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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 14 (11 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.
About Translations of Classical Logic into Polarized Linear Logic
 IN PROCEEDINGS OF THE EIGHTEENTH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 2003
"... We show that the decomposition of Intuitionistic Logic into Linear Logic along the equation A ! B = !A ( B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly we build a categorical model of classical logic (a Control Category) from a categ ..."
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Cited by 11 (0 self)
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We show that the decomposition of Intuitionistic Logic into Linear Logic along the equation A ! B = !A ( B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly we build a categorical model of classical logic (a Control Category) from a categorical model of Linear Logic by a construction similar to the coKleisli category. Secondly we analyse two standard ContinuationPassing Style (CPS) translations, the Plotkin and the Krivine's translations, which are shown to correspond to two embeddings of LLP into LL.
J.B.: A complete realisability semantics for intersection types and infinite expansion variables (2008), http://www.macs.hw.ac.uk/ ∼ fairouz/papers/drafts/compsembig.pdf
"... Abstract. Expansion was introduced at the end of the 1970s for calculating principal typings for λterms in intersection type systems. Expansion variables (Evariables) were introduced at the end of the 1990s to simplify and help mechanise expansion. Recently, Evariables have been further simplifie ..."
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Cited by 6 (6 self)
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Abstract. Expansion was introduced at the end of the 1970s for calculating principal typings for λterms in intersection type systems. Expansion variables (Evariables) were introduced at the end of the 1990s to simplify and help mechanise expansion. Recently, Evariables have been further simplified and generalised to also allow calculating other type operators than just intersection. There has been much work on semantics for intersection type systems, but only one such work on intersection type systems with Evariables. That work established that building a semantics for Evariables is very challenging. Because it is unclear how to devise a space of meanings for Evariables, that work developed instead a space of meanings for types that is hierarchical in the sense of having many degrees (denoted by indexes). However, although the indexed calculus helped identify the serious problems of giving a semantics for expansion variables, the sound realisability semantics was only complete when one single Evariable is used and furthermore, the universal type ω was not allowed. In this paper, we are able to overcome these challenges. We develop a realisability semantics where we allow an arbitrary (possibly infinite) number of expansion variables and where ω is present. We show the soundness and completeness of our proposed semantics. 1
The Scott model of Linear Logic is the extensional collapse of its relational model
, 2011
"... We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus. ..."
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Cited by 3 (2 self)
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We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus.
The Coq Proof Assistant  Reference Manual Version 6.1
, 1997
"... : Coq is a proof assistant based on a higherorder logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.enslyon.fr:/pub/LIP/COQ/V6.1 Keywords: Coq, Proof Assistant, Formal Proofs, Calculus of Inductives Constru ..."
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Cited by 2 (0 self)
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: Coq is a proof assistant based on a higherorder logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.enslyon.fr:/pub/LIP/COQ/V6.1 Keywords: Coq, Proof Assistant, Formal Proofs, Calculus of Inductives Constructions (R'esum'e : tsvp) This research was partly supported by ESPRIT Basic Research Action "Types" and by the GDR "Programmation " cofinanced by MREPRC and CNRS. Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11  T'el'ecopie : (33 1) 39 63 53 30 Manuel de r'ef'erence du syst`eme Coq version V6.1 R'esum'e : Coq est un syst`eme permettant le d'eveloppement et la v'erification de preuves formelles dans une logique d'ordre sup'erieure incluant un riche langage de d'efinitions de fonctions. Ce document constitue le manuel de r'ef'erence de la version V6.1 qui est distribu 'ee par ftp ...
Combinator Shared Reduction and Infinite Objects in Type Theory
, 1996
"... We will present a syntactical proof of correctness and completeness of shared reduction. This work is an application of type theory extended with infinite objects and coinduction. ..."
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Cited by 2 (0 self)
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We will present a syntactical proof of correctness and completeness of shared reduction. This work is an application of type theory extended with infinite objects and coinduction.
Strongly Normalising CutElimination with Strict Intersection Types
, 2003
"... This paper defines reduction on derivations in the strict intersection type assignment system of [2], by generalising cutelimination, and shows a strong normalisation result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterisation of normali ..."
