, Volume 87, Issue 2-3, pp 269-294

A Mixed λ-calculus

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Abstract

The aim of this paper is to define a λ-calculus typed in aMixed (commutative and non-commutative) Intuitionistic Linear Logic. The terms of such a calculus are the labelling of proofs of a linear intuitionistic mixed natural deduction NILL, which is based on the non-commutative linear multiplicative sequent calculus MNL [RuetAbrusci 99]. This linear λ-calculus involves three linear arrows: two directional arrows and a nondirectional one (the usual linear arrow). Moreover, the -terms are provided with seriesparallel orders on free variables.

We prove a normalization theorem which explicitly gives the behaviour of the order during the normalization procedure.

Special Issue Categorial Grammars and Pregroups Edited by Wojciech Buszkowski and Anne Preller