The operations and properties of reproducibility of the empty marking

Kurt Lautenbach presented sufficient and necessary conditions for reproducibility of the empty marking and proved that the empty marking is reproducible if and only if there is non-negative T-invariant, whose net representations have neither siphons nor traps, containing a positive entry for at least one fact and goal transition. This paper extends these results, we prove that composition, insertion, deletion and substitution do not influence reproducibility of the empty marking, and a net with reproducibility of the empty marking preserves the reproducibility in its backward net (N is a net, its backward net N-1 arises from N by reversing the direction of all arcs). We also show that 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.