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Cited by 2 (2 self)
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This paper defines reduction on derivations in the strict intersection type assignment system of [2], by generalising cutelimination, 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 using intersection types.
A system F accounting for scalars
, 2010
"... The algebraic λcalculus [40] and the linearalgebraic λcalculus [3] extend the λcalculus with the possibility of making arbitrary linear combinations of λcalculus terms (preserving ∑ αi.ti). In this paper we provide a finegrained, System Flike type system for the linearalgebraic λcalculus (L ..."
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Cited by 1 (0 self)
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The algebraic λcalculus [40] and the linearalgebraic λcalculus [3] extend the λcalculus with the possibility of making arbitrary linear combinations of λcalculus terms (preserving ∑ αi.ti). In this paper we provide a finegrained, System Flike type system for the linearalgebraic λcalculus (Lineal). We show that this scalar type system enjoys both the subjectreduction property and the strongnormalisationproperty, which constitute our main technical results. The latter yields a significant simplification of the linearalgebraic λcalculus itself, by removing the need for some restrictions in its reduction rules – and thus leaving it more intuitive. But the more important, original feature of this scalar type system is that it keeps track of ‘the amount of a type’ that this present in each term. As an example, we show how to use this type system in order to guarantee the welldefiniteness of probabilistic functions ( ∑ αi = 1) – thereby specializing Lineal into a probabilistic, higherorder λcalculus. Finally we begin to investigate the logic induced by the scalar type system, and prove a nocloning theorem expressed solely in terms of the possible proof methods in this logic. We discuss the potential connections with Linear Logic and Quantum Computation.
Developing Realisability Semantics for Intersection Types and Expansion Variables
"... Abstract. Expansion was invented at the end of the 1970s for calculating principal typings for λterms in type systems with intersection types. Expansion variables (Evariables) were invented at the end of the 1990s to simplify and help mechanize expansion. Recently, Evariables have been further si ..."
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Cited by 1 (0 self)
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Abstract. Expansion was invented at the end of the 1970s for calculating principal typings for λterms in type systems with intersection types. Expansion variables (Evariables) were invented at the end of the 1990s to simplify and help mechanize expansion. Recently, Evariables have been further simplified and generalized to also allow calculating other type operators than just intersection. There has been much work on denotational semantics for type systems with intersection types, but none whatsoever before now on type systems with Evariables. Building a semantics for Evariables turns out to be challenging. To simplify the problem, we consider only Evariables, and not the corresponding operation of expansion. We develop a realizability semantics where each use of an Evariable in a type corresponds to an independent level at which evaluation occurs in the λterm that is assigned the type. In the λterm being evaluated, the only interaction possible between portions at different levels is that higher level portions can be passed around but never applied to lower level portions. We apply this semantics to two intersection type systems. We show these systems are sound, that completeness does not hold for the first system, and completeness holds for the second system when only one Evariable is allowed (although it can be used many times and nested). As far as we know, this is the first study of a denotational semantics of intersection type systems with Evariables (using realizability or any other approach) and of the difficulties involved. 1
Machine Deduction
 In Proc. Types for Proofs and Program, LNCS 806
, 1993
"... We present in this paper a new type system which allows to extract code for an abstract machine instead of lambdaterms. Thus, we get a framework to compile correctly programs extracted from proof by translating their proof in our system and then extracting the code. Moreover, we will see that we ca ..."
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Cited by 1 (0 self)
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We present in this paper a new type system which allows to extract code for an abstract machine instead of lambdaterms. Thus, we get a framework to compile correctly programs extracted from proof by translating their proof in our system and then extracting the code. Moreover, we will see that we can associate programs to classical proofs. 1 Introduction. The proof as program paradigm, using the CurryHoward isomorphism [4], gives a way to associate a program to an intuitionistic proof. This program is almost always a functional program (in general a lambdaterm [1]) which has to be compiled before being executed [11]. This ensures some correctness about the functional program extracted from the proof. But the correctness of the compiled code is relative to the proof of the compiler. The usual way to ensure this kind of correctness is to define a semantics for the functional language, and to verify that the compiler preserves this semantics. We study in this paper a type system for th...