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Статті в журналах з теми "Lambda calculi"

Автор: Grafiati

Опубліковано: 11 березня 2023

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1

Zamdzhiev, Vladimir. "Computational Adequacy for Substructural Lambda Calculi." Electronic Proceedings in Theoretical Computer Science 333 (February 8, 2021): 322–34. http://dx.doi.org/10.4204/eptcs.333.22.

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2

KAMAREDDINE, FAIROUZ. "Typed $\lambda$-calculi with one binder." Journal of Functional Programming 15, no. 05 (June 8, 2005): 771. http://dx.doi.org/10.1017/s095679680500554x.

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3

Boudol, G. "Lambda-Calculi for (Strict) Parallel Functions." Information and Computation 108, no. 1 (January 1994): 51–127. http://dx.doi.org/10.1006/inco.1994.1003.

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4

Staples, John. "Delaying unification algorithms for lambda calculi." Theoretical Computer Science 56, no. 3 (March 1988): 277–88. http://dx.doi.org/10.1016/0304-3975(88)90135-1.

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5

BOUDOL, GÉRARD, PIERRE-LOUIS CURIEN, and CAROLINA LAVATELLI. "A semantics for lambda calculi with resources." Mathematical Structures in Computer Science 9, no. 4 (August 1999): 437–82. http://dx.doi.org/10.1017/s0960129599002893.

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Анотація:

We present the λ-calculus with resources λr, and two variants of it: a deterministic restriction λm and an extension λcr with a convergence testing operator. These calculi provide a control on the substitution process – deadlocks may arise if not enough resources are available to carry out all the substitutions needed to pursue a computation. The design of these calculi was motivated by Milner's encoding of the λ-calculus in the π-calculus. As Boudol and Laneve have shown elsewhere, the discriminating power of λm (given by the contextual observational equivalence) over λ-terms coincides with that induced by Milner's π-encoding, and coincides also with that provided by the lazy algebraic semantics (Lévy–Longo trees). The main contribution of this paper is model-theoretic. We define and solve an appropriate domain equation, and show that the model thus obtained is fully abstract with respect to λcr. The techniques used are in the line of those used by Abramsky for the lazy λ-calculus, the main departure being that the resource-consciousness of our calculi leads us to introduce a non-idempotent form of intersection types.

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6

Katayama, Susumu. "Computable Variants of AIXI which are More Powerful than AIXItl." Journal of Artificial General Intelligence 10, no. 1 (January 1, 2019): 1–23. http://dx.doi.org/10.2478/jagi-2019-0001.

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Abstract This paper presents Unlimited Computable AI, or UCAI, that is a family of computable variants of AIXI. UCAI is more powerful than AIXItl, which is a conventional family of computable variants of AIXI, in the following ways: 1) UCAI supports models of terminating computation, including typed lambda calculi, while AIXItl only supports Turing machine with timeout ˜t, which can be simulated by typed lambda calculi for any ˜t; 2) unlike UCAI, AIXItl limits the program length to some ˜l .

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7

ZORZI, MARGHERITA. "On quantum lambda calculi: a foundational perspective." Mathematical Structures in Computer Science 26, no. 7 (November 17, 2014): 1107–95. http://dx.doi.org/10.1017/s0960129514000425.

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In this paper, we propose an approach to quantum λ-calculi. The ‘quantum data-classical control’ paradigm is considered. Starting from a measurement-free untyped quantum λ-calculus calledQ, we will study standard properties such as confluence and subject reduction, and some good quantum properties. We will focus on the expressive power, analysing the relationship with other quantum computational models. Successively, we will add an explicit measurement operator toQ. On the resulting calculus, calledQ*, we will propose a complete study of reduction sequences regardless of their finiteness, proving confluence results. Moreover, since the stronger motivation behind quantum computing is the research of new results in computational complexity, we will also propose a calculus which captures the three classes of quantum polytime complexity, showing an ICC-like approach in the quantum setting.

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8

Mulmuley, Ketan. "Fully abstract submodels of typed lambda calculi." Journal of Computer and System Sciences 33, no. 1 (August 1986): 2–46. http://dx.doi.org/10.1016/0022-0000(86)90041-3.

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9

Nielson, Flemming, and Hanne Riis Nielson. "Prescriptive Frameworks for Multi-Level Lambda-Calculi." ACM SIGPLAN Notices 32, no. 12 (December 1997): 193–202. http://dx.doi.org/10.1145/258994.259018.

