 
Summary: Linear Logic Without Boxes
Georges Gonthier \Lambda Mart'in Abadi y JeanJacques L'evy \Lambda
\Lambda INRIA y Digital Equipment Corporation
Domaine de Voluceau Systems Research Center
Rocquencourt, B.P. 105 130 Lytton Avenue
78153 Le Chesnay Cedex, France Palo Alto, CA 94301, USA
Abstract
Girard's original definition of proof nets for linear
logic involves boxes. The box is the unit for erasing
and duplicating fragments of proof nets. It imposes
synchronization, limits sharing, and impedes a com
pletely local view of computation. Here we describe
an implementation of proof nets without boxes. Proof
nets are translated into graphs of the sort used in op
timal –calculus implementations; computation is per
formed by simple graph rewriting. This graph imple
mentation helps in understanding optimal reductions
in the –calculus and in the various programming lan
guages inspired by linear logic.
1 Beyond the –calculus
