Equivalence of timed state machines and safe TPN

We show that an important subclass of timed automata (Alur and Dill, 1994), called timed state machines, is weakly time equivalent to safe non-zero time Petri nets (TPNs) in the sense of Merlin and Farber (1976). We present an explicit construction for two-way translation between 1-safe TPNs and TSMs. The translation improves on the efficiency of other methods: the TSM obtained for a given net is polynomial in the size of the reachability graph, and a given TSM is translated into a net whose size grows linearly with that of the automaton model.