Petri nets and regular processes

We consider the following problems: (a) Given a labelled Petri net and a finite automaton, are they equivalent?; (b) Given a labelled Petri net, is it equivalent to some (unspecified) finite automaton? These questions are studied within the framework of trace and bisimulation equivalences, in both their strong and weak versions. (In the weak version a special τ action - likened to an ε-move in automata theory - is considered to be nonobservable.) We demonstrate that (a) is decidable for strong and weak trace equivalence and for strong bisimulation equivalence, but undecidable for weak bisimulation equivalence. On the other hand, we show that (b) is decidable for strong bisimulation equivalence, and undecidable for strong and weak trace equivalence, as well as for weak bisimulation equivalence.