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10

Joy, M. "Lambda Calculi: A Guide For Computer Scientists." Computer Journal 38, no. 1 (January 1, 1995): 78–79. http://dx.doi.org/10.1093/comjnl/38.1.78-a.

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11

Valliappan, Nachiappan, Fabian Ruch, and Carlos Tomé Cortiñas. "Normalization for fitch-style modal calculi." Proceedings of the ACM on Programming Languages 6, ICFP (August 29, 2022): 772–98. http://dx.doi.org/10.1145/3547649.

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Анотація:

Fitch-style modal lambda calculi enable programming with necessity modalities in a typed lambda calculus by extending the typing context with a delimiting operator that is denoted by a lock. The addition of locks simplifies the formulation of typing rules for calculi that incorporate different modal axioms, but each variant demands different, tedious and seemingly ad hoc syntactic lemmas to prove normalization. In this work, we take a semantic approach to normalization, called normalization by evaluation (NbE), by leveraging the possible-world semantics of Fitch-style calculi to yield a more modular approach to normalization. We show that NbE models can be constructed for calculi that incorporate the K, T and 4 axioms of modal logic, as suitable instantiations of the possible-world semantics. In addition to existing results that handle 𝛽-equivalence, our normalization result also considers 𝜂-equivalence for these calculi. Our key results have been mechanized in the proof assistant Agda. Finally, we showcase several consequences of normalization for proving meta-theoretic properties of Fitch-style calculi as well as programming-language applications based on different interpretations of the necessity modality.

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12

Dal Lago, Ugo, and Margherita Zorzi. "Wave-Style Token Machines and Quantum Lambda Calculi." Electronic Proceedings in Theoretical Computer Science 176 (February 16, 2015): 64–78. http://dx.doi.org/10.4204/eptcs.176.6.

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13

Fernández, Maribel, and Nikolaos Siafakas. "Labelled Lambda-calculi with Explicit Copy and Erase." Electronic Proceedings in Theoretical Computer Science 22 (March 30, 2010): 49–64. http://dx.doi.org/10.4204/eptcs.22.5.

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14

Di Cosmo, Roberto, and Delia Kesner. "Combining algebraic rewriting, extensional lambda calculi, and fixpoints." Theoretical Computer Science 169, no. 2 (December 1996): 201–20. http://dx.doi.org/10.1016/s0304-3975(96)00121-1.

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15

VASCONCELOS, VASCO THUDICHUM. "Lambda and pi calculi, CAM and SECD machines." Journal of Functional Programming 15, no. 1 (January 2005): 101–27. http://dx.doi.org/10.1017/s0956796804005386.

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16

Romero, Rafael, and Alejandro Díaz-Caro. "A Note on Confluence in Typed Probabilistic Lambda Calculi." Electronic Proceedings in Theoretical Computer Science 357 (April 8, 2022): 18–24. http://dx.doi.org/10.4204/eptcs.357.2.

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17

Valiron, Benoît. "On Quantum and Probabilistic Linear Lambda-calculi (Extended Abstract)." Electronic Notes in Theoretical Computer Science 270, no. 1 (February 2011): 121–28. http://dx.doi.org/10.1016/j.entcs.2011.01.011.

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18

Lago, Ugo Dal, and Francesco Gavazzo. "On Bisimilarity in Lambda Calculi with Continuous Probabilistic Choice." Electronic Notes in Theoretical Computer Science 347 (November 2019): 121–41. http://dx.doi.org/10.1016/j.entcs.2019.09.007.

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19

Arbiser, A. "The Expansion Problem in Lambda Calculi with Explicit Substitution." Journal of Logic and Computation 18, no. 6 (October 1, 2008): 849–83. http://dx.doi.org/10.1093/logcom/exn007.

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20

Dal Lago, Ugo, Francesco Gavazzo, and Ryo Tanaka. "Effectful applicative similarity for call-by-name lambda calculi." Theoretical Computer Science 813 (April 2020): 234–47. http://dx.doi.org/10.1016/j.tcs.2019.12.025.

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21

Nakazawa, Koji, Makoto Tatsuta, Yukiyoshi Kameyama, and Hiroshi Nakano. "Type checking and typability in domain-free lambda calculi." Theoretical Computer Science 412, no. 44 (October 2011): 6193–207. http://dx.doi.org/10.1016/j.tcs.2011.06.020.

