Provability in Intuitionistic Linear Logic from a New Interpretation on Petri nets

Linear logic is a logic of actions which seems well suited to various computer science applications. From its intrinsic ability to reeect computational resources, it is possible to reene diierent programming paradigms like formulae-as-types (proofs-as-programs) or formulae-as-states (proofs-as-computations) with a ner control on resource management. In the latter case, the correspondence between Intuitionistic Linear Logic (ILL) and Petri nets illustrates the interest of eecient proof search methods for proving speciications or properties of distributed systems. In contrast to existing methods, for instance based on canonical proofs, we propose here to re-visit the semantics of ILL and its interpretation on Petri nets to provide new proof techniques for proving or disproving properties. From the relationships between the notions of ordered monoid and of quantale we deene a new interpretation of ILL on Petri nets that is complete and veriies the property of nite models. Possible issues would be to derive a new algebraic semantics as basis of proof search and to propose a calculus to eeectively build counter-models for ILL.