Acta Informatica

, Volume 53, Issue 3, pp 207–246 | Cite as

A complete classification of the expressiveness of interval logics of Allen’s relations: the general and the dense cases

  • Luca Aceto
  • Dario Della Monica
  • Valentin Goranko
  • Anna Ingólfsdóttir
  • Angelo Montanari
  • Guido Sciavicco
Original Article

Abstract

Interval temporal logics take time intervals, instead of time points, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham’s modal logic of time intervals HS, which associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen’s interval relations). In this paper, we compare and classify the expressiveness of all fragments of HS on the class of all linear orders and on the subclass of all dense linear orders. For each of these classes, we identify a complete set of definabilities between HS modalities, valid in that class, thus obtaining a complete classification of the family of all 4096 fragments of HS with respect to their expressiveness. We show that on the class of all linear orders there are exactly 1347 expressively different fragments of HS, while on the class of dense linear orders there are exactly 966 such expressively different fragments.

References

  1. 1.
    Aceto, L., Della Monica, D., Ingólfsdóttir, A., Montanari, A., Sciavicco, G.: An algorithm for enumerating maximal models of Horn theories with an application to modal logics. In: Proceedings of the 19th LPAR, LNCS, vol. 8312, pp. 1–17. Springer (2013)Google Scholar
  2. 2.
    Aceto, L., Della Monica, D., Ingólfsdóttir, A., Montanari, A., Sciavicco, G.: A complete classification of the expressiveness of interval logics of Allen’s relations over dense linear orders. In: Proceedings of the 20th TIME, pp. 65–72. IEEE Computer Society (2013)Google Scholar
  3. 3.
    Aceto, L., Della Monica, D., Ingólfsdóttir, A., Montanari, A., Sciavicco, G.: On the expressiveness of the interval logic of Allen’s relations over finite and discrete linear orders. In: Fermé, E., Leite, J. (eds.) Proceedings of the 14th JELIA, LNAI, vol. 8761, pp. 267–281. Springer (2014)Google Scholar
  4. 4.
    Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)CrossRefMATHGoogle Scholar
  5. 5.
    Allen, J.F.: Towards a general theory of action and time. Artif. Intell. 23(2), 123–154 (1984)CrossRefMATHGoogle Scholar
  6. 6.
    Balbiani, P., Goranko, V., Sciavicco, G.: Two-sorted point-interval temporal logics. Electron. Notes Theor. Comput. Sci. 278, 31–45 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  8. 8.
    Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Decidable and undecidable fragments of Halpern and Shoham’s interval temporal logic: towards a complete classification. In: Proceedings of the 15th LPAR, LNCS, vol. 5330, pp. 590–604. Springer (2008)Google Scholar
  9. 9.
    Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Metric propositional neighborhood interval logics on natural numbers. Softw. Syst. Model. 12(2), 245–264 (2013)CrossRefGoogle Scholar
  10. 10.
    Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: The dark side of interval temporal logic: marking the undecidability border. Ann. Math. Artif. Intell. 71(1–3), 41–83 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bresolin, D., Della Monica, D., Montanari, A., Sala, P., Sciavicco, G.: Interval temporal logics over finite linear orders: the complete picture. In: Proceedings of the 20th ECAI, pp. 199–204 (2012)Google Scholar
  12. 12.
    Bresolin, D., Della Monica, D., Montanari, A., Sala, P., Sciavicco, G.: Interval temporal logics over strongly discrete linear orders: the complete picture. In: Proceedings of the 3rd GandALF, vol. 96, pp. 155–168. EPTCS (2012)Google Scholar
  13. 13.
    Bresolin, D., Goranko, V., Montanari, A., Sala, P.: Tableau-based decision procedures for the logics of subinterval structures over dense orderings. J. Log. Comput. 20(1), 133–166 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bresolin, D., Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood logics: expressiveness, decidability, and undecidable extensions. Ann. Pure Appl. Log. 161(3), 289–304 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bresolin, D., Montanari, A., Sala, P., Sciavicco, G.: What’s decidable about Halpern and Shoham’s interval logic? The maximal fragment abbl. In: Proceedings of the 26th LICS, pp. 387–396. IEEE Computer Society (2011)Google Scholar
