On Partially Controlled Free Choice Petri Nets

Abstract

A Petri Net (PN) is said to be live if it is possible to fire any transition from every reachable marking, although not necessarily immediately. Under appropriate conditions, a non-live PN can be made live via supervision. Under this paradigm an external-agent, the supervisor, prevents the firing of certain transitions at each reachable marking so as to enforce liveness. A PN is partially controlled if the supervisor can prevent the firing of only a subset of transitions. A Free-Choice Petri net (FCPN) is a PN where every arc from a place to a transition is either the unique output arc from that place, or, it is the unique input arc to the transition. In this paper we show the existence of a supervisory policy that enforces liveness in a partially controlled FCPN is monotone with respect to the initial marking. That is, if there is a supervisory policy that enforces liveness in a partially controlled FCPN for a particular initial marking, then there is a supervisory policy that enforces liveness for the same FCPN with a larger initial marking. Since this property is not true of the general class of partially controlled PNs, this result has several implications regarding the relationship between the class of partially controlled PNs and partially controlled FCPNs. Some implications of this result along with future research directions are also presented in this paper.