Bounds on the Number of Markings Consistent With Label Observations in Petri Nets

In this paper, we consider state estimation in discrete-event systems (DESs) modeled by labeled Petri nets and present upper bounds on the number of system states (or markings) that are consistent with an observed sequence of labels. Our analysis is applicable to Petri nets that may have nondeterministic transitions (i.e., transitions that share the same label) and/or unobservable transitions (i.e., transitions that are associated with the null label). More specifically, given knowledge of a labeled Petri net structure and its initial state, we show that the number of consistent markings in a Petri net with nondeterministic transitions is at most polynomial in the length of the observation sequence (i.e., polynomial in the number of labels observed). This polynomial dependency of the number of consistent markings on the length of the observation sequence also applies to Petri nets with unobservable transitions under the assumption that their unobservable subnets are structurally bounded. The bounds on the number of markings established in this paper imply that the state estimation problem in labeled Petri nets can be solved with complexity that is polynomial in the length of the observation sequence.