The extremal languages in supervisory control of discrete event systems with service specifications

In some application areas such as communication systems, a control specification is often defined on the specified subset of events. Such a specification is called a service specification. Kumar et al. presented a necessary and sufficient condition under which there exists a supervisor for a service specification. Given a service specification, this paper studies its sublanguages and superlanguages for which there exists a supervisor. We show that, unfortunately, neither the supremal sublanguage nor the infimal superlanguage exists in general. Motivated by the fact, we first give a subset of the sublanguages whose supremal element exists. We next present a subset of the superlanguages whose infimal element exists. Moreover, a formula for the infimal element is obtained.