We study here the class of timed automata with a single clock which is reset at each transition. We adapt for these automata the classical results for finite automata: the Kleene theorem, the closure under complementation and the Pumping Lemma. We provide an algorithm for the elimination of stuttering steps, which is essential in complementation. This algorithm relies upon the properties of the Kleene algebra of sets of real numbers, namely the existence of a normal form for sets of reals generated from intervals with rational bounds, using boolean operations, summation and star.
“Journal of Automata, Languages and Combinatorics”, vol.6, no.1, pages 3-23, 2001.