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Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λCalculi
, 1992
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Revisiting the correspondence between cutelimination and normalisation
 In Proceedings of ICALP’2000
, 2000
"... Abstract. Cutfree proofs in Herbelin’s sequent calculus are in 11 correspondence with normal natural deduction proofs. For this reason Herbelin’s sequent calculus has been considered a privileged middlepoint between Lsystems and natural deduction. However, this bijection does not extend to pro ..."
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Cited by 14 (3 self)
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Abstract. Cutfree proofs in Herbelin’s sequent calculus are in 11 correspondence with normal natural deduction proofs. For this reason Herbelin’s sequent calculus has been considered a privileged middlepoint between Lsystems and natural deduction. However, this bijection does not extend to proofs containing cuts and Herbelin observed that his cutelimination procedure is not isomorphic to βreduction. In this paper we equip Herbelin’s system with rewrite rules which, at the same time: (1) complete in a sense the cut elimination procedure firstly proposed by Herbelin; and (2) perform the intuitionistic “fragment ” of the tqprotocol a cutelimination procedure for classical logic defined by Danos, Joinet and Schellinx. Moreover we identify the subcalculus of our system which is isomorphic to natural deduction, the isomorphism being with respect not only to proofs but also to normalisation. Our results show, for the implicational fragment of intuitionistic logic, how to embed natural deduction in the much wider world of sequent calculus and what a particular cutelimination procedure normalisation is. 1
Conservative extensions of the λcalculus for the computational interpretation of sequent calculus
, 2002
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Recommended Citation Jean H. Gallier, "Constructive Logics Part I: A Tutorial on Proof Systems and Typed LambdaCalculi",. October 1991. Constructive Logics Part I: A Tutorial on Proof Systems and Typed
, 1991
"... The purpose of this paper is to give an exposition of material dealing with constructive logic, typed λcalculi, and linear logic. The emergence in the past ten years of a coherent field of research often named "logic and computation " has had two major (and related) effects: firstly, it h ..."
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The purpose of this paper is to give an exposition of material dealing with constructive logic, typed λcalculi, and linear logic. The emergence in the past ten years of a coherent field of research often named "logic and computation " has had two major (and related) effects: firstly, it has rocked vigorously the world of mathematical logic; secondly, it has created a new computer science discipline, which spans from what is traditionally called theory of computation, to programming language design. Remarkably, this new body of work relies heavily on some "old " concepts found in mathematical logic, like natural deduction, sequent calculus, and λcalculus (but often viewed in a different light), and also on some newer concepts. Thus, it may be quite a challenge to become initiated to this new body of work (but the situation is improving, there are now some excellent texts on this subject matter). This paper attempts to provide a coherent and hopefully "gentle " initiation to this new body of work. We have attempted to cover the basic material on natural deduction, sequent calculus, and typed λcalculus, but also to provide an introduction to Girard's linear logic, one of the most exciting developments in logic these past five years. The first part of these notes gives an exposition of background material (with the exception of the Girardtranslation of classical logic into