## Abstract

The common metric temporal logics for continuous time were shown to be insufficient, when it was proved in Hirshfeld and Rabinovich (1999, 2007) [7,12] that they cannot express a modality suggested by Pnueli. Moreover, no temporal logic with a finite set of modalities can express all the natural generalizations of this modality. The temporal logic with counting modalities (TLC) is the extension of until-since temporal logic TL (U, S) by "counting modalities" C_{n} (X) and over _{n} (n ∈ N); for each n the modality C_{n} (X) says that X will be true at least at n points in the next unit of time, and its dual over _{n} (X) says that X has happened n times in the last unit of time. In Hirshfeld and Rabinovich (2006) [11] it was proved that this temporal logic is expressively complete for a natural decidable metric predicate logic. In particular the Pnueli modalities Pn_{k} (X_{1}, ..., X_{k}), "there is an increasing sequence t_{1}, ..., t_{k} of points in the unit interval ahead such that X_{i} holds at t_{i}", are definable in TLC. In this paper we investigate the complexity of the satisfiability problem for TLC and show that the problem is PSPACE complete when the index of C_{n} is coded in unary, and EXPSPACE complete when the index is coded in binary. We also show that the satisfiability problem for the until-since temporal logic extended by the Pnueli modalities is PSPACE complete.

Original language | English |
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Pages (from-to) | 2331-2342 |

Number of pages | 12 |

Journal | Theoretical Computer Science |

Volume | 411 |

Issue number | 22-24 |

DOIs | |

State | Published - 17 May 2010 |

## Keywords

- Complexity
- Expressive power
- Real time temporal logics