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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 ..."
Abstract

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.
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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Cited by 5 (1 self)
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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