Expressive Completeness for 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. Hirshfeld and Rabinovich have shown that over the reals, first-order logic with binary order relation < and unary function +1 is strictly more expressive than MTL with integer constants. Indeed they prove that no temporal logic whose modalities are definable by formulas of bounded quantifier depth can be expressively complete for FO(<, + 1). In this paper we show that if we allow unary functions +q, q is an element of Q, in first-order logic and correspondingly allow rational constants in MTL, then the two logics have the same expressive power. This gives the first generalisation of Kamp's theorem on the expressive completeness of LTL for FO(<) to the quantitative setting. The proof of this result involves a generalisation of Gabbay's notion of separation to the metric setting.