A note on deciding the controllability of a language K with respect to a language L

It has been shown by P.J. Ramadge and W.M. Wonham (1987) if G is a plant automaton and K⊆L(G) is a prefix-closed language, there exists a supervisor Θ that is complete with respect to G so that L(Θ/G)=K∩L(G)= K if and only if K is controllable with respect to the plant language L(D). It is shown that if L( G) and K are represented as free-labeled Petri nets, then the controllability of K with respect to L(G ) is detectable. This result is a direct consequence of the decidability of the Petri net and reachability problem. In effect, a modeling framework capable of finitely representing a class of infinite-state systems is identified, and controllability is decidable within this framework