Revisiting Timed Logics with Automata Modalities

It is well known that (timed) $\omega$-regular properties such as `p holds at every even position' and `p occurs at least three times within the next 10 time units' cannot be expressed in Metric Interval Temporal Logic ($\mathsf{MITL}$) and Event Clock Logic ($\mathsf{ECL}$). A standard remedy to this deficiency is to extend these with modalities defined in terms of automata. In this paper, we show that the logics $\mathsf{EMITL}_{0,\infty}$ (adding non-deterministic finite automata modalities into the fragment of $\mathsf{MITL}$ with only lower- and upper-bound constraints) and $\mathsf{EECL}$ (adding automata modalities into $\mathsf{ECL}$) are already as expressive as $\mathsf{EMITL}$ (full $\mathsf{MITL}$ with automata modalities). In particular, the satisfiability and model-checking problems for $\mathsf{EMITL}_{0,\infty}$ and $\mathsf{EECL}$ are PSPACE-complete, whereas the same problems for $\mathsf{EMITL}$ are EXPSPACE-complete. We also provide a simple translation from $\mathsf{EMITL}_{0,\infty}$ to diagonal-free timed automata, which enables practical satisfiability and model checking based on off-the-shelf tools.