Abstract
The reachability tree cannot, in general, be used to solve the reachability problem because of the existence of the ω symbol. The central issue of this paper is to limit the reachability tree to a finite one without using the ω symbol so that it can be used to solve the reachability problem. For this purpose, we consider the marking space in a Petri net as a Euclidean metric space, and limit the reachability tree in a hypersphere in the metric space based on the following ideas: Let II be the set of all executable marking sequences from the initial marking to the final marking. Let aπ be the maximum value of the norms of all the markings included in such a marking sequence π which belongs to II, and let a be the minimum value of all those aπ (if II is empty, let a be 0). Then, the final marking is reachable if and only if it is included in the reachability tree in the hypersphere Ur with its center at the origin and radius r which is an upper bound fo a. This paper also presents the way to get such an upper bound r of a for two classes of Petri nets (one of them includs the well-known trap circuit Petri net). Then, by creating the reachability tree in such a hypersphere Ur, we can determine not only whether the final marking is reachable or not, but also at least one firing sequence of transitions leading to it at the same time.
