A Strict Minimal Siphon-Based Necessary and Sufficient Condition for Liveness Verification in General Petri Nets

To describe deadlocks occurring in general nets, some characterizations have been proposed to establish a relationship between liveness and siphons. Among them, the most prominent characterizations are max-controlled siphons and deadly marked siphons. Nevertheless, the former is only a sufficient condition while the latter may lead to some unreasonable operations, i.e., modified markings are necessary if applicable. In this paper, we elaborately discuss the related concepts of deadly marked siphons when liveness analysis is conducted. Subsequently, a new siphon-based characterization of liveness, i.e., undermarked siphons that explain liveness in the original markings, is proposed to demonstrate its more general applicability. Finally, a necessary and sufficient condition is developed to show the relationship between the liveness and undermarked siphons in general nets.