On the complexity of supervisory control design in the RW framework

The time complexity of supervisory control design for a general class of problems is studied. It is shown to be very unlikely that a polynomial-time algorithm can be found when either (1) the plant is composed of m components running concurrently or (2) the set of legal behaviors is given by the intersection of n legal specifications. That is to say, in general, there is no way to avoid constructing a state space which has size exponential in m+n. It is suggested that, rather than discouraging future work in the field, this result should point researchers to more fruitful directions, namely, studying special cases of the problem, where certain structural properties possessed by the plant or specification lend themselves to more efficient algorithms for designing supervisory controls. As no background on the subject of computational complexity is assumed, we have tried to explain all the borrowed material in basic terms, so that this paper may serve as a tutorial for a system engineer not familiar with the subject.