The necessary and sufficient condition for liveness of some timed Petri nets as well as Petri nets without time

The initial-marking-based necessary and sufficient condition for liveness of special timed Petri nets as well as ordinary Petri nets without time is given by the direct and net-structural approach. It is shown that by using the properties and theorems, the new initial-marking-based necessary and sufficient condition for liveness of special timed Petri nets as well as Petri nets without time is derived.<>