A note on deciding controllability in pushdown systems

Consider an event alphabet Σ. The supervisory control theory of Ramadge and Wonham asks the question, given a plant model G, with language K L_M(G)⊆Σ^* and another language K⊆ L_M(G), is there a supervisor ψ such that L_M(ψ/G)=K. Ramadge and Wonham showed that a necessary condition for this to be true is the so-called controllability of K with respect to L_M(G). They showed that when G is a finite state automaton and K is a regular language (also generated by a finite state automaton), then the controllability property was decidable for K. The class of languages generated by pushdown automata properly includes the regular languages. They are accepted by finite state machines coupled with pushdown stack memory. This makes them interesting candidates as supervisory languages, since the supervisor will have nonfinite memory. In this note, we show the following: i) If S is a specification given by a deterministic pushdown automaton and L is generated by a finite state machine, then there is an algorithm to decide whether K=S∩L is controllable with respect to L. ii) It is undecidable for an arbitrary specification S generated by a nondeterministic pushdown automaton and plant language L generated by a finite state machine whether K=S∩L is controllable with respect to L.