  We solve some decision problems for timed automata which were recently raised
by S. Tripakis in [ Folk Theorems on the Determinization and Minimization of
Timed Automata, in the Proceedings of the International Workshop FORMATS'2003,
LNCS, Volume 2791, p. 182-188, 2004 ] and by E. Asarin in [ Challenges in Timed
Languages, From Applied Theory to Basic Theory, Bulletin of the EATCS, Volume
83, p. 106-120, 2004 ]. In particular, we show that one cannot decide whether a
given timed automaton is determinizable or whether the complement of a timed
regular language is timed regular. We show that the problem of the minimization
of the number of clocks of a timed automaton is undecidable. It is also
undecidable whether the shuffle of two timed regular languages is timed
regular. We show that in the case of timed B\"uchi automata accepting infinite
timed words some of these problems are Pi^1_1-hard, hence highly undecidable
(located beyond the arithmetical hierarchy).
