Behavioral permissiveness is an important criterion for the design of a livenessenforcing Petri net supervisor. However, given a Petri net model, there may not exist a behaviorally optimal liveness-enforcing pure Petri net supervisor expressed by a set of control places (monitors). For example, the Petri net model of an AMS shown in Fig. 6.1 is an instance that cannot be optimally controlled by adding control places without self-loops. It has 39 reachable markings, 23 of which are legal, as shown in Fig. 6.2. For the optimal control purposes, an optimal supervisor should make the system live with all the 23 legal markings. However, it can be verified by the theory of regions in (Ghaari et al., 2003) and (Uzam, 2002) that the net has 10 MTSIs and six of them have no monitor solutions, implying that there is no optimal pure Petri net supervisor that can lead to a live system with all the 23 legal markings.