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

  • Paritosh K. Pandya
  • Simoni S. Shah
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7561)


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.


Normal Form Temporal Logic Time Stamp Regular Language Unary Modality 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Paritosh K. Pandya
    • 1
  • Simoni S. Shah
    • 1
  1. 1.Tata Institute of Fundamental ResearchMumbaiIndia

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