  We consider the model-checking problem for freeze LTL on one-counter automata
(OCA). Freeze LTL extends LTL with the freeze quantifier, which allows one to
store different counter values of a run in registers so that they can be
compared with one another. As the model-checking problem is undecidable in
general, we focus on the flat fragment of freeze LTL, in which the usage of the
freeze quantifier is restricted. In a previous work, Lechner et al. showed that
model checking for flat freeze LTL on OCA with binary encoding of counter
updates is decidable and in 2NEXPTIME. In this paper, we prove that the problem
is, in fact, NEXPTIME-complete no matter whether counter updates are encoded in
unary or binary. Like Lechner et al., we rely on a reduction to the
reachability problem in OCA with parameterized tests (OCA(P)). The new aspect
is that we simulate OCA(P) by alternating two-way automata over words. This
implies an exponential upper bound on the parameter values that we exploit
towards an NP algorithm for reachability in OCA(P) with unary updates. We
obtain our main result as a corollary. As another application, relying on a
reduction by Bundala and Ouaknine, one obtains an alternative proof of the
known fact that reachability in closed parametric timed automata with one
parametric clock is in NEXPTIME.
