The Horn Fragment of Branching Algebra

Authors Alessandro Bertagnon , Marco Gavanelli , Alessandro Passantino , Guido Sciavicco , Stefano Trevisani

Thumbnail PDF


  • Filesize: 0.54 MB
  • 16 pages

Document Identifiers

Author Details

Alessandro Bertagnon
  • Department of Engineering, University of Ferrara, Italy
Marco Gavanelli
  • Department of Engineering, University of Ferrara, Italy
Alessandro Passantino
  • Department of Mathematics and Computer Science, University of Ferrara, Italy
Guido Sciavicco
  • Department of Mathematics and Computer Science, University of Ferrara, Italy
Stefano Trevisani
  • Department of Mathematics and Computer Science, University of Ferrara, Italy

Cite AsGet BibTex

Alessandro Bertagnon, Marco Gavanelli, Alessandro Passantino, Guido Sciavicco, and Stefano Trevisani. The Horn Fragment of Branching Algebra. In 27th International Symposium on Temporal Representation and Reasoning (TIME 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 178, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Branching Algebra is the natural branching-time generalization of Allen’s Interval Algebra. As in the linear case, the consistency problem for Branching Algebra is NP-hard. Being relatively new, however, not much is known about the computational behaviour of the consistency problem of its sub-algebras, except in the case of the recently found subset of convex branching relations, for which the consistency of a network can be tested via path consistency and it is therefore deterministic polynomial. In this paper, following Nebel and Bürckert, we define the Horn fragment of Branching Algebra, and prove that it is a sub-algebra of the latter, being closed under inverse, intersection, and composition, that it strictly contains both the convex fragment of Branching Algebra and the Horn fragment of Interval Algebra, and that its consistency problem can be decided via path consistency. Finally, we experimentally prove that the Horn fragment of Branching Algebra can be used as an heuristic for checking the consistency of a generic network with a considerable improvement over the convex subset.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constraint and logic programming
  • Constraint programming
  • Consistency
  • Branching time
  • Horn Fragment


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. J.F. Allen. Maintaining knowledge about temporal intervals. Communications of the ACM, 26(11):832-843, 1983. Google Scholar
  2. J.F. Allen and P. J. Hayes. Short time periods. In Proc. of IJCAI 1987: 10th International Joint Conference on Artificial Intelligence, pages 981-983, 1987. Google Scholar
  3. J.F. Condotta, D. D'Almeida, C. Lecoutre, and L. Saïs. From qualitative to discrete constraint networks. In Proc. of KI 2006: Workshop on Qualitative Constraint Calculi, pages 54-64, 2006. Google Scholar
  4. S. Darabi, S.C.C. Blom, and M. Huisman. A verification technique for deterministic parallel programs. In Proc. of NFM 2017: 9th International Symposium on NASA Formal Methods, volume 10227 of Lecture Notes in Computer Science, pages 247-264. Springer, 2017. Google Scholar
  5. S. Durhan and G. Sciavicco. Allen-like theory of time for tree-like structures. Information and Computation, 259(3):375-389, 2018. Google Scholar
  6. M. Gavanelli, A. Passantino, and G. Sciavicco. Deciding the consistency of branching time interval networks. In Proc. of TIME 2018: 25th International Symposium on Temporal Representation and Reasoning, volume 120 of LIPIcs, pages 12:1-12:15, 2018. Google Scholar
  7. L. Henshen and L. Wos. Unit refutation and Horn sets. Journal of the ACM, 21:590-605, 1974. Google Scholar
  8. P. Jonsson and V. Lagerkvist. An initial study of time complexity in infinite-domain constraint satisfaction. Artificial Intelligence, 245:115-133, 2017. Google Scholar
  9. A. Krokhin, P. Jeavons, and P. Jonsson. Reasoning about temporal relations: The tractable subalgebras of Allen’s interval algebra. Journal of the ACM, 50(5):591-640, 2003. Google Scholar
  10. P.B. Ladkin and A. Reinefeld. Fast algebraic methods for interval constraint problems. Annals of Mathematics and Artificial Intelligence, 19(3-4):383-411, 1997. Google Scholar
  11. M. Mantle, S. Batsakis, and G. Antoniou. Large scale reasoning using Allen’s Interval Algebra. In Proc. of the 15th Mexican International Conference on Artificial Intelligence, volume 11062 of Lecture Notes in Computer Science, pages 29-41. Springer, 2017. Google Scholar
  12. L. Mudrová and N. Hawes. Task scheduling for mobile robots using interval algebra. In Proc. of ICRA 2015: International Conference on Robotics and Automation, pages 383-388. IEEE, 2015. Google Scholar
  13. B. Nebel. Solving hard qualitative temporal reasoning problems: Evaluating the efficiency of using the ORD-Horn class. Constraints, 1(3):175-190, 1997. Google Scholar
  14. B. Nebel and H.J. Bürckert. Reasoning about temporal relations: A maximal tractable subclass of allen’s interval algebra. Journal of the ACM, 42(1):43-66, 1995. Google Scholar
  15. M. Ragni and S. Wölfl. Branching Allen. In Proc. of ISCS 2004: 4th International Conference on Spatial Cognition, volume 3343 of Lecture Notes in Computer Science, pages 323-343. Springer, 2004. Google Scholar
  16. A.J. Reich. Intervals, points, and branching time. In Proc. of TIME 1994: 9th International Symposium on Temporal Representation and Reasoning, pages 121-133. IEEE, 1994. Google Scholar
  17. J. Renz and B. Nebel. Efficient methods for qualitative spatial reasoning. Journal of Artificial Intelligence Resoning, 15:289-318, 2001. Google Scholar
  18. J. Renz and B. Nebel. Qualitative spatial reasoning using constraint calculi. In Handbook of Spatial Logic, pages 161-215. Springer, 2007. Google Scholar
  19. E. Rishes, S.M. Lukin, D.K. Elson, and M.A. Walker. Generating different story tellings from semantic representations of narrative. In Proc. of ICIDS 2013: 6th International Conference on Interactive Storytelling, volume 8230 of Lecture Notes in Computer Science, pages 192-204. Springer, 2013. Google Scholar
  20. G. Rosu and S. Bensalem. Allen linear (interval) temporal logic - translation to LTL and monitor synthesis. In Proc. of CAV 2006: 18th International Conference on Computer Aided Verification, volume 4144 of Lecture Notes in Computer Science, pages 263-277. Springer, 2006. Google Scholar
  21. J. Schimpf and K. Shen. Ecl^ips^e - from LP to CLP. Theory and Practice of Logic Programming, 12(1-2):127-156, 2012. Google Scholar
  22. M. Theune, K. Meijs, D. Heylen, and R.Ordelman. Generating expressive speech for storytelling applications. IEEE Transactions on Audio, Speech & Language Processing, 14(4):1137-1144, 2006. Google Scholar
  23. P. van Beek and R. Cohen. Exact and approximate reasoning about temporal relations. Computational Intelligence, 6:132-144, 1990. Google Scholar
  24. A.K. Zaidi and L.W. Wagenhals. Planning temporal events using point-interval logic. Mathematical and Computer Modelling, 43(9):1229-1253, 2006. Google Scholar