Proving Symmetries by Model Transformation
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.
Unable to display preview. Download preview PDF.
- 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
- 5.Harel, D.: Effective transformations on infinite trees with applications to high undecidability, dominoes and fairness. J. ACM, 224–248 (1986)Google Scholar
- 7.Mears, C.: Automatic Symmetry Detection and Dynamic Symmetry Breaking for Constraint Programming. Ph.D. thesis, Monash University (2009)Google Scholar
- 8.Mears, C., Garcia de la Banda, M., Wallace, M.: On implementing symmetry detection. Constraints 14 (2009)Google Scholar
- 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
- 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
- 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