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 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. 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 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.Finally, we consider general metric temporal logic allowing quantification over algebraic variables and free mixing of algebraic formulae and temporal propositional symbols.

General (acm F.4.0), Mathematical Logic (acm F.4.1), Deduction and Theorem Proving (acm I.2.3), Knowledge Representation Formalisms and Methods (acm I.2.4)
Modal logic (including the logic of norms) (msc 03B45), Abstract model theory (msc 03C95), Finite structures (msc 03C13)
Department of Computer Science [CS]

Montanari, A, & de Rijke, M. (1995). Two-sorted metric temporal logics. Department of Computer Science [CS]. CWI.