Intuitionistic differential nets and lambda-calculus

doi:10.1016/J.TCS.2010.12.022

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Intuitionistic differential nets and lambda-calculus

Journal Article•DOI•

Intuitionistic differential nets and lambda-calculus

Paolo Tranquilli¹•Institutions (1)

École normale supérieure de Lyon¹

01 Apr 2011-Theoretical Computer Science (Elsevier)-Vol. 412, Iss: 20, pp 1979-1997

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 ?

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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.

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Proceedings Article•DOI•

Collapsing non-idempotent intersection types

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Thomas Ehrhard¹•Institutions (1)

Pierre-and-Marie-Curie University¹

01 Jan 2012

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.

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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.

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49 citations

Book Chapter•DOI•

Solvability in resource lambda-calculus

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Michele Pagani¹, Simona Ronchi Della Rocca¹•Institutions (1)

University of Turin¹

20 Mar 2010

TL;DR: This work defines a term solvable whenever there is a simple head context reducing the term into a sum where at least one addend is the identity, and gives a syntactical, operational and logical characterization of this kind of solvability.

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Abstract: The resource calculus is an extension of the λ-calculus allowing to model resource consumption. Namely, the argument of a function comes as a finite multiset of resources, which in turn can be either linear or reusable, giving rise to non-deterministic choices, expressed by a formal sum. Using the λ-calculus terminology, we call solvable a term that can interact with the environment: solvable terms represent meaningful programs. Because of the non-determinism, different definitions of solvability are possible in the resource calculus. Here we study the optimistic (angelical, or may) notion, and so we define a term solvable whenever there is a simple head context reducing the term into a sum where at least one addend is the identity. We give a syntactical, operational and logical characterization of this kind of solvability.

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43 citations

Cites background from "Intuitionistic differential nets an..."

...The outer -reduction o −→ is the closure to linear contexts of the steps given in Definition 3....
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...Definition 5....
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...Partially founded by the Italian MIUR project CONCERTO, and the French ANR projet blanc CHOCO, ANR-07-BLAN-0324. showed a Curry-Howard correspondence between this calculus and Ehrhard and Regnier’s differential nets [3]....
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...This paper deals extensively with the weakest notion of solvability, which asks that a term is solvable whenever a suitable context filled with it reduces to a sum, where at least one addend is the identity....
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...head-normal form....
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Journal Article•DOI•

Linearity, Non-determinism and Solvability

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Michele Pagani¹, Simona Ronchi Della Rocca²•Institutions (2)

University of Paris¹, University of Turin²

01 Jan 2010-Fundamenta Informaticae

TL;DR: This work studies the notion of solvability in the resource calculus, an extension of the λ-calculus modelling resource consumption, and gives a syntactical, operational and logical characterization for the may-solvability and only a partial characterization of the must-solvable.

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Abstract: We study the notion of solvability in the resource calculus, an extension of the λ-calculus modelling resource consumption. Since this calculus is non-deterministic, two different notions of solvability arise, one optimistic (angelical, may) and one pessimistic (demoniac, must). We give a syntactical, operational and logical characterization for the may-solvability and only a partial characterization of the must-solvability. Finally, we discuss the open problem of a complete characterization of the must-solvability.

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34 citations

Journal Article•DOI•

Proof nets and the call-by-value λ-calculus

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Beniamino Accattoli¹•Institutions (1)

École Polytechnique¹

16 Nov 2015-Theoretical Computer Science

TL;DR: This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets, and identifies a subcalculus that is shown to be as expressive as the full calculus.

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Abstract: This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets. The presentation is carefully tuned in order to realize an isomorphism between the two systems: every single rewriting step on the calculus maps to a single step on proof nets, and vice-versa. In this way, we obtain an algebraic reformulation of proof nets. Moreover, we provide a simple correctness criterion for our proof nets, which employ boxes in an unusual way, and identify a subcalculus that is shown to be as expressive as the full calculus.

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34 citations

Cites background from "Intuitionistic differential nets an..."

...[38] Paolo Tranquilli (2009): Nets Between Determinism and Nondeterminism....
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...Lionel Vaux [40] and Paolo Tranquilli [38, 39] study the relationship between the differential λ-calculus and differential proof nets....
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...[39] Paolo Tranquilli (2011): Intuitionistic differential nets and lambda-calculus....
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Book Chapter•DOI•

Parallel Reduction in Resource Lambda-Calculus

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Michele Pagani¹, Paolo Tranquilli²•Institutions (2)

University of Turin¹, Paris Diderot University²

03 Dec 2009

TL;DR: This work defines parallel reduction in resource calculus and applies the technique by Tait and Martin-Lof to achieve confluence, and slightly generalizes a technique by Takahashi to obtain a standardization result.

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Abstract: We study the resource calculus --- the non-lazy version of Boudol's *** -calculus with resources. In such a calculus arguments may be finitely available and mixed, giving rise to nondeterminism, modelled by a formal sum. We define parallel reduction in resource calculus and we apply, in such a nondeterministic setting, the technique by Tait and Martin-Lof to achieve confluence. Then, slightly generalizing a technique by Takahashi, we obtain a standardization result.

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33 citations

Cites background or methods from "Intuitionistic differential nets an..."

