Probabilistic marking estimation in labeled Petri nets

Given a labeled Petri net, possibly with silent transitions, we are interested in performing current marking estimation in a probabilistic setting. We assume a known initial marking or a known finite set of initial markings, each with some a priori probability, and our goal is to obtain the conditional probabilities of possible current markings, conditioned on an observed sequence of labels. Under the assumptions that (i) the reachability set of the unobservable subnet of the given Petri net (starting from any possible initial state) is bounded (i.e., it has a finite number of reachable states), and (ii) a characterization of the a priori probabilities for occurrence of each transition enabled at each reachable marking is available, we develop a recursive algorithm that can perform current marking estimation online to efficiently obtain the probabilities of current states (conditioned on a sequence of observations). The proposed algorithm can be used in conjunction with a variety of supervisory control and fault diagnosis algorithms, in order to relax stringent constraints imposed by existing methodologies that typically rely solely on binary information regarding the possibility or impossibility of current states (but not their probabilities).