Petri nets are a well-known and intensively studied model of concurrency often used to specify distributed or parallel systems. Vector addition systems with states are simply Petri nets provided control states. In this paper we introduce a natural partial order semantics for vector addition systems with states which extends the process semantics of Petri nets. The addition of control states to Petri nets leads to undecidable problems, namely the equality of two process languages given by two systems. However we show that basic problems about the set of markings reached along the processes of a VASS, such as boundedness, covering and reachability, can be reduced to the analogous problems for Petri nets. We show also how to check effectively any MSO property of these partial orders, provided that the system is bounded. This result generalizes known results and techniques for the model checking of compositional message sequence graphs.
