 holds not only for the relational semantics but also for any model of the bang calculus coming from a model of differential LL which satisfies a version of the Taylor formula [17, 18]....

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  • ...1 below) in a purely combinatorial way (along the lines of [10, 17]), without using a reducibility argument....

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  • ...Our definition of Taylor expansion is a generalization of analogous deeply studied notions for CBN [20, 22, 33] and CBV [10, 17] λ-calculi....

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  • ...It can be shown that both call-by-name [21, 22] and call-by-value [10, 17] resource calculi can be embedded in the resource bang calculus....

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  • ...1 is a generalization of System R [14] for CBN λ-calculus and the type system used in [17] for CBV λ-calculus....

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Book ChapterDOI
02 Dec 2018
TL;DR: Operational and implementative studies of proposed call-by-value lambda-calculus extensions are provided, showing that they are equivalent with respect to termination, and also at the level of time cost models.
Abstract: The good properties of Plotkin’s call-by-value lambda-calculus crucially rely on the restriction to weak evaluation and closed terms. Open call-by-value is the more general setting where evaluation is weak but terms may be open. Such an extension is delicate and the literature contains a number of proposals. Recently, we provided operational and implementative studies of these proposals, showing that they are equivalent with respect to termination, and also at the level of time cost models.

29 citations


Cites background or methods from "Collapsing non-idempotent intersect..."

  • ...For Plotkin’s original CbV λ-calculus, it has been introduced by Ehrhard [23]....

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  • ...Here we introduce Ehrhard’s multi type system for CbV [23] and show that—with respect to it—the fireball calculus λfire fails the denotational test of the benchmark sketched in Sect....

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  • ...We connect the size of type derivations for a term with its evaluation via rewriting, and the size of elements in its denotation with the size of its normal form, in a model coming from the linear logic interpretation of CbV and presented as a type system: Ehrhard’s relational semantics for CbV [23]....

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  • ...Among them, Ehrhard’s [23], Diaz-Caro, Manzonetto, and Pagani’s [22], Carraro and Guerrieri’s [13], Ehrhard and Guerrieri’s [24], and Guerrieri’s [31] deal with CbV, while de Carvalho’s [14,16], Bernadet and Lengrand’s [8], de Carvalho, Pagani, and Tortora de Falco’s [17], Accattoli, Graham-Lengrand, and Kesner’s [2] provide exact bounds....

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  • ...It recasts in CbV de Carvalho’s work for CbN [14,16], building on a type system introduced by Ehrhard [23] for Plotkin’s original CbV λ-calculus λv [45]....

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01 Jan 2017
TL;DR: Two resource aware typing systems for the λμ-calculus based on non-idempotent intersection and union types are defined and typability provides upper bounds for the length of head-reduction sequences and maximal reduction sequences.
Abstract: We define two resource aware typing systems for the λμ-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments –based on decreasing measures of type derivations– to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the length of head-reduction sequences and maximal reduction sequences. 1998 ACM Subject Classification F.4.1 Mathematical Logic

27 citations

References
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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

Journal ArticleDOI
TL;DR: Calculi are introduced, based on a categorical semantics for computations, that provide a correct basis for proving equivalence of programs for a wide range of notions of computation.
Abstract: The λ-calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with λ-terms. However, if one goes further and uses βη-conversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with total functions from values to values ) that may jeopardise the applicability of theoretical results. In this paper we introduce calculi, based on a categorical semantics for computations , that provide a correct basis for proving equivalence of programs for a wide range of notions of computation .

1,825 citations

Journal ArticleDOI
TL;DR: This work presents an extension of the lambda-calculus with differential constructions, and state and prove some basic results (confluence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-Calculus.

307 citations

01 Jan 2009
TL;DR: In this article, the authors voulu guider le lecteur a travers la thematique, en lui tracant un chemin progressif et raisonne, which part des mecanismes symboliques de l'elimination des coupures, pour aboutir a leur transcription algebrique en diagrammes de coherence dans les categories monoidales.
Abstract: La theorie de la demonstration est issue d'une histoire courte et tumultueuse, construite en marge des mathematiques traditionnelles. Aussi, son langage reste souvent idiosyncratique: calcul des sequents, elimination des coupures, propriete de la sous-formule, etc. Dans cet article, nous avons voulu guider le lecteur a travers la thematique, en lui tracant un chemin progressif et raisonne, qui part des mecanismes symboliques de l'elimination des coupures, pour aboutir a leur transcription algebrique en diagrammes de coherence dans les categories monoidales. Cette promenade spirituelle au point de convergence de l'algebre et de la linguistique est ardue parfois, mais aussi pleine d'attraits: car a ce jour, aucune langue (formelle ou informelle) n'a ete autant etudiee par les mathematiciens que la langue des demonstrations logiques.

199 citations

Journal ArticleDOI
06 Nov 2006
TL;DR: This work introduces interaction nets for a fragment of the differential lambda-calculus and exhibits in this framework a new symmetry between the of course and the why not modalities of linear logic, which is completely similar to the symmetries between the tensor and par connectives oflinear logic.
Abstract: We introduce interaction nets for a fragment of the differential lambda-calculus and exhibit in this framework a new symmetry between the of course and the why not modalities of linear logic, which is completely similar to the symmetry between the tensor and par connectives of linear logic. We use algebraic intuitions for introducing these nets and their reduction rules, and then we develop two correctness criteria (weak typability and acyclicity) and show that they guarantee strong normalization. Finally, we outline the correspondence between this interaction nets formalism and the resource lambda-calculus.

157 citations