The Unary Fragments of Metric Interval Temporal Logic: Bounded versus Lower Bound Constraints

Abstract

We study two unary fragments of the well-known metric interval temporal logic $\mathit{MITL[\textsf{U}_I,\textsf{S}_I]}$ that was originally proposed by Alur and Henzinger, and we pin down their expressiveness as well as satisfaction complexities. We show that $\mbox{$\mathit{MITL[\textsf{F}_\infty,\textsf{P}_\infty]}$}$ which has unary modalities with only lower-bound constraints is (surprisingly) expressively complete for Partially Ordered 2-Way Deterministic Timed Automata (po2DTA) and the reduction from logic to automaton gives us its NP-complete satisfiability. We also show that the fragment $\mbox{$\mathit{MITL[\textsf{F}_b,\textsf{P}_b]}$}$ having unary modalities with only bounded intervals has NEXPTIME-complete satisfiability. But strangely, $\mathit{MITL[\textsf{F}_b,\textsf{P}_b]}$ is strictly less expressive than $\mathit{MITL[\textsf{F}_\infty,\textsf{P}_\infty]}$. We provide a comprehensive picture of the decidability and expressiveness of various unary fragments of MITL.