We study timed systems in which some timing features are unknown parameters. First we consider Upper-bound Parametric Timed Automata (U-PTAs), one of the simplest extensions of timed automata with parameters, in which parameters are only used as clock upper bounds. Up to now, there have been several decidability results for the existence of parameter values in U-PTAs such that flat TCTL formulas are satisfied. We prove here that this does not extend to the full logic and that only one level of nesting leads to undecidability. This provides, to the best of our knowledge, the first problem decidable for Timed Automata with an undecidable parametric emptiness version for U-PTAs. Second we study Lower/Upper-bound Parametric Timed Automata (L/U-PTAs) in which parameters are used either as clock lower bound, or as clock upper bound, but not both. We prove that without invariants, flat TCTL is decidable for L/U-PTAs by resolving the last non investigated liveness properties.
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