Counterexamples to "liveness-enforcing supervision of bounded ordinary Petri nets using partial-order methods"

This note shows by means of simple counterexamples that some key results presented by He and Lemmon on the liveness verification and enforcing of Petri nets using unfolding are incorrect. As a result, the applicability of unfolding for Petri net supervision is still an open issue. 1 Introduction Recently, He and Lemmon have presented an original approach for the analysis and control of bounded Petri nets based on unfolding. This technique allows one to describe the set of reachable marking of a given net N by means of a finite occurrence net (called the unfolding of N) without the necessity of explicitly computing the reachability graph of N [4]. This often leads to significant computational advantages. In a series of papers [1-3] He and Lemmon have used unfolding for liveness verification and enforcing. We show in this note through a series of counterexamples that some key results of these papers are incorrect. As a result, although we still strongly believe that unfolding is an interesting and potentially fruitful technique for Petri net control, the applicability of unfolding for Petri net supervision is still an open issue. 2 Counterexamples for [1] Let us first consider the approach proposed in [1]. The unfolding βc used in this approach is an extension of the well-known McMillan unfolding [4] and includes cut-off transitions not preceding any other cut-off transitions. As an example, the unfolding βc of the net N1 in Figure 1.(a) is shown in part (b) of the same figure. Note that here we are conventionally using a double arrowed arc to represent two arcs with opposite directions. The approach proposed in [1] is based on three concepts: • deadlocked configurations; • cut graph; • cut cycles. (*) Paper to appear as a correspondence on the IEEE Transactions on Automatic Control, 2004.