Trends in Temporal Reasoning: Constraints, Graphs and Posets

  • Jacqueline W. DaykinEmail author
  • Mirka Miller
  • Joe Ryan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)


Temporal reasoning finds many applications in numerous fields of artificial intelligence – frameworks for representing and analyzing temporal information are therefore important. Allen’s interval algebra is a calculus for temporal reasoning that was introduced in 1983. Reasoning with qualitative time in Allen’s full interval algebra is NP-complete. Research since 1995 identified maximal tractable subclasses of this algebra via exhaustive computer search and also other ad-hoc methods. In 2003, the full classification of complexity for satisfiability problems over constraints in Allen’s interval algebra was established algebraically. We review temporal reasoning concepts including a method for deciding tractability of temporal constraint satisfaction problems based on the theory of algebraic closure operators for constraints. Graph-based temporal representations such as interval and sequence graphs are discussed. We also propose novel research for scheduling algorithms based on the Fishburn-Shepp inequality for posets.


Algebraic closure Allen’s interval algebra Artificial intelligence Constraint satisfaction problem Fishburn-Shepp inequality Graph Poset Qualitative temporal reasoning Tractable satisfiability 


  1. [A-83]
    Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)CrossRefzbMATHGoogle Scholar
  2. [A-91]
    Allen, J.F.: Temporal reasoning and planning. In: Allen, J.F., Kautz, H.A., Pelavin, R.N., Tenenberg, J.D. (eds.) Reasoning About Plans, Chapter 1, pp. 1–67. Morgan Kaufmann, San Mateo (1991)Google Scholar
  3. [CS-73]
    Coombs, C.H., Smith, J.E.K.: On the detection of structures in attitudes and developmental processses. Psych. Rev. 80, 337–351 (1973)CrossRefGoogle Scholar
  4. [DMP-91]
    Dechter, R., Meiri, I., Pearl, J.: Temporal constraint networks. Artif. Intell. 49(1–3), 61–95 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [D-92]
    Dorn, J.: Temporal reasoning in sequence graphs. In: AAAI 1992, pp. 735–740 (1992)Google Scholar
  6. [DJ-97]
    Drakengren, T., Jonsson, P.: Eight maximal tractable subclasses of Allen’s algebra with metric time. J. Artif. Intell. Res. 7, 25–45 (1997)MathSciNetzbMATHGoogle Scholar
  7. [DJ-97ii]
    Drakengren, T., Jonsson, P.: Twenty-one large tractable subclasses of Allen’s algebra. Artif. Intell. 93(1–2), 297–319 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [DJ-98]
    Drakengren, T., Jonsson, P.: A complete classification of tractability in Allen’s algebra relative to subsets of basic relations. Artif. Intell. 106(2), 205–219 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [F-84]
    Fishburn, P.C.: A correlational inequality for linear extensions of a poset. Order 1(2), 127–137 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [GSS-93]
    Gerevini, A., Schubert, L.K., Schaeffer, S.: Temporal reasoning in Timegraph I-II. SIGART Bull. 4(3), 21–25 (1993)CrossRefGoogle Scholar
  11. [GKS-94]
    Golumbic, M.C., Kaplan, H., Shamir, R.: On the complexity of DNA physical mapping. Adv. Appl. Math. 15, 251–261 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [GS-92]
    Golumbic, M.C., Shamir, R.: Algorithms and complexity for reasoning about time. In: AAAI 1992, pp. 741–747 (1992)Google Scholar
  13. [GS-93]
    Golumbic, M.C., Shamir, R.: Complexity and algorithms for reasoning about time: a graph theoretic approach. J. ACM 40(5), 1108–1133 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [JCG-97]
    Jeavons, P., Cohen, D.A., Gyssens, M.: Closure properties of constraints. J. ACM 44(4), 527–548 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [JB-96]
    Jonsson, P., Bäckström, C.: A linear-programming approach to temporal reasoning. In: AAAI 1996, pp. 1235–1240 (1996)Google Scholar
  16. [JCMPM-10]
    Juarez, J.M., Campos, M., Morales, A., Palma, J., Marin, R.: Applications of temporal reasoning to intensive care units. J. Healthc. Eng. 1(4), 615–636 (2010)CrossRefGoogle Scholar
  17. [K-93]
    Karp, R.M.: Mapping the genome: some combinatorial problems arising in molecular biology. In: STOC 1993, pp. 278–285 (1993)Google Scholar
  18. [KL-91]
    Kautz, H.A., Ladkin, P.B.: Integrating metric and qualitative temporal reasoning. In: AAAI 1991, pp. 241–246 (1991)Google Scholar
