MELL proof-nets in the category of graphs

We present a formalization of proof-nets (and more generally, proof- structures) for the multiplicative-exponential fragment of linear logic, with a novel treatment of boxes: instead of integrating boxes into to the graphical structure (e.g., by adding explicit auxiliary doors, plus a mapping from boxed nodes to the main door, and ad hoc conditions on the nesting of boxes), we fix a graph morphism from the underlying graph of the proof-structure to the tree of boxes given by the nesting order. This approach allows to apply tools and notions from the theory of algebraic graph transformations, and obtain very synthetic presentations of sophisticated operations: for instance, each element of the Taylor expansion of a proof-structure is obtained by a pull- back along a morphism representing a thick subtree of the tree of boxes. A treatment of cut elimination in this framework currently under development.