Lambda Calculus and Intuitionistic Linear Logic
 Simona Ronchi della Rocca,
 Luca Roversi
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Abstract
The introduction of Linear Logic extends the CurryHoward Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources.
This paper introduces a typed functional language Λ! and a categorical model for it.
The terms of Λ! encode a version of natural deduction for Intuitionistic Linear Logic such that linear and non linear assumptions are managed multiplicatively and additively, respectively. Correspondingly, the terms of Λ! are built out of two disjoint sets of variables. Moreover, the λabstractions of Λ! bind variables and patterns. The use of two different kinds of variables and the patterns allow a very compact definition of the onestep operational semantics of Λ!, unlike all other extensions of CurryHoward Isomorphism to Intuitionistic Linear Logic. The language Λ! is ChurchRosser and enjoys both Strong Normalizability and Subject Reduction.
The categorical model induces operational equivalences like, for example, a set of extensional equivalences.
The paper presents also an untyped version of Λ! and a type assignment for it, using formulas of Linear Logic as types. The type assignment inherits from Λ! all the good computational properties and enjoys also the PrincipalType Property.
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 Title
 Lambda Calculus and Intuitionistic Linear Logic
 Journal

Studia Logica
Volume 59, Issue 3 , pp 417448
 Cover Date
 19971101
 DOI
 10.1023/A:1005092630115
 Print ISSN
 00393215
 Online ISSN
 15728730
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 linear logic
 CurryHoward isomorphism
 typed λcalculi
 Authors

 Simona Ronchi della Rocca ^{(1)}
 Luca Roversi ^{(1)}
 Author Affiliations

 1. Dipartimento di Informatica, Università degli studi di Torino, C.so Svizzera n.185, 10149, Torino, Italy