  We study timed systems in which some timing features are unknown parameters.
Parametric timed automata (PTAs) are a classical formalism for such systems but
for which most interesting problems are undecidable. Notably, the parametric
reachability emptiness problem, i.e., the emptiness of the parameter valuations
set allowing to reach some given discrete state, is undecidable.
Lower-bound/upper-bound parametric timed automata (L/U-PTAs) achieve
decidability for reachability properties by enforcing a separation of
parameters used as upper bounds in the automaton constraints, and those used as
lower bounds.
  In this paper, we first study reachability. We exhibit a subclass of PTAs
(namely integer-points PTAs) with bounded rational-valued parameters for which
the parametric reachability emptiness problem is decidable. Using this class,
we present further results improving the boundary between decidability and
undecidability for PTAs and their subclasses such as L/U-PTAs.
  We then study liveness. We prove that:
  (1) deciding the existence of at least one parameter valuation for which
there exists an infinite run in an L/U-PTA is PSpace-complete;
  (2) the existence of a parameter valuation such that the system has a
deadlock is however undecidable;
  (3) the problem of the existence of a valuation for which a run remains in a
given set of locations exhibits a very thin border between decidability and
undecidability.
