Abstract
The Still-Life problem is challenging for CP techniques because the basic constraints of the game of Life are loose and give poor propagation for Still-Life. In this paper, we show how ad hoc global \({\tt case}\) constraints can be customized to construct various models to provide much stronger propagation with CP solvers. Since we use custom ad hoc constraints of high arity where the number of tuples to define the constraint are large, the actual constraint representation becomes important to avoid excessive space consumption. We demonstrate how to use BDDs to construct good representations for the \({\tt case}\) constraint which is critical for efficiency. Our results seem comparable to hybrid CP/IP models even though we are only using propagation albeit on ad hoc global constraints. This paper shows an extensive example of how to systematically build models using different kinds of ad hoc constraints. It also demonstrates the solving potential of ad hoc global constraints.
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References
Beldiceanu, N. (2000). Global constraints as graph properties on structured networks of elementary constraints of the same type. TR T2000-01, SICS.
Bessière, C., & Règin, J.-C. (1997). Arc consistency for general constraint networks: Preliminary results. In International Joint Conference on Artificial Intelligence (IJCAI), (pp. 398–404).
Bosch, R., & Trick, M. (2004). Constraint programming and hybrid formulations for three Life designs. Annals of Operations Research, 130, 41–56.
Bryant, R. E. (1986). Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers, 35(8), 667–691.
Carlsson, M., et al., (2005). SICStus Prolog User’s Manual. Swedish Institute of Computer Science, release 3.12.1 edition. ISBN 91-630-3648-7.
Cheng, C. K., Lee, J. H. M., & Stuckey, P. J. (2003). Box constraint collections for ad-hoc constraints. In Principles and Practice of Constraint Programming (CP), (pp. 214–228).
Clarke, E. M., Grumberg, O., & Peled, D. A. (1999). Model Checking. The MIT Press.
Dao, T. B. H., Lallouet, A., Legtchenko, A., & Martin, L. (2002). Indexical-based solver learning. In Principles and Practice of Constraint Programming (CP), (pp. 541–555). (September).
Gardner, M. (1970). The fantastic combinations of John Conway’s new solitaire game. Scientific American, 223, 120–123.
Harvey, W. D., & Ginsberg, M. L. (1995). Limited discrepancy search. In International Joint Conference on Artificial Intelligence (IJCAI).
Larrosa, J., Morancho, E., & Niso, D. (2005). On the practical applicability of bucket elimination: Still-life as a case study. Journal of Artificial Intelligence Research, 23, 21–440.
Petrie, K. E., Smith, B. M., & Yorke-Smith, N. (2004). Dynamic symmetry breaking in constraint programming and linear programming hybrids. In European Starting AI Researcher Symp.
Smith, B. M. (2002). A dual graph translation of a problem in ‘life’. In Principles and Practice of Constraint Programming (CP), (pp. 402–414).
Smith, B. M., & Sturdy, P. (2005). Value ordering for finding all solutions. In International Joint Conference on Artificial Intelligence (IJCAI), (pp. 311–316).
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Cheng, K.C.K., Yap, R.H.C. Applying Ad-hoc Global Constraints with the \({\tt case}\) Constraint to Still-Life. Constraints 11, 91–114 (2006). https://doi.org/10.1007/s10601-006-8058-9
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DOI: https://doi.org/10.1007/s10601-006-8058-9