On Supervisory Policies That Enforce Liveness in a Class of Completely Controlled Petri Nets Obtained via Reenement Accepted, Ieee Transactions on Automatic Control

ion as suggested by the conditions of this paper is not computationally expensive. For anygiven re ned PN bN there might be many possible candidates for the abstracted PN N and the live,MGPN eN that satis es the requirements of this paper. However, nding a candidate abstractedPN N with the smallest coverability graph can be computationally expensive.The coverability graph of the PN bN shown in gure 1(iii) has eighty vertices, while that of thePN N has only four vertices. Clearly, testing the existence of a supervisory policy that enforcesliveness in the CCPN M is easier than the corresponding test for the CCPN cM . There exists asupervisory policy that enforces liveness in the CCPN M . This can be inferred by theorem 3.2 andthe fact that in the PN N , m0 ! t2t3t0t1 ! m0 and all transitions in T appear once in the stringt2t3t0t1. The supervisory policy P of preventing the ring of transition t1 when there is a tokenin p2 enforces liveness in M . From the proof of lemma 3.5 we infer the supervisory policy bP ofpreventing the ring of t1 when there is a token in p2 also enforces liveness in cM .4 ConclusionsReferences [6, 7] introduce a necessary and su cient condition for the existence of a supervisorypolicy that enforces liveness in a Completely Controlled Petri net (CCPN). This procedure has atime-complexity that is polynomially related to the number of vertices in the coverability graph ofthe underlying Petri net (PN) of the CCPN. However, the number of vertices in the coverabilitygraph of a PN can be exponentially related to the number of places and transitions. Using there nement/abstraction procedure of Suzuki and Murata [8] we presented a procedure of reducingthe computational burden of this test. The re nement procedure of Suzuki and Murata involves thesubstitution of a single transition in a PN N by another PN eN . The abstraction procedure involvesreversal of this process. We showed that if the underlying PN bN of the original CCPN cM is obtainedby re ning a transition in a PN N by a live, marked-graph PN eN with some additional restrictions,then testing the existence of a supervisory policy that enforces liveness in cM is equivalent to thecorresponding test for the CCPN M whose underlying PN is the abstracted PN N . Using anexample we illustrated the computational savings of this procedure. As a future research directionwe suggest investigations into weakening the restrictions on the PN eN that yields a similar result.Towards this end, it might be worthwhile to investigate the application of the transformationsthat preserve liveness such as those listed in reference [1] to the synthesis of supervisory policies in16 complex, non-live CCPNs from similar policies for a simpler, abstracted CCPN that is also non-live.References[1] G. Berthelot. Checking properties of nets using transformations. In Advances in Petri Nets,pages 19{40, Germany, 1985. Lecture Notes in Computer Science, Vol. 222, Springer-Verlag.[2] N.D. Jones, L.H. Landweber, and Y.E. Lien. Complexity of some problems in Petri nets.Theoretical Computer Science, 4:277{299, 1977.[3] T. Murata. Petri nets: properties, analysis and applications. Proc. of the IEEE, 77(4):541{580,April 1989.[4] T. Murata, V.B. Le, and D.J. Leu. A method for realizing the synchronic distance matrix asa marked graph. In IEEE International Conference on Circuits and Systems, pages 609{612,May 1982. Rome, Italy.[5] J.L. Peterson. Petri net theory and the modeling of systems. Prentice-Hall, Englewood Cli s,NJ, 1981.[6] R.S. Sreenivas. Enforcing liveness via supervisory control in discrete event dynamic systemsmodeled by completely controlled Petri nets. In WODES-96: International Workshop on Dis-crete Event Systems, pages 296{301, August 1996. University of Edinburgh, UK (IEE Press).[7] R.S. Sreenivas. On the existence of supervisory policies that enforce liveness in discrete event dy-namic systems modeled by controlled Petri nets. IEEE Trans. on Automatic Control, 42(7):928{945, July 1996.[8] I. Suzuki and T. Murata. A method for stepwise re nement and abstraction of Petri nets.Journal of Computer and System Sciences, 27:51{76, 1983.17