Petri Nets Step Transitions and Proofs in Partially Commutative Linear Logic

We encode the execution of Petri nets in Partially Commutative Linear Logic, an intuitionistic logic introduced by Ph. de Groote which contains both commutative and non commutative connectives. We are thus able to faithfully represent the concurrent firing of Petri nets as long as it can be depicted by a series-parallel order. This coding is inspired from the description of context-free languages by Lambek grammars. This report is an extended version (with complete proofs) of an article to appear in the proceedings of the Logic Colloquium 1999 (Utrecht)