Metric Temporal Logic with Counting

Ability to count number of occurrences of events within a specified time interval is very useful in specification of resource bounded real time computation. In this paper, we study an extension of Metric Temporal Logic ($\mathsf{MTL}$) with two different counting modalities called $\mathsf{C}$ and $\mathsf{UT}$ (until with threshold), which enhance the expressive power of $\mathsf{MTL}$ in orthogonal fashion. We confine ourselves only to the future fragment of $\mathsf{MTL}$ interpreted in a pointwise manner over finite timed words. We provide a comprehensive study of the expressive power of logic $\mathsf{CTMTL}$ and its fragments using the technique of EF games extended with suitable counting moves. Finally, as our main result, we establish the decidability of $\mathsf{CTMTL}$ by giving an equisatisfiable reduction from $\mathsf{CTMTL}$ to $\mathsf{MTL}$. The reduction provides one more example of the use of temporal projections with oversampling introduced earlier for proving decidability. Our reduction also implies that $\mathsf{MITL}$ extended with $\mathsf{C}$ and $\mathsf{UT}$ modalities is elementarily decidable.