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Computation with classical sequents
 MATHEMATICAL STRUCTURES OF COMPUTER SCIENCE
, 2008
"... X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X ..."
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Cited by 16 (16 self)
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X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X, we will show how elaborate calculi can be embedded, like the λcalculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.
CutElimination in the Strict Intersection Type Assignment System is Strongly Normalising
 NOTRE DAME J. OF FORMAL LOGIC
, 2004
"... This paper defines reduction on derivations (cutelimination) in the Strict Intersection Type Assignment System of [1] and shows a strong normalisation result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterisation of normalisability of term ..."
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Cited by 15 (12 self)
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This paper defines reduction on derivations (cutelimination) in the Strict Intersection Type Assignment System of [1] and shows a strong normalisation result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterisation of normalisability of terms, using intersection types.
A Theory of Explicit Substitutions with Safe and Full Composition
 Logical Methods in Computer Science
"... Vol. 5 (3:1) 2009, pp. 1–29 ..."
Reducibility: a ubiquitous method in lambda calculus with intersection types
, 2002
"... A general reducibility method is developed for proving reduction properties of lambda terms typeable in intersection type systems with and without the universal type #. Su#cient conditions for its application are derived. This method leads to uniform proofs of confluence, standardization, and weak h ..."
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Cited by 3 (1 self)
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A general reducibility method is developed for proving reduction properties of lambda terms typeable in intersection type systems with and without the universal type #. Su#cient conditions for its application are derived. This method leads to uniform proofs of confluence, standardization, and weak head normalization of terms typeable in the system with the type #. The method extends Tait's reducibility method for the proof of strong normalization of the simply typed lambda calculus, Krivine's extension of the same method for the strong normalization of intersection type system without #, and StatmanMitchell's logical relation method for the proof of confluence of ##reduction on the simply typed lambda terms. As a consequence, the confluence and the standardization of all (untyped) lambda terms is obtained.
Resource operators for λcalculus
 INFORM. AND COMPUT
, 2007
"... We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties ..."
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Cited by 3 (2 self)
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We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simplytyped terms, step by step simulation of βreduction and full composition.
Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with de Bruijn Indices
, 2009
"... The λcalculus with de Bruijn indices assembles each αclass of λterms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable λterms, that is a term is normalisable if and only if it is typeable. To be clos ..."
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Cited by 1 (1 self)
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The λcalculus with de Bruijn indices assembles each αclass of λterms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable λterms, that is a term is normalisable if and only if it is typeable. To be closer to computations and to simplify the formalisation of the atomic operations involved in βcontractions several calculi of explicit substitution were developed and some of them are written with de Bruijn indices. Versions of explicit substitutions calculi without types and with simple type systems are well investigated in contrast to versions with more elaborated type systems such as intersection types. In previous work, we introduced a de Bruijn version of the λcalculus with an intersection type system and proved it preserves the subject reduction, a basic type system property. In this paper a version with de Bruijn indices of an intersection type system originally introduced to characterise principal typings for βnormal forms (βnf for short) is presented. We present the characterisation in this new system and the corresponding versions for the type inference and the reconstruction of normal forms from principal typings algorithms. We briefly discuss about the failure of the subject reduction property and some possible solutions for it. ∗ Supported by a PhD scholarship at the Universidade de Brasília. † Supported by the Fundação de Apoio à Pesquisa do Distrito Federal [FAPDF 8004/2007] 1 1
circuits, computations and Classical Logic
, 2005
"... X is an untyped language for describing circuits by composition of basic components. This language is well suited to describe structures which we call “circuits ” and which are made of parts that are connected by wires. Moreover X gives an expressive platform on which algebraic objects and many diff ..."
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X is an untyped language for describing circuits by composition of basic components. This language is well suited to describe structures which we call “circuits ” and which are made of parts that are connected by wires. Moreover X gives an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and some its potential uses. To demonstrate the expressive power of X, we will show how, even in an untyped setting, elaborate calculi can be embedded, like the naturals, the λcalculus, Bloe and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ˜µ. Keywords: Language design, mobility, circuits, classical logic, CurryHoward correspondance Résumé X est un langage non typ é conçu pour d écrire les circuits par composition de «briques » de base. Ce langage s’adapte parfaitement à la description des structures que nous appelons «circuits » et qui sont faites de composants connect és par des fils. De plus, X fournit une plateforme expressive sur laquelle des objets alg ébriques et de nombreux paradigmes de programmation (applicative) de toutes sortes peuvent être appliqu és. Dans ce rapport, nous pr ésenterons la syntaxe de X, ses règles de r éduction et certaines de ses utilisations potentielles. Pour mettre en lumière le pouvoir expressif de X, nous montrerons comment, même dans un cadre non typ é, on peut y plonger des calculs relativement sophistiqu és, comme les entiers naturels, le λcalcul, le calcul de substitutions explicites λx de Bloe et Rose, le calcul λµ de Parigot et le calcul λµ˜µ de Curien et Herbelin. Motsclés: Conception de langage, mobilit é, circuits, logique classique,
Explicit Alpha Conversion and Garbage Collection in X
, 2006
"... This paper will present improvements on the term graph rewriting model of implementation for the (untyped) calculus X as presented in [3]. X is a new style calculus which embodies both substitution and context call, that has first been defined in [13] and was later extensively studied in [2]. ..."
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This paper will present improvements on the term graph rewriting model of implementation for the (untyped) calculus X as presented in [3]. X is a new style calculus which embodies both substitution and context call, that has first been defined in [13] and was later extensively studied in [2].