On recognizable timed languages

In this work we generalize the fundamental notion of recognizability from untimed to timed languages. The essence of our definition is the existence of a right-morphism from the monoid of timed words into a bounded subset of itself. We show that the recognizable languages are exactly those accepted by deterministic timed automata and argue that this is, perhaps, the right class of timed languages, and that the closure of untimed regular languages under projection is a positive accident that cannot be expected to hold beyond the finite-state case.