Completeness Results for Two-sorted Metric Temporal Logics

Temporal logic has been successfully used for modeling and analyzing the behavior of reactive and concurrent systems. One short-coming of (standard) temporal logic is that it is inadequate for real-time applications, because it only deals with qualitative timing properties. This is overcome by metric temporal logics which offer a uniform logical framework in which both qualitative and quantitative timing properties can be expressed by making use of a parameterized operator of relative temporal realization. We view metric temporal logics as two-sorted formalisms having formulae ranging over time instants and parameters ranging over an (ordered) abelian group of temporal displacements. In this paper we deal with completeness results for basic systems of metric temporal logic — such issues have largely been ignored in the literature. We first provide an axiomatization of the pure metric fragment of the logic, and prove its soundness and completeness.Then, we show how to obtain the metric temporal logic of linear orders by adding an ordering over displacements.