Results 1 
5 of
5
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 ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
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.
On a localstep cutelimination procedure for the intuitionistic sequent calculus
 Proc. of the 13th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning (LPAR’06), volume 4246 of LNCS
, 2006
"... Abstract. In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cutelimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cutelimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. Then we define a reduction relation on those proofs that exactly corresponds to normalization in natural deduction. The reduction relation is simulated soundly and completely by a cutelimination procedure which consists of local proof transformations. It follows that the sequent calculus with our cutelimination procedure is a proper extension that is conservative over natural deduction with normalization. 1
Structural proof theory as rewriting
"... Abstract. The multiary version of the λcalculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cutelimination/normalisation rules) and permutation rules, t ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. The multiary version of the λcalculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cutelimination/normalisation rules) and permutation rules, typical of sequent calculus and of natural deduction with generalised elimination rules. We argue that this system is a suitable tool for doing structural proof theory as rewriting. As an illustration, we investigate combinations of reduction and permutation rules and whether these combinations induce rewriting systems which are confluent and terminating. In some cases, the combination allows the simulation of nonterminating reduction sequences known from explicit substitution calculi. In other cases, we succeed in capturing interesting classes of derivations as the normal forms w.r.t. wellbehaved combinations of rules. We identify six of these “combined ” normal forms, among which are two classes, due to Herbelin and Mints, in bijection with normal, ordinary natural deductions. A computational explanation for the variety of “combined ” normal forms is the existence of three ways of expressing multiple application in the calculus. 1
Towards a canonical classical natural deduction system José Espírito Santo
"... This paper studies a new classical natural deduction system, presented as a typed calculus named λµlet. It is designed to be isomorphic to Curien and Herbelin’s λµ˜µcalculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left ..."
Abstract
 Add to MetaCart
This paper studies a new classical natural deduction system, presented as a typed calculus named λµlet. It is designed to be isomorphic to Curien and Herbelin’s λµ˜µcalculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. leftintroduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot’s λµcalculus with the idea of “coercion calculus ” due to Cervesato and Pfenning, accommodating letexpressions in a surprising way: they expand Parigot’s syntactic class of named terms. This calculus and the mentioned isomorphism Θ offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of “readback ” of calculi in the sequent calculus format into natural deduction syntax; a formalization of the usual semantics of the λµ˜µcalculus, that explains coterms and cuts as, respectively, contexts and holefilling instructions. λµlet is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus; and Θ provides the “readback ” map and the formalized semantics, based on the precise notions of context and “holeexpression” provided by λµlet. We use “readback ” to achieve a precise connection with Parigot’s λµ, and to derive λcalculi for callbyvalue combining control and letexpressions in a logically founded way. Finally, the semantics Θ, when fully developed, can be inverted at each syntactic category. This development gives us license to see sequent calculus as the semantics of natural deduction; and uncovers a new syntactic concept in λµ˜µ (“cocontext”), with which one can give a new definition of ηreduction.