When Do You Start Counting? Revisiting Counting and Pnueli Modalities in Timed Logics

Pnueli first noticed that certain simple 'counting' properties appear to be inexpressible in popular timed temporal logics such as Metric Interval Temporal Logic (MITL). This interesting observation has since been studied extensively, culminating in strong timed logics that are capable of expressing such properties yet remain decidable. A slightly more general case, namely where one asserts the existence of a sequence of events in an arbitrary interval of the form <a, b> (instead of an upper-bound interval of the form [0, b>, which starts from the current point in time), has however not been addressed satisfactorily in the existing literature. We show that counting in [0, b> is in fact as powerful as counting in <a, b>; moreover, the general property 'there exist x', x'' in I such that x' <= x'' and phi(x', x'') holds' can be expressed in Extended Metric Interval Temporal Logic (EMITL) with only [0, b>.