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22

Mouri, Motohiko, and Norihiro Kamide. "Strong Normalizability of Typed Lambda-Calculi for Substructural Logics." Logica Universalis 2, no. 2 (September 24, 2008): 189–207. http://dx.doi.org/10.1007/s11787-008-0036-0.

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23

Dybkjær, Hans, and Austin Melton. "Comparing Hagino's categorical programming language and typed lambda-calculi." Theoretical Computer Science 111, no. 1-2 (April 1993): 145–89. http://dx.doi.org/10.1016/0304-3975(93)90186-w.

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24

Abramsky, Samson, and Marina Lenisa. "Linear realizability and full completeness for typed lambda-calculi." Annals of Pure and Applied Logic 134, no. 2-3 (July 2005): 122–68. http://dx.doi.org/10.1016/j.apal.2004.08.003.

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25

Hermida, Claudio, and Bart Jacobs. "Fibrations with indeterminates: contextual and functional completeness for polymorphic lambda calculi." Mathematical Structures in Computer Science 5, no. 4 (December 1995): 501–31. http://dx.doi.org/10.1017/s0960129500001213.

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Анотація:

Lambek used categories with indeterminates to capture explicit variables in simply typed λ-calculus. He observed that such categories with indeterminates can be described as Kleisli categories for suitable comonads. They account for ‘functional completeness’ for Cartesian (closed) categories.Here we refine this analysis, by distinguishing ‘contextual’ and ‘functional’ completeness, and extend it to polymorphic λ-calculi. Since the latter are described as certain fibrations, we are lead to consider indeterminates, not only for ordinary categories, but also for fibred categories. Following a 2-categorical generalisation of Lambek's approach, such fibrations with indeterminates are presented as 'simple slices' in suitable 2-categories of fibrations; more precisely, as Kleisli objects. It allows us to establish contextual and functional completeness results for some polymorphic calculi.

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26

CAPRETTA, VENANZIO, та SILVIO VALENTINI. "A general method for proving the normalization theorem for first and second order typed λ-calculi". Mathematical Structures in Computer Science 9, № 6 (грудень 1999): 719–39. http://dx.doi.org/10.1017/s0960129599002923.

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Анотація:

In this paper we describe a method for proving the normalization property for a large variety of typed lambda calculi of first and second order, which is based on a proof of equivalence of two deduction systems. We first illustrate the method on the elementary example of simply typed lambda calculus, and then we show how to extend it to a more expressive dependent type system. Finally we use it to prove the normalization theorem for Girard's system F.

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27

Espírito Santo, José, Ralph Matthes, and Luís Pinto. "A coinductive approach to proof search through typed lambda-calculi." Annals of Pure and Applied Logic 172, no. 10 (December 2021): 103026. http://dx.doi.org/10.1016/j.apal.2021.103026.

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28

League, Christopher. "Lambda Calculi: A Guide for Computer Scientists by Chris Hankin." ACM SIGACT News 31, no. 1 (March 2000): 8–13. http://dx.doi.org/10.1145/346048.568490.

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29

Neergaard, P. "Conservation and Uniform Normalization in Lambda Calculi with Erasing Reductions." Information and Computation 178, no. 1 (October 10, 2002): 149–79. http://dx.doi.org/10.1016/s0890-5401(02)93153-6.

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30

Díaz-Caro, Alejandro, Pablo Arrighi, Manuel Gadella, and Jonathan Grattage. "Measurements and Confluence in Quantum Lambda Calculi With Explicit Qubits." Electronic Notes in Theoretical Computer Science 270, no. 1 (February 2011): 59–74. http://dx.doi.org/10.1016/j.entcs.2011.01.006.

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31

Neergaard, Peter Møller, and Morten Heine Sørensen. "Conservation and Uniform Normalization in Lambda Calculi with Erasing Reductions." Information and Computation 178, no. 1 (October 2002): 149–79. http://dx.doi.org/10.1006/inco.2002.3153.

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32

Goldberg, Mayer. "A construction of one-point bases in extended lambda calculi." Information Processing Letters 89, no. 6 (March 2004): 281–86. http://dx.doi.org/10.1016/j.ipl.2003.12.005.

