On asymptotically efficient solutions for a class of supervisory control problems

We consider restricted sequences of forbidden-string and forbidden-state problems and show the limiting problem instance defined by a sequence has a solution if and only if each problem instance in the sequence has a solution. Furthermore, the supervisor that enforces the desired behavior in the limiting problem instance also enforces the desired behavior in each problem instance in the sequence. In essence, the storage requirement for supervisory control of each instance is bounded above by that of the limiting problem instance. In addition, if the limiting problem instance is available, a similar observation can be made regarding the complexity of supervisory control. This observation coupled with recent results on the solvability of the supervisory control problem involving certain classes of nonregular behaviors and finite-representations of infinite-state systems is suggested as an effective procedure to combat the computational and storage issues in supervisory control. We conclude with an incomplete list of future research directions.