  We consider pushdown timed automata (PTAs) that are timed automata (with
dense clocks) augmented with a pushdown stack. A configuration of a PTA
includes a control state, dense clock values and a stack word. By using the
pattern technique, we give a decidable characterization of the binary
reachability (i.e., the set of all pairs of configurations such that one can
reach the other) of a PTA. Since a timed automaton can be treated as a PTA
without the pushdown stack, we can show that the binary reachability of a timed
automaton is definable in the additive theory of reals and integers. The
results can be used to verify a class of properties containing linear relations
over both dense variables and unbounded discrete variables. The properties
previously could not be verified using the classic region technique nor
expressed by timed temporal logics for timed automata and CTL$^*$ for pushdown
systems. The results are also extended to other generalizations of timed
automata.
