Reachability criterion for Petri nets with known firing count vectors

A formal necessary and sufficient condition on the general Petri net reachability problem is presented by eliminating all spurious solutions among known nonnegative integer solutions of state equation and unifying all the causes of those spurious solutions into a maximal-strongly-connected and siphon-and-trap subnet N-w. This result is based on the decomposition of a given net (N, M-o) with M-d and the concepts of "no immature siphon at the reduced initial marking M-wo" and "no immature trap at the reduced end marking M-wd" on N-w which are both extended from "no token-free siphon at the initial marking M-o" and "no token-free trap at the end marking M-d" on N, respectively, which have been both effectively, explicitly or implicitly, used in the well-known fundamental and simple subclasses.