We study the problem of local and asynchronous computation in the context of multiplicative exponential linear logic (MELL) proof nets. The main novelty isin a complete set of rewriting rules for cut-elimination in presence of weakening (which requires garbage collection). The proposed reduction system is strongly normalizing and confluent.

Sharing graphs are an implementation of linear logic proof-nets in such a way that their reduction never duplicate a redex. In their usual formulations, proof-nets present a problem of coherence: if the proof-net N reduces by standard cut-elimination to N 0, then, by reducing the sharing graph of N we donot obtain the sharing graph of N 0.Wesolve this problem by changing the way the information is coded into sharing graphs and introducing a new reduction rule (absorption). The rewriting system is con uent and terminating. The proof of this fact exploits an algebraic semantics for sharing graphs. 1

We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proof-nets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simply-typed terms, step by step simulation of β-reduction and full composition.

Sharing graphs are a brilliant solution to the implementation of Lévy optimal reductions of λ-calculus. Sharing graphs are interesting on their own and optimal sharing reductions are just a particular reduction strategy of a more general sharing reduction system. The paper is a gentle introduction to sharing graphs and tries to confute some of the common myths on the difficulty of sharing implementations.