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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 22 (21 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.
Conservative extensions of the λcalculus for the computational interpretation of sequent calculus
, 2002
"... ..."
CurryHoward Term Calculi for GentzenStyle Classical Logic
, 2008
"... This thesis is concerned with the extension of the CurryHoward Correspondence to classical logic. Although much progress has been made in this area since the seminal paper by Griffin, we believe that the question of finding canonical calculi corresponding to classical logics has not yet been resolv ..."
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This thesis is concerned with the extension of the CurryHoward Correspondence to classical logic. Although much progress has been made in this area since the seminal paper by Griffin, we believe that the question of finding canonical calculi corresponding to classical logics has not yet been resolved. We examine computational interpretations of classical logics which we keep as close as possible to Gentzen’s original systems, equipped with general notions of reduction. We present a calculus X i which is based on classical sequent calculus and the stronglynormalising cutelimination procedure defined by Christian Urban. We examine how the notion of shallow polymorphism can be adapted to the moregeneral setting of this calculus. We show that the intuitive adaptation of these ideas fails to be sound, and give a novel solution. In the setting of classical natural deduction, we examine the lambdamu calculus of Parigot. We show that the underlying logic is incomplete in various ways, compared with a standard Gentzenstyle presentation of classical natural deduction. We relax the identified
Classical Callbyneed and duality
"... Abstract. We study callbyneed from the point of view of the duality between callbyname and callbyvalue. We develop sequentcalculus style versions of callbyneed both in the minimal and classical case. As a result, we obtain a natural extension of callbyneed with control operators. This lea ..."
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Abstract. We study callbyneed from the point of view of the duality between callbyname and callbyvalue. We develop sequentcalculus style versions of callbyneed both in the minimal and classical case. As a result, we obtain a natural extension of callbyneed with control operators. This leads us to introduce a callbyneed λµcalculus. Finally, by using the dualities principles of λµ˜µcalculus, we show the existence of a new callbyneed calculus, which is distinct from callbyname, callbyvalue and usual callbyneed theories. 1
On Zucker's isomorphism for LJ and its extension to Pure Type Systems
, 2003
"... It is shown how the sequent calculus LJ can be embedded into a simple extension of the calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is prove ..."
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Cited by 2 (0 self)
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It is shown how the sequent calculus LJ can be embedded into a simple extension of the calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is proved strongly normalizing, thus showing strong normalization for Gentzen's cut elimination steps. This re nes previous results by Zucker, Pottinger and Herbelin on the isomorphism between natural deduction and sequent calculus.
Characterising strongly normalising intuitionistic sequent terms
"... Abstract. This paper gives a characterisation, via intersection types, of the strongly normalising terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary λcalc ..."
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Abstract. This paper gives a characterisation, via intersection types, of the strongly normalising terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary λcalculus. The completeness of the typing system is obtained from subject expansion at root position. This paper’s sequent term calculus integrates smoothly the λterms with generalised application or explicit substitution. Strong normalisability of these terms as sequent terms characterises their typeability in certain “natural ” typing systems with intersection types. The latter are in the natural deduction format, like systems previously studied by Matthes and Lengrand et al., except that they do not contain any extra, exceptional rules for typing generalised applications or substitution.
language with control operators
, 2004
"... Characterizing strong normalization in a language with control operators ..."
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The language Χ: 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.
Classical Callbyneed and duality
, 2011
"... We study callbyneed from the point of view of the duality between callbyname and callbyvalue. We develop sequentcalculus style versions of callbyneed both in the minimal and classical case. As a result, we obtain a natural extension of callbyneed with control operators. This leads us to ..."
Abstract
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We study callbyneed from the point of view of the duality between callbyname and callbyvalue. We develop sequentcalculus style versions of callbyneed both in the minimal and classical case. As a result, we obtain a natural extension of callbyneed with control operators. This leads us to introduce a callbyneed λµcalculus. Finally, by using the dualities principles of λµ˜µcalculus, we show the existence of a new callbyneed calculus, which is distinct from callbyname, callbyvalue and usual callbyneed theories.