On the Existence of Supervisory Policies that Enforce Liveness in Discrete-Event Dynamic Systems Modeled by Controlled Petri Nets

We consider discrete-state plants represented by controlled Petri nets (CtlPN’s), where a subset of transitions can be prevented from firing by a supervisor. A transition in a CtlPN can fire at a marking if there are sufficient tokens in its input places and it is permitted to fire by the supervisor. A CtlPN is live if it is possible to fire any transition from every marking that is reachable under supervision. In this paper we derive a necessary and sufficient condition for the existence of a supervisory policy that enforces liveness in CtlPN’s. We show this condition cannot be tested for an arbitrary CtlPN. However, for bounded CtlPN’s, or CtlPN’s where each transition is individually controllable, we show the existence of a supervisory policy which enforces that liveness is decidable. We also show the existence of a supervisory policy that enforces liveness is necessary and sufficient for the existence of a minimally restrictive supervisory policy.