On a weaker notion of controllability of a language K with respect to a language L

A necessary and sufficient condition for a prefix-closed language K contained in Sigma * to be controllable with respect to another prefix-closed language L contained in Sigma * is that K contained in L. A weaker notion of controllability where it is not required that K contained in L is considered here. If L is the prefix-closed language generated by a plant automaton G, then essentially there exists a supervisor Theta that is complete with respect to G such that L( Theta mod G)=K intersection L if and only if K is weakly controllable with respect to L. For an arbitrary modeling formalism it is shown that the inclusion problem is reducible to the problem of deciding the weaker notion of controllability. Therefore, removing the requirement that K contained in L from the original definition of controllability does not help the situation from a decidability viewpoint. This observation is then used to identify modeling formalisms that are not viable for supervisory control of the untimed behaviors of discrete-event dynamic systems.<
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