Modal algebra and Petri nets

We use the by now well established setting of modal semirings to derive a modal algebra for Petri nets. It is based on a relation-algebraic calculus for separation logic that enables calculations of properties in a pointfree fashion and at an abstract level. Basically, we start from an earlier logical approach to Petri nets that in particular uses modal box and diamond operators for stating properties about the state space of such a net. We provide relational translations of the logical formulas which further allow the characterisation of general behaviour of transitions in an algebraic fashion. From the relational structure an algebra for frequently used properties of Petri nets is derived. In particular, we give connections to typical used assertion classes of separation logic. Moreover, we demonstrate applicability of the algebraic approach by calculations concerning a standard example of a mutex net.