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33

Arrial, Victor, Giulio Guerrieri, and Delia Kesner. "Quantitative Inhabitation for Different Lambda Calculi in a Unifying Framework." Proceedings of the ACM on Programming Languages 7, POPL (January 9, 2023): 1483–513. http://dx.doi.org/10.1145/3571244.

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Анотація:

We solve the inhabitation problem for a language called λ!, a subsuming paradigm (inspired by call-by-push-value) being able to encode, among others, call-by-name and call-by-value strategies of functional programming. The type specification uses a non-idempotent intersection type system, which is able to capture quantitative properties about the dynamics of programs. As an application, we show how our general methodology can be used to derive inhabitation algorithms for different lambda-calculi that are encodable into λ!.

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34

FAGORZI, SONIA, and ELENA ZUCCA. "A calculus of open modules: call-by-need strategy and confluence." Mathematical Structures in Computer Science 17, no. 4 (August 2007): 675–751. http://dx.doi.org/10.1017/s0960129507006238.

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We present a simple module calculus where selection and execution of a component is possible onopenmodules, that is, modules that still need to import some external definitions. Hence, it provides a kernel model for a computational paradigm in which standard execution (that is, execution of a single computation described by a fragment of code) can be interleaved with operations at the meta-level, which can manipulate in various ways the context in which this computation takes place. Formally, this is achieved by introducingconfigurationsas basic terms. These are, roughly speaking, pairs consisting of an (open, mutually recursive) collection of named components and a term representing a program running in the context of these components. Configurations can be manipulated by classical module/fragment operators, hence reduction steps can be either execution steps of the program or steps that perform module operations (calledreconfigurationsteps).Since configurations combine the features of lambda abstractions (first-class functions), records, environments with mutually recursive definitions and modules, the calculus extends and integrates both traditional module calculi and recursive lambda calculi. We state confluence of the calculus, and propose different ways to prevent errors arising from the lack of some required component, either by a purely static type system or by a combination of static and run-time checks. Moreover, we define a call-by-need strategy that performs module simplificationonly when needed and only once, leading to a generalisation of call-by-need lambda calculi that includes module features. We prove the soundness and completeness of this strategy using an approach based on information content, which also allows us to preserve confluence, even when local substitution rules are added to the calculus.

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35

Broda, Sabine, and Luís Damas. "Compact bracket abstraction in combinatory logic." Journal of Symbolic Logic 62, no. 3 (September 1997): 729–40. http://dx.doi.org/10.2307/2275570.

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AbstractTranslations from Lambda calculi into combinatory logics can be used to avoid some implementational problems of the former systems. However, this scheme can only be efficient if the translation produces short output with a small number of combinators, in order to reduce the time and transient storage space spent during reduction of combinatory terms. In this paper we present a combinatory system and an abstraction algorithm, based on the original bracket abstraction operator of Schönfinkel [9]. The algorithm introduces at most one combinator for each abstraction in the initial Lambda term. This avoids explosive term growth during successive abstractions and makes the system suitable for practical applications. We prove the correctness of the algorithm and establish some relations between the combinatory system and the Lambda calculus.

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36

Bruce, Kim B., Roberto Di Cosmo, and Giuseppe Longo. "Provable isomorphisms of types." Mathematical Structures in Computer Science 2, no. 2 (June 1992): 231–47. http://dx.doi.org/10.1017/s0960129500001444.

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A constructive characterization is given of the isomorphisms which must hold in all models of the typed lambda calculus with surjective pairing. Using the close relation between closed Cartesian categories and models of these calculi, we also produce a characterization of those isomorphisms which hold in all CCC's. Using the correspondence between these calculi and proofs in intuitionistic positive propositional logic, we thus provide a characterization of equivalent formulae of this logic, where the definition of equivalence of terms depends on having “invertible” proofs between the two terms. Work of Rittri (1989), on types as search keys in program libraries, provides an interesting example of use of these characterizations.

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37

Ahn, Ki Yung. "Mechanized Proof of Type Preservation for Polymorphic Lambda Calculi Using Abella." Journal of KIISE 47, no. 5 (May 31, 2020): 496–503. http://dx.doi.org/10.5626/jok.2020.47.5.496.

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38

Matthes, Ralph. "Tarski's Fixed-Point Theorem And Lambda Calculi With Monotone Inductive Types." Synthese 133, no. 1/2 (October 2002): 107–29. http://dx.doi.org/10.1023/a:1020831825964.

