Decremental Consistency Checking of Temporal Constraints: Algorithms for the Point Algebra and the ORD-Horn Class

  • Massimo Bono
  • Alfonso Emilio GereviniEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11008)


Deciding consistency of a set of temporal constraints (CSP) over either the Point Algebra (PA) or the Interval Algebra (IA) is a fundamental problem in qualitative temporal reasoning. Given an inconsistent temporal CSP and a sequence of constraint relaxations to perform, incrementally decided by the user or an application, decremental consistency checking is the problem of determining if the revised CSP becomes consistent after each relaxation. We propose new algorithms for decremental consistency checking of a CSP over either PA or the ORD-Horn subalgebra of IA. These techniques exploit a graph representation of the CSP and some data structures that are maintained at each relaxation. An experimental analysis shows that solving decremental consistency checking using our algorithms can be significantly faster than using existing static algorithms.


  1. 1.
    Allen, J.: Maintaining knowledge about temporal intervals. Commun. ACM 26(1), 832–843 (1983)CrossRefGoogle Scholar
  2. 2.
    Bakker, R.R., Dikker, F., Tempelman, F., Wognum, P.M.: Diagnosing and solving over-determined constraint satisfaction problems. IJCAI 93, 276–281 (1993)Google Scholar
  3. 3.
    van Beek, P.: Reasoning about qualitative temporal information. In: Proceedings of the Eighth National Conference of the American Association for Artificial Intelligence (AAAI-1990), Boston, MA, pp. 728–734 (1990)Google Scholar
  4. 4.
    van Beek, P.: Reasoning about qualitative temporal information. Artif. Intell. 58(1–3), 297–321 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cohn, A.G., Renz, J.: Qualitative spatial representation and reasoning. In: Handbook of Knowledge Representation, pp. 551–596 (2008)CrossRefGoogle Scholar
  6. 6.
    Condotta, J., Kaci, S., Schwind, N.: A framework for merging qualitative constraints networks. In: Proceedings of the Twenty-First International Florida Artificial Intelligence, pp. 586–591 (2008)Google Scholar
  7. 7.
    Cormen, T., Leiserson, C., Rivest, R.: Introduction to Algorithms. The MIT Press, Cambridge (1990)zbMATHGoogle Scholar
  8. 8.
    Delgrande, J., Gupta, A.: Updating \(<=,<\)-chains. Inf. Process. Lett. 83(5), 261–268 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Delgrande, J., Gupta, A., Allen, T.: A comparison of point-based approaches to qualitative temporal reasoning. Artif. Intell. 131(1–2), 135–170 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Drakengren, T., Jonsson, P.: Twenty-one large tractable subclasses of Allen’s algebra. Artif. Intell. 93(1–2), 297–319 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Eppstein, D., Galil, Z., Italiano, G.: Dynamic graph algorithms. In: Atallah, M. (ed.) Algorithms and Theory of Computation Handbook. CRC Press, Boca Raton (1999)Google Scholar
  12. 12.
    Freuder, E.C., Wallace, R.J.: Partial constraint satisfaction. Artif. Intell. 58(1–3), 21–70 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Frigioni, D., Marchetti-Spaccamela, A., Nanni, U.: Dynamic algorithms for classes of constraint satisfaction problems. Theor. Comput. Sci. 259(1–2), 287–305 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Georgiadis, L., Hansen, T.D., Italiano, G.F., Krinninger, S., Parotsidis, N.: Decremental data structures for connectivity and dominators in directed graphs. In: 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, pp. 42:1–42:15 (2017)Google Scholar
  15. 15.
    Gerevini, A., Renz, J.: Combining topological and size information for spatial reasoning. Artif. Intell. 137, 1–42 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gerevini, A., Schubert, L.: Efficient algorithms for qualitative reasoning about time. Artif. Intell. 74, 207–248 (1995)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gerevini, A.E.: Incremental qualitative temporal reasoning: algorithms for the point algebra and the ORD-Horn class. Artif. Intell. 166(1–2), 37–80 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gerevini, A.E., Saetti, A.: Computing the minimal relations in point-based qualitative temporal reasoning through metagraph closure. Artif. Intell. 175(2), 556–585 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Golumbic, C., Shamir, R.: Complexity and algorithms for reasoning about time: a graph-theoretic approach. J. Assoc. Comput. Mach. (ACM) 40(5), 1108–1133 (1993)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, Boston (1972). PlenumCrossRefGoogle Scholar
  21. 21.
    Krokhin, A., Jeavons, P., Jonsson, P.: The tractable subalgebras of Allen’s interval algebra. J. Assoc. Comput. Mach. (ACM) 50(5), 591–640 (2003)CrossRefGoogle Scholar
  22. 22.
    Ladkin, P., Maddux, R.: On binary constraint problems. J. Assoc. Comput. Mach. (ACM) 41(3), 435–469 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Łącki, J.: Improved deterministic algorithms for decremental reachability and strongly connected components. ACM Trans. Algorithms 9(3), 27:1–27:15 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Long, D., Fox, M.: The 3rd international planning competition: results and analysis. J. Artif. Intell. Res. 20, 1–59 (2003)CrossRefGoogle Scholar
  25. 25.
    Nau, D.S., Au, T.C., Ilghami, O., Kuter, U., Murdock, J.W., Wu, D., Yaman, F.: Shop2: an htn planning system. J. Artif. Intell. Res. 20, 379–404 (2003)CrossRefGoogle Scholar
  26. 26.
    Nebel, B., Bürckert, H.J.: Reasoning about temporal relations: a maximal tractable subclass of Allen’s interval algebra. J. Assoc. Comput. Mach. (ACM) 42(1), 43–66 (1995)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Penberthy, J.S.: Planning with continuous change (1993), Technical report UW-CSE-93-12-01Google Scholar
  28. 28.
    Sharir, M.: A strong-connectivity algorithm and its applications in data flow analysis. Comput. Math. Appl. 7(1), 67–72 (1981)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tarjan, R.: Depth first search and linear graph algorithms. SIAM J. Comput. 1(2), 215–225 (1972)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Vilain, M., Kautz, H.: Constraint propagation algorithms for temporal reasoning. In: Proceedings of the Fifth National Conference of the American Association for Artificial Intelligence (AAAI-1986), pp. 377–382. Morgan Kaufmann (1986)Google Scholar
  31. 31.
    Vilain, M., Kautz, H., van Beek, P.: Constraint propagation algorithms for temporal reasoning: a revised report. In: Readings in Qualitative Reasoning about Physical Systems, pp. 373–381. Morgan Kaufman, San Mateo (1990)CrossRefGoogle Scholar
  32. 32.
    Yang, Q.: Intelligent Planning. Springer, Heidelberg (1997). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria dell’InformazioneUniversità degli Studi di BresciaBresciaItaly

Personalised recommendations