We investigate the decidability and complexity status of model-checking problems on unlabelled reachability graphs of Petri nets by considering first-order, modal and pattern-based languages without labels on transitions or atomic propositions on markings. We consider several parameters to separate decidable problems from undecidable ones. Not only are we able to provide precise borders and a systematic analysis, but we also demonstrate the robustness of our proof techniques.

Separability in Petri nets means the property for a net k · N with an initial marking k · M to behave in the same way as k parallel instances of the same net N with an initial marking M, thus divided by k. We prove the separability of plain, bounded,

The unconstrained step semantics of Petri nets is impractical for simulating and modelling applications. In the past, this inadequacy has been alleviated by introducing various flavours of maximally concurrent semantics, as well as priority orders. In this paper, we introduce a general way of controlling step semantics of Petri nets through step firing policies that restrict the concurrent behaviour of Petri nets and so improve their execution and modelling features. In a nutshell, a step firing policy disables at each marking a subset of enabled steps which could otherwise be executed. We discuss various examples of step firing policies and then investigate the synthesis problem for Petri nets controlled by such policies. Using generalised regions of step transition systems, we provide an axiomatic characterisation of those transition systems which can be realised as reachability graphs of Petri nets controlled by a given step firing policy. We also provide two different decision and synthesis algorithms for PT-nets and step firing policies based on linear rewards of steps, where the reward for firing a single transition is either fixed or it depends on the current net marking. The simplicity of the algorithms supports our claim that the proposed approach is practical.

Abstract. The synthesis problem for Petri nets consists in deciding constructively the existence of a Petri net with sequential state graph isomorphic to a given graph. If events are attached to locations, one may set as an additional requirement that the synthesised net should be distributable; i.e. such that events at different locations have no common input place, whence distributed conflicts are avoided. Distributable nets are easily implemented by finite families of automata (one per location) communicating with each other by asynchronous message passing. We show that the general Petri net synthesis problem and its distributed version may both be solved in time polynomial in the size of the given graph. We report on some preliminary experiments of Petri net synthesis applied to the distribution of reactive automata using the tool SYNET .