Half-order modal logic: How to prove real-time properties

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 of [AH891: 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 "rear time. We extend our proof system to be complete for TPTL. and demonstrate how it can be used to derive real-time properties.