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The Importance of the Past in Interval Temporal Logics: The Case of Propositional Neighborhood Logic

  • Dario Della Monica
  • Angelo Montanari
  • Pietro Sala
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7360)

Abstract

In our contribution, we study the effects of adding past operators to interval temporal logics. We focus our attention on the representative case of Propositional Neighborhood Logic (\(\mathsf{A \overline A}\) for short), taking into consideration different temporal domains. \(\mathsf{A \overline A}\) is the proper fragment of Halpern and Shoham’s modal logic of intervals with modalities for Allen’s relations meets (future modality) and met by (past modality). We first prove that, unlike what happens with point-based linear temporal logic, \(\mathsf{A \overline A}\) is strictly more expressive than its future fragment A. Then, we show that there is a log-space reduction from the satisfiability problem for \(\mathsf{A \overline A}\) over ℤ to its satisfiability problem over ℕ. Compared to the corresponding reduction for point-based linear temporal logic, the one for \(\mathsf{A \overline A}\) turns out to be much more involved. Finally, we prove that \(\mathsf{A \overline A}\) is able to separate ℚ and ℝ, while A is not.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Dario Della Monica
    • 1
  • Angelo Montanari
    • 2
  • Pietro Sala
    • 3
  1. 1.Department of Computer ScienceUniversity of SalernoItaly
  2. 2.Department of Mathematics and Computer ScienceUniversity of UdineItaly
  3. 3.Department of Computer ScienceUniversity of VeronaItaly

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