On the liveness problem of 1-place-unbounded Petri nets

An augmented reachability tree (ART) is proposed to extend the ability of the classical reachability tree for solving the liveness problem of a class of Petri nets where there exists at most one unbounded place. The idea is based on the computation of the minimal token number of the unbounded place. An algorithm for obtaining the minimal token number is shown. Examples are given to illustrate the technique. The proposed method can also be used to solve the shortest path problem in graphs with negative costs.