Observable liveness of Petri nets

Whereas, for Petri nets, the traditional liveness property guarantees that each transition of a Petri net can always occur again, observable liveness requires that, from any reachable marking, each observable transition can be forced to fire by choosing appropriate controllable transitions; hence it is defined for Petri nets with distinguished observable and controllable transitions. We introduce observable liveness and show that this new notion generalizes traditional liveness in various ways. In particular, liveness of a 1-bounded Petri net implies observable liveness, provided the only conflicts that can appear are between controllable transitions. This assumption refers to applications where the uncontrollable part models a deterministic machine (or several deterministic machines), whereas the user of the machine is modeled by the controllable part and can behave arbitrarily.