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39

Henglein, Fritz, and Harry G. Mairson. "The complexity of type inference for higher-order typed lambda calculi." Journal of Functional Programming 4, no. 4 (October 1994): 435–77. http://dx.doi.org/10.1017/s0956796800001143.

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Анотація:

AbstractWe analyse the computational complexity of type inference for untyped λ-terms in the second-order polymorphic typed λ-calculus (F2) invented by Girard and Reynolds, as well as higher-order extensions F3, F4, …, Fω proposed by Girard. We prove that recognising the F2-typable terms requires exponential time, and for Fω the problem is non-elementary. We show as well a sequence of lower bounds on recognising the Fk-typable terms, where the bound for Fk+1 is exponentially larger than that for Fk.The lower bounds are based on generic simulation of Turing Machines, where computation is simulated at the expression and type level simultaneously. Non-accepting computations are mapped to non-normalising reduction sequences, and hence non-typable terms. The accepting computations are mapped to typable terms, where higher-order types encode reduction sequences, and first-order types encode the entire computation as a circuit, based on a unification simulation of Boolean logic. A primary technical tool in this reduction is the composition of polymorphic functions having different domains and ranges.These results are the first nontrivial lower bounds on type inference for the Girard/Reynolds system as well as its higher-order extensions. We hope that the analysis provides important combinatorial insights which will prove useful in the ultimate resolution of the complexity of the type inference problem.

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40

Valiron, Benoit, and Steve Zdancewic. "Modeling Simply-Typed Lambda Calculi in the Category of Finite Vector Spaces." Scientific Annals of Computer Science 24, no. 2 (2014): 325–68. http://dx.doi.org/10.7561/sacs.2014.2.325.

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41

Qian, Zhenyu, and Tobias Nipkow. "Reduction and unification in lambda calculi with a general notion of subtype." Journal of Automated Reasoning 12, no. 3 (1994): 389–406. http://dx.doi.org/10.1007/bf00885767.

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42

Fiore, Marcelo, Roberto Di Cosmo, and Vincent Balat. "Remarks on isomorphisms in typed lambda calculi with empty and sum types." Annals of Pure and Applied Logic 141, no. 1-2 (August 2006): 35–50. http://dx.doi.org/10.1016/j.apal.2005.09.001.

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43

Ehrhard, Thomas, Yves Lafont, and Laurent Regnier. "Foreword." Mathematical Structures in Computer Science 8, no. 6 (December 1998): 541. http://dx.doi.org/10.1017/s0960129598002618.

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Анотація:

This special issue is devoted to complete and revised versions of papers presented at the Logique et Modèles du Calcul (Logic and Models of Computation) Conference held at CIRM in Marseilles in September 1996.The conference was organized by Marie-Renée Donnadieu, and the members of the programme committee were Gérard Berry, Jean-Yves Girard, Max Kanovich, Jean-Louis Krivine, Yves Lafont and François Lamarche.The scope of the conference was quite large, including, for instance, digital circuits, process calculi, abstract machines for lambda-calculus, computational interpretations of linear logic and logical aspects of proof search. Only the last three topics are represented in this special issue of Mathematical Structures in Computer Science, but we think that these papers are quite representative of the liveliness of research in this area on the borderline between mathematics and computing.We wish to thank Giuseppe Longo, who kindly proposed publishing this issue. For editorial reasons, a few contributions could not be included – they should appear in later issues.

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44

Martini, Simone. "Categorical models for non-extensional λ-calculi and combinatory logic". Mathematical Structures in Computer Science 2, № 3 (вересень 1992): 327–57. http://dx.doi.org/10.1017/s096012950000150x.

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The notions of weak Cartesian closed category and very weak CCC are introduced by dropping the extensionality (and the naturality) requirements in the adjunction defining the closed structure of a CCC. A number of specific examples of these categories are given. The weak notions are shown to be equivalent from both the semantic and syntactic standpoint to the typed non-extensional lambda-calculus and to the typed Combinatory Logic, extended with surjective pairs. Type-free models are characterized as reflexive objects in wCCCs. Finally, categorical models for the second-order non-extensional calculus are defined, by introducing a simple generalization of the notion of PL-category.

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45

ECHAHED, RACHID. "Foreword: special issue on term and graph rewriting." Mathematical Structures in Computer Science 28, no. 8 (July 6, 2018): 1287–89. http://dx.doi.org/10.1017/s0960129518000191.

