Thomas A. Henzinger
We introduce a novel extension of propositional modal logic that is interpreted over Kripke structures in which a value is associated with every possible world. These values are, however, not treated as full first-order objects; they can be accessed only by a very restricted form of quantification: the freeze quantifier binds a variable to the value of the current world. We present a complete proof system for this half-order modal logic.
As a special case, we obtain the real-time temporal logic TPTL: the models are restricted to infinite sequences of states, whose values are monotonically increasing natural numbers. The ordering relation between states is interpreted as temporal precedence, while the value associated with a state is interpreted as its "real" time. We extend our proof system to be complete for TPTL, and demonstrate how it can be used to derive real-time properties.
Ninth Annual Symposium on Principles of Distributed Computing (PODC), ACM Press, 1990, pp. 281-296.