On Liveness Enforcing Supervisory Policies for Arbitrary Petri Nets

Chen Chen,Ramavarapu S Sreenivas,Arun Raman,Hesuan Hu

doi:10.1109/tac.2020.2970060

Abstract

Neither the existence nor the nonexistence of a liveness enforcing supervisory policy (LESP) for an arbitrary Petri net (PN) is semidecidable. In an attempt to identify decidable instances, we explore the decidability of certain properties of the set of initial markings for which an LESP exists, and the decidability of the existence of a specific class of LESPs. We first prove that for an arbitrary PN structure, determining if there is an initial marking, or there are no initial markings, for which there is an LESP, is not semidecidable. Then, we characterize the class of PN structures for which the set of all initial markings for which an LESP exists is right-closed . We show that testing membership, or nonmembership, of an arbitrary PN in this class of PNs is not semidecidable. We then consider a restricted class of LESPs, called marking monotone LESPs (MM-LESPs). We show that the existence of an MM-LESP for an arbitrary PN is decidable.

Highlights

A DISCRETE-EVENT dynamic system (DEDS) is a discrete-state, event-driven system, where the discrete state changes at a discrete-time instant due to the occurrence of events
After introducing K-class of Petri net (PN) structures in Section VII, in Section VIII we prove that the existence of an MM-liveness enforcing supervisory policy (LESP) for an arbitrary PN is decidable
This article is about the existence of an LESP for an arbitrary PN N (m0), where N = (Π, T, Φ, Γ), and m0 : Π → N is the initial marking

Summary

INTRODUCTION

A DISCRETE-EVENT dynamic system (DEDS) is a discrete-state, event-driven system, where the discrete state changes at a discrete-time instant due to the occurrence of events. If N ∈ H, Δ(N ) = ΔM (N ) [7] These results in the literature provide pointers on a possible approach to expand the class of PNs for which the existence of an LESP is decidable, first, by restricting the properties of the set Δ(N ) (for example, right-closure) and second, by restricting the nature of the LESP (for example, MM-LESPs). If a supervisory policy P is such that R(N, m, P) (which can have an unbounded number of markings) can be reduced to a reachability graph with a finite number of appropriately defined symbolic markings such that the liveness property is preserved, the existence of P is likely to be decidable

NOTATIONS AND DEFINITIONS

VIII. MM-LESP FOR ARBITRARY PNS

3: Replace mi by the set:

CONCLUSION