RAPPORT Two - sorted metric temporal logics

Temporal logic has been successfully used for modeling and analyzing the behavior of reactive and concurrent systems. One shortcoming 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 ooer 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. In this paper we deal with completeness issues for basic systems of metric temporal logic | despite their relevance, such issues have been ignored or only partially addressed in the literature. 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. We rst 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. Finally, we consider general metric temporal logic allowing quantiication over algebraic variables and free mixing of algebraic formulae and temporal propositional symbols. The rst part of the paper is a revised and extended version of 13].