A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure
Résumé
We consider a lambda-calculus for which applicative terms have no longer the form (...((u u_1) u_2) ... u_n) but the form (u [u_1 ; ... ; u_n]), for which [u_1 ; ... ; u_n] is a list of terms. While the structure of the usual lambda-calculus is isomorphic to the structure of natural deduction, this new structure is isomorphic to the structure of Gentzen-style 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 cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version of this lambda-calculus, a substitution operator and a list concatenation operator. By this way, each elementary step of cut-elimination exactly matches with a beta-reduction, a substitution propagation step or a concatenation computation step. Though it is possible to extend the isomorphism to classical logic and to other connectives, we do not treat of it in this paper.
Domaines
Logique en informatique [cs.LO]
Origine : Fichiers produits par l'(les) auteur(s)
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