Proving Symmetries by Model Transformation

  • Christopher Mears
  • Todd Niven
  • Marcel Jackson
  • Mark Wallace
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6876)

Abstract

The presence of symmetries in a constraint satisfaction problem gives an opportunity for more efficient search. Within the class of matrix models, we show that the problem of deciding whether some well known permutations are model symmetries (solution symmetries on every instance) is undecidable. We then provide a new approach to proving the model symmetries by way of model transformations. Given a model M and a candidate symmetry σ, the approach first syntactically applies σ to M and then shows that the resulting model σ(M) is semantically equivalent to M. We demonstrate this approach with an implementation that reduces equivalence to a sentence in Presburger arithmetic, using the modelling language MiniZinc and the term re-writing language Cadmium, and show that it is capable of proving common symmetries in models.

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References

  1. 1.
    Cohen, D., Jeavons, P., Jefferson, C., Petrie, K., Smith, B.: Symmetry definitions for constraint satisfaction problems. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 17–31. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Enderton, H.: A Mathematical Introduction to Logic, 2nd edn. Academic Press, Inc., London (2001)MATHGoogle Scholar
  3. 3.
    Flener, P., Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Matrix modelling. In: Proc. Formul 2001, CP 2001 Workshop on Modelling and Problem Formulation (2001)Google Scholar
  4. 4.
    Flener, P., Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I., Pearson, J., Walsh, T.: Breaking row and column symmetries in matrix models. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 187–192. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Harel, D.: Effective transformations on infinite trees with applications to high undecidability, dominoes and fairness. J. ACM, 224–248 (1986)Google Scholar
  6. 6.
    Mancini, T., Cadoli, M.: Detecting and breaking symmetries by reasoning on problem specifications. In: Zucker, J.-D., Saitta, L. (eds.) SARA 2005. LNCS (LNAI), vol. 3607, pp. 165–181. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Mears, C.: Automatic Symmetry Detection and Dynamic Symmetry Breaking for Constraint Programming. Ph.D. thesis, Monash University (2009)Google Scholar
  8. 8.
    Mears, C., Garcia de la Banda, M., Wallace, M.: On implementing symmetry detection. Constraints 14 (2009)Google Scholar
  9. 9.
    Mears, C., Garcia de la Banda, M., Wallace, M., Demoen, B.: A novel approach for detecting symmetries in CSP models. In: Fifth International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (2008)Google Scholar
  10. 10.
    Puget, J.-F.: Automatic detection of variable and value symmetries. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 475–489. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Pugh, W.: The omega test: a fast and practical integer programming algorithm for dependence analysis. Communications of the ACM, 102–114 (1992)Google Scholar
  12. 12.
    Robinson, R.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Math. 12, 177–209 (1971)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Roy, P., Pachet, F.: Using symmetry of global constraints to speed up the resolution of constraint satisfaction problems. In: ECAI 1998 Workshop on Non-binary Constraints (1998)Google Scholar
  14. 14.
    Van Hentenryck, P., Flener, P., Pearson, J., Agren, M.: Compositional derivation of symmetries for constraint satisfaction. In: Zucker, J.-D., Saitta, L. (eds.) SARA 2005. LNCS (LNAI), vol. 3607, pp. 234–247. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Christopher Mears
    • 1
  • Todd Niven
    • 1
  • Marcel Jackson
    • 2
  • Mark Wallace
    • 1
  1. 1.Faculty of ITMonash UniversityAustralia
  2. 2.Department of MathematicsLa Trobe UniversityAustralia

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