Checking Partial-Order Properties of Vector Addition Systems with States

Message Sequence Graphs (MSGs) form a popular model often used for the documentation of telecommunication protocols. They consist of typical scenarios of message exchanges depicted as partial-orders of events that lead from one control state to another. On the other hand Petri nets are a well-known formalism for distributed or parallel systems based on the notion of token game. Both approaches profit by a visual presentation and are the subject of numerous formal verification techniques and tools. In this paper we investigate a formalism which provides MSGs with the notion of token game and extends Petri nets with both control states and partial orders. Providing Petri nets with control states corresponds precisely to the model of Vector Addition Systems with States (VASSs). Thus we need to define first a partial-order semantics for VASSs which adopts the basic features of communication scenarios. To do so we extend simply the process semantics of Petri nets. We obtain a formal model that enjoys several interesting properties in terms of expressiveness and concision. The addition of control states to Petri nets under the partial-order semantics leads to undecidable problems. Similarly to MSGs, one cannot decide in particular whether two given VASSs describe the same process language. 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. This relies on a new technique that simulates all prefixes of all processes. In this way Petri net tools can be used to verify the properties of a VASS under the process semantics. We present also a technique to check effectively any MSO property of these partial orders, provided that the given system is bounded. This enables us to tackle more verification problems and subsumes known results for the model checking of MSGs. All algorithms presented in this paper have been implemented - n a prototype tool available online.