  This paper contains two results on timed extensions of pushdown automata
(PDA). As our first result we prove that the model of dense-timed PDA of
Abdulla et al. collapses: it is expressively equivalent to dense-timed PDA with
timeless stack. Motivated by this result, we advocate the framework of
first-order definable PDA, a specialization of PDA in sets with atoms, as the
right setting to define and investigate timed extensions of PDA. The general
model obtained in this way is Turing complete. As our second result we prove
NEXPTIME upper complexity bound for the non-emptiness problem for an expressive
subclass. As a byproduct, we obtain a tight EXPTIME complexity bound for a more
restrictive subclass of PDA with timeless stack, thus subsuming the complexity
bound known for dense-timed PDA.