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Rewriting techniques constitute a foundational theory of computing science. They are being investigated for several structures, such as lambda-terms, strings, first-order terms or graphs, and have been successfully used in many areas such as programming languages, automated reasoning, program verification, security, etc. The growing interest in this research area is witnessed by the leading international events, such as ICGT (International Conference on Graph Transformation) and the recent FSCD conference (International Conference on Formal Structures for Computation and Deduction), which gathers all topics of the former international conferences RTA (Rewriting Techniques and Applications) and TLCA (Typed Lambda Calculi and Applications). During the last decade, a particular interest has been devoted to the study of the impact of shared structures in term and graph rewriting through the international workshops editions of TERMGRAPH and GCM (Graph Computation Models).

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46

GOGUEN, HEALFDENE, and JEAN GOUBAULT-LARRECQ. "Sequent combinators: a Hilbert system for the lambda calculus." Mathematical Structures in Computer Science 10, no. 1 (February 2000): 1–79. http://dx.doi.org/10.1017/s0960129599002911.

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This paper introduces Hilbert systems for λ-calculus, called sequent combinators, addressing many of the problems of Hilbert systems that have led to the more widespread adoption of natural deduction systems in computer science. This suggests that Hilbert systems, with their uniform approach to meta-variables and substitution, may be a more suitable framework than λ-calculus for type theories and programming languages. Two calculi are introduced here. The calculus SKIn captures λ-calculus reduction faithfully, is confluent even in the presence of meta-variables, is normalizing but not strongly normalizing in the typed case, and standardizes. The sub-calculus SKInT captures λ-reduction in slightly less obvious ways, and is a language of proof-terms not directly for intuitionistic logic, but for a fragment of S4 that we name near-intuitionistic logic. To our knowledge, SKInT is the first confluent, first-order calculus to capture λ-calculus reduction fully and faithfully and be strongly normalizing in the typed case. In particular, no calculus of explicit substitutions has yet achieved this goal.

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47

LEVIN, MICHAEL Y., and BENJAMIN C. PIERCE. "TinkerType: a language for playing with formal systems." Journal of Functional Programming 13, no. 2 (March 2003): 295–316. http://dx.doi.org/10.1017/s0956796802004550.

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TinkerType is a pragmatic framework for compact and modular description of formal systems (type systems, operational semantics, logics, etc.). A family of related systems is broken down into a set of clauses – individual inference rules – and a set of features controlling the inclusion of clauses in particular systems. Simple static checks are used to help maintain consistency of the generated systems. We present TinkerType and its implementation and describe its application to two substantial repositories of typed lambda-calculi. The first repository covers a broad range of typing features, including subtyping, polymorphism, type operators and kinding, computational effects, and dependent types. It describes both declarative and algorithmic aspects of the systems, and can be used with our tool, the TinkerType Assembler, to generate calculi either in the form of typeset collections of inference rules or as executable ML typecheckers. The second repository addresses a smaller collection of systems, and provides modularized proofs of basic safety properties.

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48

Spreen, Dieter. "On functions preserving levels of approximation: A refined model construction for various lambda calculi." Theoretical Computer Science 212, no. 1-2 (February 1999): 261–303. http://dx.doi.org/10.1016/s0304-3975(98)00144-3.

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49

Mann, Matthias, and Manfred Schmidt-Schauß. "Similarity implies equivalence in a class of non-deterministic call-by-need lambda calculi." Information and Computation 208, no. 3 (March 2010): 276–91. http://dx.doi.org/10.1016/j.ic.2009.11.003.

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50

Geuvers, Herman, and Mark-Jan Nederhof. "Modular proof of strong normalization for the calculus of constructions." Journal of Functional Programming 1, no. 2 (April 1991): 155–89. http://dx.doi.org/10.1017/s0956796800020037.

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AbstractWe present a modular proof of strong normalization for the Calculus of Constructions of Coquand and Huet (1985, 1988). This result was first proved by Coquand (1986), but our proof is more perspicious. The method consists of a little juggling with some systems in the cube of Barendregt (1989), which provides a fine structure of the calculus of constructions. It is proved that the strong normalization of the calculus of constructions is equivalent with the strong normalization of Fω.In order to give the proof, we first establish some properties of various type systems. Therefore, we present a general framework of typed lambda calculi, including many well-known ones.

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