Linear Logic Without Boxes

Girard's original deenition 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 completely 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 optimal calculus implementations; computation is performed by simple graph rewriting. This graph implementation helps in understanding optimal reductions in the-calculus and in the various programming languages inspired by linear logic. 1 Beyond the-calculus The-calculus is not entirely explicit about the operations of erasing and duplicating arguments. These operations are important both in the theory of the-calculus and in its implementations, yet they are typically treated somewhat informally, implicitly. The proof nets of linear logic 1] provide a reenement of the-calculus where these operations become explicit; they are even reeected in the type system for proof nets (that is, in linear logic). Abramsky, Wadler, and others have suggested that this new expressive-ness makes linear logic a good basis for principled and useful improvements in functional-programming systems (e.g., 2, 3]). In some sense, however, linear logic could go further. The usual formulation of proof nets involves boxes. The box is the unit for discarding and copying fragments of proof nets. It works as a synchronization mark. The disappearance, reproduction, opening , and movement of boxes remain global operations; full boxes are handled at once, not incrementally, so for example it is not possible to copy a box gradually , in little pieces. As Girard points out, boxes are a bridle to parallelism. They are also an obstacle to sharing: the box formalism does not support some sophisticated mechanisms for \partial sharing" of common subexpressions available in-calculus implementations such as Lamping's 4] and Kathail's 5]. These sharing mechanisms are essential for optimality in reductions , and we believe that they can be of practical value. Moreover, boxes complicate the proof theory of linear logic; with boxes, linear logic falls short of giving a fully local account of computation. In this paper we describe a translation of proof nets into a system of sharing graphs. Proof-net reduction is simulated with graph rewriting. Sharing graphs are interaction nets, in the sense of Lafont 6]; hence rewriting is obviously Church-Rosser, and a naive implementation is straightforward. Everything in the graph system is entirely local. In particular, there are …