Metric Temporal Logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL--which includes invariance and time-bounded response properties--is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature.
@cite_3: Metric temporal logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL-which includes invariance and time-bounded response properties-is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature. The question of the decidability of MTL over infinite words remains open.
All the decidability results presented in this paper concern timed alternating automata over finite timed words, including the results that are ostensibly about infinite timed words. In particular, our model-checking procedure for the safety fragment of MTL over infinite timed words depends on the fact that any infinite timed word violating a safety property has a finite , that is, a finite prefix none of whose extensions satisfies the property. Since writing the extended abstract of this paper @cite_3 , we have obtained some positive and negative decidability results about the language emptiness problem for timed alternating automata over infinite words. We discuss these results in the conclusion, .