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27
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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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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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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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.
A Note on Intersection Types
, 1995
"... : Following J.L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters  there is no special constant !. We show here that the wellknown result on D, stating th ..."
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: Following J.L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters  there is no special constant !. We show here that the wellknown result on D, stating that any term which possesses a type in D strongly normalises does not need a new reducibility argument, but is a mere consequence of strong normalization for natural deduction restricted to the conjunction and implication. The proof of strong normalization for natural deduction, and therefore our result, as opposed to reducibility arguments, can be carried out within primitive recursive arithmetic. On the other hand, this enlightens the relation between & and & that G. Pottinger has already wondered about, and can be applied to other situations, like the lambda calculus with multiplicities of G. Boudol. Keywords: Lambda calculus , intersection types , strong normalization. Logic, proof th...
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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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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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.
Completeness and Soundness results forX with Intersection and Union Types
"... This paper defines intersection and union type assignment for the sequent calculus X, a substitutionfree language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is complete (i.e. closed for subjectexpansion), and ..."
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This paper defines intersection and union type assignment for the sequent calculus X, a substitutionfree language that enjoys the CurryHoward correspondence with respect to Gentzen’s sequent calculus for classical logic. We show that this notion is complete (i.e. closed for subjectexpansion), and show that the nonlogical nature of both intersection and union types disturbs the soundness (i.e. closed for reduction) properties. This implies that this notion of intersectionunion type assignment needs to be restricted to satisfy soundness as well, making it unsuitable to define a semantics. We will look at two (confluent) notions of reduction, called CallbyName and CallbyValue, and prove soundness results for those.