More than 0s and 1s: metric quantifiers and counting over timed words

We study the expressiveness of the pointwise interpretations (i.e. over timed words) of some predicate and temporal logics with metric and counting features. We show that counting in the unit interval (0, 1) is strictly weaker than counting in (0, b) with arbitrary b ≥ 0; moreover, allowing the latter indeed leads to expressive completeness for the metric predicate logic Q2MLO, recovering the corresponding result for the continuous interpretations (i.e. over signals). Exploiting this connection, we show that in contrast to the continuous case, adding "punctual" predicates into Q2MLO is still insufficient for the full expressive power of the Monadic First-Order Logic of Order and Metric (FO[<,+1]). Finally, we propose a generalisation of the recently proposed Pnueli automata modalities and show that the resulting metric temporal logic is expressively complete for FO[<,+1].