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Interval Logics and ωB-Regular Languages

  • Angelo Montanari
  • Pietro Sala
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7810)

Abstract

In the recent years, interval temporal logics are emerging as a workable alternative to more standard point-based ones. In this paper, we establish an original connection between these logics and ωB-regular languages. First, we provide a logical characterization of regular (resp., ω-regular) languages in the interval logic \(A\mspace{-0.3mu}B\bar{B}\) of Allen’s relations meets, begun by, and begins over finite linear orders (resp., ℕ). Then, we lift such a correspondence to ωB-regular languages by substituting \(A\mspace{-0.3mu}B\bar{B}\bar{A}\) for \(A\mspace{-0.3mu}B\bar{B}\) (\(A\mspace{-0.3mu}B\bar{B}\bar{A}\) is obtained from \(A\mspace{-0.3mu}B\bar{B}\) by adding a modality for Allen’s relation met by). In addition, we show that new classes of extended (ω-)regular languages can be naturally defined in \(A\mspace{-0.3mu}B\bar{B}\bar{A}\).

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References

  1. 1.
    Allen, J.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(11), 832–843 (1983)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bojańczyk, M.: A Bounding Quantifier. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 41–55. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Bojańczyk, M.: Weak MSO with the unbounding quantifier. Theory of Computing Systems 48(3), 554–576 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bojańczyk, M., Colcombet, T.: ω-regular expressions with bounds. In: LICS, pp. 285–296. IEEE Computer Society (2006)Google Scholar
  5. 5.
    Bresolin, D., Goranko, V., Montanari, A., Sala, P.: Tableaux for logics of subinterval structures over dense orderings. J. of Logic and Comp. 20(1), 133–166 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bresolin, D., Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions. Annals of Pure and Applied Logic 161(3), 289–304 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bresolin, D., Montanari, A., Sala, P., Sciavicco, G.: What’s decidable about Halpern and Shoham’s interval logic? The maximal fragment \(\mathsf{AB \overline{BL}}\). In: LICS, pp. 387–396. IEEE Computer Society (2011)Google Scholar
  8. 8.
    Goranko, V., Montanari, A., Sciavicco, G.: A road map of interval temporal logics and duration calculi. J. of Applied Non-Classical Logics 14(1-2), 9–54 (2004)zbMATHCrossRefGoogle Scholar
  9. 9.
    Halpern, J., Shoham, Y.: A propositional modal logic of time intervals. J. of the ACM 38(4), 935–962 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Lodaya, K.: Sharpening the Undecidability of Interval Temporal Logic. In: Kleinberg, R.D., Sato, M. (eds.) ASIAN 2000. LNCS, vol. 1961, pp. 290–298. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Montanari, A., Puppis, G., Sala, P.: A Decidable Spatial Logic with Cone-Shaped Cardinal Directions. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 394–408. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Montanari, A., Puppis, G., Sala, P.: Maximal Decidable Fragments of Halpern and Shoham’s Modal Logic of Intervals. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 345–356. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Montanari, A., Puppis, G., Sala, P., Sciavicco, G.: Decidability of the interval temporal logic \(AB\bar{B}\) on natural numbers. In: STACS, pp. 597–608 (2010)Google Scholar
  14. 14.
    Moszkowski, B.: Reasoning about digital circuits. Tech. rep. stan-cs-83-970, Dept. of Computer Science, Stanford University, Stanford, CA (1983)Google Scholar
  15. 15.
    Venema, Y.: A modal logic for chopping intervals. J. of Logic and Comp. 1(4), 453–476 (1991)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Angelo Montanari
    • 1
  • Pietro Sala
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of UdineItaly
  2. 2.Department of PharmacologyUniversity of VeronaItaly

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