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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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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
Completing Herbelin’s programme
"... In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts t ..."
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In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts the associativity of “applicative terms”. Herbelin worked with a fragment of sequent calculus with constraints on left introduction. In this paper we complete Herbelin’s programme for full sequent calculus, that is, sequent calculus without the mentioned constraints, but where permutative conversions necessarily show up. This requires the introduction of a lambdalike calculus for full sequent calculus and an extension of natural deduction that gives meaning to “applicative contexts” and “applicative terms”. Such extension is a calculus with modus ponens and primitive substitution that refines von Plato’s natural deduction; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. The prooftheoretical outcome is noteworthy: the puzzling relationship between cut and substitution is settled; and cutelimination in sequent calculus is proven isomorphic to normalisation in the proposed natural deduction system. The isomorphism is the mapping that inverts the associativity of applicative terms.
Lectures on the curryhoward isomorphism
, 1998
"... The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent ..."
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The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent types, secondorder logic corresponds to polymorphic types, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is much more to the isomorphism than this. For instance, it is an old idea—due to Brouwer, Kolmogorov, and Heyting, and later formalized by Kleene’s realizability interpretation—that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the CurryHoward isomorphism gives syntactic representations of such procedures. These notes give an introduction to parts of proof theory and related
Cut Formulae and Logic Programming
"... . In this paper we present a mechanism to define names for proofwitnesses of formulae and thus to use Gentzen's cutrule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proofsearch ..."
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. In this paper we present a mechanism to define names for proofwitnesses of formulae and thus to use Gentzen's cutrule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proofsearch by indicating when an instance of the cutrule should be attempted. By using the cutrule there are proofs that can be made dramatically shorter. We explain how this idea of using the cutrule can be applied to the logic of hereditary Harrop formulae. 1 Introduction The computation mechanisms both for logic and for functional programming are searches for cutfree proofs. First, in pure logic programming the achievement of a goal G w.r.t. a program P can be seen 1 as the search for a proof in Gentzen's intuitionistic sequent calculus LJ [Gen69], of the sequent P ) G, that by Gentzen's cutelimination theorem can be cutfree [Bee89], [Mil90]; a term found as a witness to a proof contains among...
Acalculus Structure Isomorphic to Gentzenstyle Sequent Calculus Structure
, 1995
"... . We consider a calculus for which applicative terms have no longer the form (:::((u u1) u2 )::: un) but the form (u [u1 ; :::; un ]), for which [u1 ; :::; un ] is a list of terms. While the structure of the usual calculus is isomorphic to the structure of natural deduction, this new structure is ..."
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. We consider a calculus for which applicative terms have no longer the form (:::((u u1) u2 )::: un) but the form (u [u1 ; :::; un ]), for which [u1 ; :::; un ] is a list of terms. While the structure of the usual calculus is isomorphic to the structure of natural deduction, this new structure is isomorphic to the structure of Gentzenstyle sequent calculus. To express the basis of the isomorphism, we consider intuitionistic logic with the implication as sole connective. However we do not consider Gentzen's calculus LJ, but a calculus LJT which leads to restrict the notion of cutfree proofs in LJ. We need also to explicitly consider, in a simply typed version of this calculus, a substitution operator and a list concatenation operator. By this way, each elementary step of cutelimination exactly matches with a fireduction, a substitution propagation step or a concatenation computation step. Though it is possible to extend the isomorphism to classical logic and to other connectives,...