...In [4,10] this reduction is called the giant-step one (hence the name of the rule) to distinguish it from the baby-step one we will discuss in Section 5....
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...Indeed, the second author shows in [4] that resource calculus corresponds to the intuitionistic minimal fragment of differential nets with promotion [5], exactly as λ-calculus corresponds to the intuitionistic minimal fragment of linear logic proof-nets [6]....
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...We conclude the paper by discussing another, more atomic reduction of resource terms, called baby-step reduction in [4] (here Definition 10)....
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...This way of writing the application comes from Girard’s linear logic [6]: indeed !-marked arguments (called perpetual) correspond exactly to exponential boxes (see [4]), the synchronized areas of proofs viable for non-linear operations (duplication and erasing)....
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References

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Journal Article•DOI•

Linear logic

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Jean-Yves Girard¹•Institutions (1)

University of Paris¹

30 Jan 1987

3,947 citations

Journal Article•DOI•

Linear logic

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Patrick Lincoln

01 May 1992-Sigact News

TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.

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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 .

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2,304 citations

"Intuitionistic differential nets an..." refers background in this paper

...Introduction Twenty years ago Jean-Yves Girard introduced Linear Logic (LL, [13]) starting from a fine analysis of the coherent semantics he had introduced for system F....
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Journal Article•DOI•

Fiftieth volume of theoretical computer science

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Maurice Nivat

01 Jan 1988-Theoretical Computer Science

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.

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Abstract: The collection of TCS issues is about 1 meter high, 17,000 pages long and it contains 1100 papers. When in 1974 Einar Fredriksson and myself started talking about the creation of a journal dedicated to Theoretical Computer Science we were very far from even dreaming that it could take such an extension within twelve years. We were also a bit shy: what could such a journal, very theoretical indeed and hard to read, be useful to, and who would read it? Fortunately, some people encouraged us and indeed helped us a lot, Mike Paterson who was at that time President of EATCS and who accepted to become Associate Editor, Albert Meyer who was a very active editor at the beginning, Arto Salomaa, who was to become President of EATCS shortly afterwards. Indeed, I should mention all the first members of the Editorial Board, for TCS would never have come to existence without them. Theoretical Computer Science is not a clearly defined discipline with neat borderlines: it is more a state of mind, the conviction that the observed computation phenomena can be formally described and analysed as any physical phenomenon; the conviction that such a formal description helps to understand these phenomena and to master them in order to design better algorithms, better computers, better systems. Our fundamental activity is not to prove theorems in strange mathematical theories, it is to model a complicated reality and in this respect it has to be compared with theoretical physics or what we call in French “Mecanique rationnelle”. This comparison can be pursued rather far, for we also use all possible mathematical concepts and methods and when we do not find appropriate ones in traditional mathematics we create them. The aim is quite clear: using the compact and unambiguous language of mathematics brings to life concepts and methods which will be useful to all designers, builders and users of computer systems, exactly in the same way as matrix calculus or Fourier series and transforms are useful to all engineers and technicians in the electric and electronic industry. And when one thinks about the amount of time it took to build the mathematical theory of matrices and to polish and simplify it up to the state in which it could be taught to all future engineers and become a tool in daily use, we can be extremely satisfied by the development of Theoretical Computer Science. It is true that concepts and methods which were still vague and unclear when TCS was created became essential tools for all industrial designers and manufacturers, in algorithmics, in semantics, in automata theory and control, etc. . . . Certainly, TCS can be proud to have contributed to this development. Coming back to what I was saying a few minutes ago, 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

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1,480 citations

Journal Article•DOI•

The structure of multiplicatives

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Vincent Danos¹, Laurent Regnier¹•Institutions (1)

Centre national de la recherche scientifique¹

01 Oct 1989-Archive for Mathematical Logic

TL;DR: Investigating Girard's new propositionnal calculus, which aims at a large scale study of computation, there is a stumble quickly on that question: What is a multiplicative connective?

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Abstract: Investigating Girard's new propositionnal calculus which aims at a large scale study of computation, we stumble quickly on that question: What is a multiplicative connective? We give here a detailed answer together with our motivations and expectations.

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402 citations

"Intuitionistic differential nets an..." refers background in this paper

...Though structures already have computational meaning, we define the correctness criterion following the Danos-Regnier one for LL proof nets [6]....
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Journal Article•DOI•

The differential Lambda-calculus

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Thomas Ehrhard¹, Laurent Regnier¹•Institutions (1)

Centre national de la recherche scientifique¹

02 Dec 2003-Theoretical Computer Science

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.

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Abstract: We present an extension of the lambda-calculus with differential constructions. We 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.

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307 citations

"Intuitionistic differential nets an..." refers background in this paper

...Non linear and linear substitutions enjoy the same properties found in [11], though due to the simpler syntax proofs are somewhat easier....
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...Differential λ-calculus and differential nets Anatural direction of investigation arising from [12] and [11] is the questionwhether differential λ-calculus can be translated into differential nets....
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...The second problem arises from the particular syntax presented in [11]....
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...The substitutions employed are those found in differential λ-calculus [11], most notably the linear one....
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...Recently Ehrhard has refined the coherent semantics by means of topological vector spaces and continuous linear maps [9,10], and again from such semantical refinement the same author and Regnier presented extensions with syntactic differential operators for both Linear Logic [12] and λ-calculus [11]....
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