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The theory of the natural numbers with linear order and monadic predicates underlies propositional linear temporal logic. To study temporal logics that are suitable for reasoning about real-time systems, we combine this classical theory of infinite state sequences with a theory of discrete time, via a monotonic function that maps every state to its time. The resulting theory of timed state sequences is shown to be decidable, albeit nonelementary, and its expressive power is characterized by ω-regular sets. Several more expressive variants are proved to be highly undecidable. This framework allows us to classify a wide variety of real-time logics according to their complexity and expressiveness. Indeed, it follows that most formalisms proposed in the literature cannot be decided. We are, however, able to identify two elementary real-time temporal logics as expressively complete fragments of the theory of timed state …