In [Gue95] we investigated the properties of (Levy) optimality for ?-graph reductions ` la Lamping from a graph theoretic point of view, introducing some new concepts as the ones of sharing-graph and sharing-morphism, and adapting some methods already used by Geometry of Interaction. The previous approach succeeds in the study ot the graph-rewriting system, giving an extended set of rules w.r.t. the ones suggested by Gonthier et al, and Asperti. Such an approach is not completely satisfactory from a logical point view, as it does not help in the comprehension of the logical meaning of the intermediate graphs which arise during the computations. We suggest an interpretation of the sharing-notes (muxes) as tuple constructors, and we plan to study a calculus and the corresponding deduction system by means of suitable proof-nets among which there are the sharing-graphs.
We also plan to extend the methods to MELL and other calculi. [Gue95] S. Guerrini. Sharing-graphs, sharing-morphims, and (optimal) ?-graph reductions. Accepted at The Tbilisi Symposium on Language, Logic and Computation, Tbilisi, October 1995.