Expressive Completeness of Metric Temporal Logic

Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. In this paper we show that over the reals MTL has the same expressive power as monadic first-order logic with binary order relation < and an infinite collection of unary functions +q, where q is a non-negative rational. The proof of this result involves a generalisation of Gabbay’s notion of separation to the metric setting.