A Powerful Technique to Eliminate Isomorphism in Finite Model Search

  • Xiangxue Jia
  • Jian Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4130)


We propose a general-purpose technique, called DASH (Decision Assignment Scheme Heuristic), to eliminate isomorphic subspaces when generating finite models. Like LNH, DASH is based on inherent isomorphism in first order clauses on finite domains. Unlike other methods, DASH can completely eliminate isomorphism during the search. Therefore, DASH can generate all the models none of which are isomorphic. And DASH is an efficient technique for finite model enumeration. The main idea is to cut the branch of the search tree which is isomorphic to a branch that has been searched. We present a new method to describe the class of isomorphic branches. We implemented this technique by modifying SEM1.7B, and the new tool is called SEMD. This technique proves to be very efficient on typical problems like the generation of finite groups, rings and quasigroups. The experiments show that SEMD is much faster than SEM on many problems, especially when generating all the models and when there is no model. SEMD can generate all the non-isomorphic models with little extra cost, while other tools like MACE4 will spend more time.


Isomorphism scheme symmetry breaking LNH DASH 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Jackson, D., Jha, S., Damon, C.A.: Isomorph-free model enumeration: A new method for checking relational specifications. ACM Transactions on Programming Languages and Systems 20(2), 302–343 (1998)CrossRefGoogle Scholar
  2. 2.
    Boy de la Tour, T., Countcham, P.: An isomorph-free SEM-like enumeration of models. Electr. Notes Theor. Comput. Sci. 125(2), 91–113 (2005)CrossRefGoogle Scholar
  3. 3.
    Moskewicz, M., et al.: Chaff: Engineering an efficient SAT solver. In: Proc. 39th Design Automation Conference, pp. 530–535 (2001)Google Scholar
  4. 4.
    Audemard, G., Henocque, L.: The extended least number heuristic. In: Proc. of the 1st Int’l Joint Conference on Automated Reasoning, pp. 427–442 (2001)Google Scholar
  5. 5.
    Sutcliffe, G., Suttner, C.B.: The TPTP problem library – CNF release v1. 2.1. Journal of Automated Reasoning 21(2), 177–203 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Zhang, H.: An efficient propositional prover. In: McCune, W. (ed.) CADE 1997. LNCS, vol. 1249, pp. 272–275. Springer, Heidelberg (1997)Google Scholar
  7. 7.
    Gent, I., Smith, B.: Symmetry breaking in constraint programming. In: Proc. ECAI 2000, pp. 599–603 (2000)Google Scholar
  8. 8.
    Gent, I., Harvey, W., Kelsey, T., Linton, S.: Generic SBDD using computational group theory. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 333–347. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Crawford, J.: A theoretical analysis of reasoning by symmetry in first order logic. Technical report, AT&T Bell Laboratories (1996)Google Scholar
  10. 10.
    Crawford, J., Ginsberg, M., Luks, E., Roy, A.: Symmetry-breaking predicates for search problems. In: Proc. KR 1996, pp. 149–159 (1996)Google Scholar
  11. 11.
    Slaney, J.: Finite domain enumerator. system description. In: Bundy, A. (ed.) CADE 1994. LNCS, vol. 814, Springer, Heidelberg (1994)Google Scholar
  12. 12.
    Zhang, J.: Constructing finite algebras with FALCON. Journal of Automated Reasoning 17(1), 1–22 (1996)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Zhang, J.: Computer search for counterexamples to Wilkie’s identity. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 441–451. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Zhang, J., Zhang, H.S.: a system for enumerating models. In: Proc. 14th Int’l Joint Conf. on Artificial Intelligence (IJCAI), pp. 298–303 (1995)Google Scholar
  15. 15.
    Claessen, K., Sörensson, N.: New techniques that improve mace-style finite model finding. In: Proceedings of the CADE-19 Workshop: Model Computation - Principles, Algorithms, Applications (Miami, USA) (2003)Google Scholar
  16. 16.
    Fujita, M., Slaney, J., Bennett, F.: Automatic generation of some results in finite algebra. In: Proc. 13th Int’l Joint Conf. on Artificial Intelligence (IJCAI), pp. 52–57 (1993)Google Scholar
  17. 17.
    Eén, N., Sörensson, N.: The MiniSat page. Webpage, Chalmers University (2005),
  18. 18.
    Peltier, N.: A new method for automated finite model building exploiting failures and symmetries. J. of Logic and Computation 8(4), 511–543 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Burris, S., Lee, S.: Small models of the high school identities. Intl. J. of Algebra and Computatio 2, 139–178 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fahle, T., Schamberger, S., Sellmann, M.: Symmetry breaking. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 93–107. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  21. 21.
    McCune, W.: MACE 2.0 reference manual and guide. Technical Report No. 249, Argonne National Laboratory, Argonne, IL, USA (2001)Google Scholar
  22. 22.
    McCune, W.: Mace4 reference manual and guide. Technical Report No. 264, Argonne National Laboratory, Argonne, IL, USA (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Xiangxue Jia
    • 1
    • 2
  • Jian Zhang
    • 1
  1. 1.Laboratory of Computer Science, Institute of SoftwareChinese Academy of Sciences 
  2. 2.Chinese Academy of SciencesGraduate University 

Personalised recommendations