Removing " - Transitions in Timed

Timed automata are among the most widely studied models for real-time systems. Silent transitions, i.e., "-transitions, have already been proposed in the original paper on timed automata by Alur and Dill 2]. In 7] it is shown that "-transitions can be removed, if they do not reset clocks; moreover "-transitions strictly increase the power of timed automata, if there is a self-loop containing "-transitions which reset some clocks. The authors of 7] left open the problem about the power of the "-transitions which reset clocks, if they do not lie on any cycle. The present paper settles this open question. Precisely, we prove that a timed automaton such that no "-transition with nonempty reset set lies on any directed cycle can be eeectively transformed into a timed automaton without "-transitions. Interestingly, this main result holds under the assumption of non-Zenoness and it is false otherwise. Besides, we develop a promising new technique based on a notion of precise time which allows to show that some timed languages are not recognizable by any "-free timed automaton.