  16. 16.
    Bresolin, D., Sala, P., Sciavicco, G.: On begins, meets, and before. Int. J. Found. Comput. Sci. 23(3), 559–583 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Conradie, W., Sciavicco, G.: On the expressive power of first order-logic extended with Allen’s relations in the strict case. In: Proceedings of the 14th CAEPIA, LNCS, vol. 7023, pp. 173–182. Springer (2011)Google Scholar
  18. 18.
    Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Expressiveness of the interval logics of Allen’s relations on the class of all linear orders: complete classification. In: Proceedings of the 22nd IJCAI, pp. 845–850 (2011)Google Scholar
  19. 19.
    Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Interval temporal logics: a journey. Bull. Eur. Assoc. Theor. Comput. Sci. 105, 73–99 (2011)MathSciNetMATHGoogle Scholar
  20. 20.
    Emerson, E.A.: Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B: Formal Models and Semantics, pp. 995–1072. MIT Press, Cambridge (1990)Google Scholar
  21. 21.
    Gennari, R., Tonelli, S., Vittorini, P.: An AI-based process for generating games from flat stories. In: Proceedings of the 33rd SGAI, pp. 337–350 (2013)Google Scholar
  22. 22.
    Halpern, J., Shoham, Y.: A propositional modal logic of time intervals. J. ACM 38(4), 935–962 (1991)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM 32(1), 137–161 (1985)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Laban, S., El-Desouky, A.: RISMA: A rule-based interval state machine algorithm for alerts generation, performance analysis and monitoring real-time data processing. In: Proceedings of the EGU General Assembly 2013, Geophysical Research Abstracts, vol. 15 (2013)Google Scholar
  25. 25.
    Marcinkowski, J., Michaliszyn, J.: The undecidability of the logic of subintervals. Fundam. Inform. 131(2), 217–240 (2014)MathSciNetMATHGoogle Scholar
  26. 26.
    Montanari, A., Puppis, G., Sala, P.: Maximal decidable fragments of Halpern and Shoham’s modal logic of intervals. In: Proceedings of the 37th ICALP, LNCS, vol. 6199, pp. 345–356. Springer (2010)Google Scholar
  27. 27.
    Montanari, A., Puppis, G., Sala, P., Sciavicco, G.: Decidability of the interval temporal logic \({\sf AB} \overline{{\sf B}}\) over the natural numbers. In: Proceedings of the 31st STACS, pp. 597–608 (2010)Google Scholar
  28. 28.
    Moszkowski, B.: Reasoning About Digital Circuits. Technical report. stan-cs-83-970, Dept. of Computer Science, Stanford University, Stanford, CA (1983)Google Scholar
  29. 29.
    Pratt-Hartmann, I.: Temporal prepositions and their logic. Artif. Intell. 166(1–2), 1–36 (2005)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rosenstein, J.: Linear Orderings. Academic Press, Waltham (1982)MATHGoogle Scholar
  31. 31.
    Stirling, C.: Modal and temporal properties of processes. Springer, Berlin (2001)CrossRefMATHGoogle Scholar
  32. 32.
    Terenziani, P., Snodgrass, R.T.: Reconciling point-based and interval-based semantics in temporal relational databases: a treatment of the telic/atelic distinction. IEEE Trans. Knowl. Data Eng. 16(5), 540–551 (2004)CrossRefGoogle Scholar
  33. 33.
    Venema, Y.: Expressiveness and completeness of an interval tense logic. Notre Dame J. Form. Log. 31(4), 529–547 (1990)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Zhou, C., Hansen, M.R.: Duration Calculus: A Formal Approach to Real-Time Systems. EATCS Monographs in Theoretical Computer Science. Springer, Berlin (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Luca Aceto
    • 1
  • Dario Della Monica
    • 1
  • Valentin Goranko
    • 2
    • 3
  • Anna Ingólfsdóttir
    • 1
  • Angelo Montanari
    • 4
  • Guido Sciavicco
    • 5
  1. 1.ICE-TCS, School of Computer ScienceReykjavik UniversityReykjavíkIceland
  2. 2.Department of PhilosophyStockholm UniversityStockholmSweden
  3. 3.Department of MathematicsUniversity of JohannesburgJohannesburgSouth Africa
  4. 4.Department of Mathematics and Computer ScienceUniversity of UdineUdineItaly
  5. 5.Department of Information Engineering and CommunicationsUniversity of MurciaMurciaSpain

Personalised recommendations