  19. [KKD-08]
    Krieger, H.-U., Kiefer, B., Declerck, T.: A framework for temporal representation and reasoning in business intelligence applications. In: AAAI Spring Symposium: AI Meets Business Rules and Process Management, pp. 59–70 (2008)Google Scholar
  20. [KJJ-03]
    Krokhin, A., Jeavons, P., Jonsson, P.: Reasoning about temporal relations: the tractable subalgebras of Allen’s interval algebra. J. ACM 50(5), 591–640 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [LM-88]
    Ladkin, P.B., Maddux, R.D.: On binary constraint networks, Technical report KES. U.88.8, Kestrel Institute, Palo Alto, CA (1988)Google Scholar
  22. [L-97]
    Ligozat, G.: Figures for thought: temporal reasoning with pictures, AAAI Technical report WS-97-11, pp. 31–36 (1997)Google Scholar
  23. [L-98]
    Ligozat, G.: “Corner" relations in Allen’s algebra. Constraints 3(2–3), 165–177 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [M-77]
    Mackworth, A.K.: Consistency in networks of relations. Artif. Intell. 8(1), 99–118 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [MZ-08]
    Mandoiu, I., Zelikovsky, A. (eds.): Bioinformatics Algorithms: Techniques and Applications. Wiley Series in Bioinformatics. Wiley Interscience, Hoboken (2008)zbMATHGoogle Scholar
  26. [M-91]
    Meiri, I.: Combining qualitative and quantitative constraints in temporal reasoning. In: AAAI 1991, pp. 260–267 (1991)Google Scholar
  27. [NB-95]
    Nebel, B., Bürckert, H.-J.: Reasoning about temporal relations: a maximal tractable subclass of Allen’s interval algebra. J. ACM 42(1), 43–66 (1995)CrossRefzbMATHGoogle Scholar
  28. [N-91]
    Nökel, K. (ed.): Temporally Distributed Symptoms in Technical Diagnosis. LNCS (LNAI), vol. 517. Springer, Heidelberg (1991)zbMATHGoogle Scholar
  29. [PS-97]
    Pe’er, I., Shamir, R.: Satisfiability problems on intervals and unit intervals. Theoret. Comput. Sci. 175(2), 349–372 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [PR-94]
    Pruesse, G., Ruskey, F.: Generating linear extensions fast. SIAM J. Comput. 23(2), 373–386 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [S-82]
    Shepp, L.A.: The XYZ conjecture and the FKG inequality. Ann. Probab. 10(3), 824–827 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [SC-88]
    Song, F., Cohen, R.: The interpretation of temporal relations in narrative. In: AAAI 1988, pp. 745–750 (1988)Google Scholar
  33. [T-76]
    Tarjan, R.E.: Edge-disjoint spanning trees and depth-first search. Acta Inform. 6(2), 171–185 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [VB-89]
    van Beek, P.: Approximation algorithms for temporal reasoning. In: IJCAI 1989, pp. 1291–1296 (1989)Google Scholar
  35. [VB-90]
    van Beek, P.: Reasoning about qualitative temporal information. In: AAAI 1990, pp. 728–734 (1990)Google Scholar
  36. [VB-92]
    van Beek, P.: Reasoning about qualitative temporal information. Artif. Intell. 58(1–3), 297–326 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  37. [VBC-90]
    van Beek, P., Cohen, R.: Exact and approximate reasoning about temporal relations. Comput. Intell. 6(3), 132–144 (1990)CrossRefGoogle Scholar
  38. [V-82]
    Vilain, M.B.: A system for reasoning about time. In: AAAI 1982, pp. 197–201 (1982)Google Scholar
  39. [VK-86]
    Vilain, M., Kautz, H.: Constraint propagation algorithms for temporal reasoning. In: AAAI 1986, pp. 377–382 (1986)Google Scholar
  40. [VKVB-89]
    Vilain, M., Kautz, H., van Beek, P.: Constraint propagation algorithms for temporal reasoning: a revised report. In: Weld, D.S., de Kleer, J. (eds.) Readings in Qualitative Reasoning about Physical Systems, pp. 373–381. Morgan Kaufmann, California (1989)Google Scholar
  41. [WH-90]
    Ward, S.A., Halstead, R.H.: Computation Structures. MIT Press, Cambridge, Mass (1990)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jacqueline W. Daykin
    • 1
    • 2
    Email author
  • Mirka Miller
    • 3
    • 4
  • Joe Ryan
    • 3
  1. 1.Department of Computer ScienceRoyal Holloway, University of LondonEghamUK
  2. 2.Department of InformaticsKing’s College LondonLondonUK
  3. 3.School of Electrical Engineering and Computer ScienceUniversity of NewcastleNew South WalesAustralia
  4. 4.Department of MathematicsUniversity of West BohemiaPilsenCzech Republic

Personalised recommendations