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Classification of solutions of matrix equation related to parallel structure of a Petri net

In order to solve the reachability problem of Petri nets, two approaches, reachability tree and matrix equation, are usually applied. Firing count vector which is a solution of matrix equation will give us available information when using the method of matrix equation. The major problem using matrix analysis is the lack of information of firing sequences and the existence of spurious solutions. In ordinary cases, an incidence matrix does not have full rank that is necessary condition to obtain the inverse matrix for the solutions of the matrix equation. Research must be done to determine which solutions should be tried as firing count vectors which number may be possibly infinite. This paper takes aim at the classification of solutions of matrix equation related to parallel structure of a Petri net. A firing counter vector can be obtained as a solution of the matrix equation and a solution must be expressed by a form of one special solution and an arbitrary linear combination of its fundamental solutions. Because the fundamental solutions are the transition invariant, the spurious solutions must be related to the structure of the Petri net. The authors classify the solutions only related to parallel structure of a Petri net and show the number of these solutions must be finite. Certainly, we will also discuss the other solutions related to cycle structure and give a hint to solve those solutions.