A method for structural analysis of Petri net models

Reachability tree and firing count vector provide information for solving the reachability problem in Petri nets. Firing count vector can be obtained as a solution of the equation of incidence matrix. In ordinary cases, an incidence matrix does not have full rank which is a necessary condition to obtain the inverse matrix for the solutions of the matrix equation. A method to make full rank matrix from an incidence matrix is to add some counter-places into the incidence matrix. However, in this method, the cycle structure and/or parallel structure of a given Petri net model must be identified and the total number of cycles and parallels must be known. Up to the present, there is no way to count cycle and parallel numbers in a given Petri net. In this paper, we prove that the number of cycles and/or parallels of a Petri net model must be n-r. Further, taking the advantage of the solutions of the homogeneous equation of transition invariants, we propose a method for identifying the cycle structure and/or parallel structure in that model