Decidability and expressive power of real time logics

Temporal Logic based on the two modalities "Since" and "Until" (TL) is popular among computer scientists as the framework for reasoning about a system evolving in time. This logic is usually referred to as the standard linear time temporal logic. The "linear time" appears in the name of this logic probably because when it was introduced by Pnueli its intended models were linear orders; more precisely, ω-models. The adjective "standard" is presumable used due to the Kamp theorem which states that it is expressively equivalent (over the w-models) to first order monadic logic of order - a very fundamental logic. The two logics are expressively equivalent whether the system evolves in discrete steps or in continuous time; however, for continuous time, both logics can not express properties like: "X will happen exactly after one unit of time." Unfortunately, the extension of TL by this modality is undecidable. Two of the most important characteristics of specification formalisms are expressive power and the decidability/complexity of the validity and model checking problems. Over the years, different extensions of TL which can specify metric properties of real time were suggested. The goal of this talk is to survey expressiveness and decidability results for temporal and predicate logics for the specification of metric properties of real time.