Operating and Analyzing the Reproducibility of Empty Marking Nets

To reproduce the empty marking, there is a necessary and sufficient condition in which a non-negative T-invariant exists, whose net representations have neither siphons nor traps, containing a positive entry for at least one fact and goal transition. This result is extended and it is proved that the operations of composition, insertion, deletion and substitution do not influence the reproducibility of the empty marking. Moreover, some properties related to the empty marking net are discussed, For example, a net with reproducibility of the empty marking has preserved the reproducibility in its inversed net, and the empty marking in acyclic P/T nets with a positive entry for at least one fact and goal transition is reproducible if and only if the net is covered by T-invariant. In particular, if a Horn-net satisfies all the above conditions and it is acyclic, then the T-invariant is realizable. These theoretical results show that there are interesting connections to other notions, for example, to the forward and backward liveness of the empty marking. On the other hand, there are application oriented aspects of those results. Examples can be found in proving complex logic inference and checking throughness of workflow logic net. Finally its algorithm is proposed.