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Probabilistic timed automata (PTAs) are timed automata (TAs) extended with discrete probability distributions. They serve as a mathematical model for a wide range of applications that involve both stochastic and timed behaviours. In this work, we consider the problem of model-checking linear dense-time properties over PTAs. In particular, we study linear dense-time properties that can be encoded by TAs with infinite acceptance criterion. First, we show that the problem of model-checking PTAs against deterministic-TA specifications can be solved through a product construction. Based on the product construction, we prove that the computational complexity of the problem with deterministic-TA specifications is EXPTIME-complete. Then we show that when relaxed to general (nondeterministic) TAs, the model-checking problem becomes undecidable. Our results substantially extend state of the art with both the dense-time feature and the nondeterminism in TAs.

Verifying Probabilistic Timed Automata Against Omega-Regular Dense-Time Properties