Towards an Efficient SAT Encoding for Temporal Reasoning

  • Duc Nghia Pham
  • John Thornton
  • Abdul Sattar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4204)


In this paper, we investigate how an IA network can be effectively encoded into the SAT domain. We propose two basic approaches to modelling an IA network as a CSP: one represents the relations between intervals as variables and the other represents the relations between end-points of intervals as variables. By combining these two approaches with three different SAT encoding schemes, we produced six encoding schemes for converting IA to SAT. These encodings were empirically studied using randomly generated IA problems of sizes ranging from 20 to 100 nodes. A general conclusion we draw from these experimental results is that encoding IA into SAT produces better results than existing approaches. Further, we observe that the phase transition region maps directly from the IA encoding to each SAT encoding, but, surprisingly, the location of the hard region varies according to the encoding scheme. Our results also show a fixed performance ranking order over the various encoding schemes.


Encode Scheme Temporal Reasoning Direct Encode Atomic Relation Point Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Allen, J.F.: Maintaining knowledge about temporal intervals. Communications of the ACM 26, 832–843 (1983)MATHCrossRefGoogle Scholar
  2. 2.
    Cheeseman, P., Kanefsky, B., Taylor, W.M.: Where the really hard problems are. In: IJCAI 1991, pp. 331–337 (1991)Google Scholar
  3. 3.
    Freuder, E.C.: Synthesizing constraint expressions. Communication of ACM 21(11), 958–966 (1978)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Frisch, A.M., Peugniez, T.J.: Solving non-Boolean satisfiability problems with stochastic local search. In: IJCAI 2001, pp. 282–290 (2001)Google Scholar
  5. 5.
    Gent, I.P.: Arc consistency in SAT. In: ECAI 2002, pp. 121–125 (2002)Google Scholar
  6. 6.
    Ghiathi, K., Ghassem-Sani, G.: Using satisfiability in temporal planning. WSEAS Transactions on Computers 3(4), 963–969 (2004)Google Scholar
  7. 7.
    Golumbic, M.C., Shamir, R.: Complexity and algorithms for reasoning about time: A graph-theoretic approach. Journal of ACM, 1108–1133 (1993)Google Scholar
  8. 8.
    Hoos, H.H.: SAT-encodings, search space structure, and local search performance. In: IJCAI 1999, pp. 296–302 (1999)Google Scholar
  9. 9.
    Kautz, H., McAllester, D., Selman, B.: Encoding plans in propositional logic. In: KR 1996, pp. 374–384 (1996)Google Scholar
  10. 10.
    Ladkin, P., Reinefeld, A.: Effective solution of qualitative interval constraint problems. Artificial Intelligence 57(1), 105–124 (1992)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Mackworth, A.K.: Consistency in networks of relations. Artificial Intelligence 8(1), 99–118 (1977)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: DAC 2001, pp. 530–535 (2001)Google Scholar
  13. 13.
    Nebel, B.: Solving hard qualitative temporal reasoning problems: Evaluating the efficiency of using the ORD-Horn class. Constraints 1(3), 175–190 (1997)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nebel, B., Bürckert, H.-J.: Reasoning about temporal relations: A maximal tractable subclass of Allen’s Interval Algebra. Journal of ACM 42(1), 43–66 (1995)MATHCrossRefGoogle Scholar
  15. 15.
    Prestwich, S.: Local search on SAT-encoded colouring problems. In: SAT 2003, pp. 105–119 (2003)Google Scholar
  16. 16.
    Smith, B.M., Dyer, M.E.: Locating the phase transition in binary constraint satisfaction problems. Artificial Intelligence 81(1-2), 155–181 (1996)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Thornton, J., Beaumont, M., Sattar, A., Maher, M.: A local search approach to modelling and solving Interval Algebra problems. Journal of Logic and Computation 14(1), 93–112 (2004)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    van Beek, P.: Reasoning about qualitative temporal information. Artificial Intelligence 58, 297–326 (1992)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    van Beek, P., Cohen, R.: Exact and approximate reasoning about temporal relations. Computational Intelligence 6, 132–144 (1990)CrossRefGoogle Scholar
  20. 20.
    van Beek, P., Manchak, D.W.: The design and experimental analysis of algorithms for temporal reasoning. Journal of Artificial Intelligence Research 4, 1–18 (1996)MATHGoogle Scholar
  21. 21.
    Vilain, M., Kautz, H.: Constraint propagation algorithms for temporal reasoning. In: AAAI 1986, pp. 377–382 (1986)Google Scholar
  22. 22.
    Walsh, T.: SAT v CSP. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 441–456. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Duc Nghia Pham
    • 1
    • 2
  • John Thornton
    • 1
    • 2
  • Abdul Sattar
    • 1
    • 2
  1. 1.Safeguarding Australia ProgramNational ICT Australia Ltd.Australia
  2. 2.Institute for Integrated and Intelligent SystemsGriffith UniversityAustralia

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