Remarks on the difficulty of top-down supervisor synthesis

This paper shows that language based top-down supervisor synthesis of Ramadge-Wonham supervisory control theory is in general not feasible. We show this as a direct consequence of the undecidability result of Decomposable Subset problem (and its prefix closed version), which in turn is a corollary of the undecidability result of Trace Closed Subset problem. Essentially, it is in general not possible to decide in finite amount of time whether there exists a string in a regular language such that all those strings indistinguishable from it are contained in the same language. We bring these results to the attention of control community and investigate the decidability status of some other related problems.