A Dual Graph Translation of a Problem in ‘Life’

  • Barbara M. Smith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2470)


Conway’s game of Life provides interesting problems in which modelling issues in constraint programming can be explored. The problem of finding a maximum density stable pattern (‘still-life’) is discussed. A formulation of this problem as a constraint satisfaction problem with 0-1 variables and non-binary constraints is compared with its dual graph translation into a binary CSP. The success of the dual translation is surprising, from previously-reported experience, since it has as many variables as the non-binary CSP and very large domains. An important factor is the identification of many redundant constraints: it is shown that these can safely be removed from a dual graph translation if arc consistency is maintained during search.


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  1. [1]
    F. Bacchus and P. van Beek. On the Conversion Between Non-Binary and Binary Constraint Satisfaction Problems. In Proceedings AAAI’98, pages 311–318, 1998.Google Scholar
  2. [2]
    C. Bessière and J.-C. Régin. Arc consistency for general constraint networks: preliminary results. In Proceedings IJCAI’97, volume 1, pages 398–404, 1997.Google Scholar
  3. [3]
    R. Bosch and M. Trick. Constraint programming and hybrid formulations for three life designs. In N. Jussien and F. Laburthe, editors, Proceedings of the Fourth International Workshop on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimisation Problems (CP-AI-OR’02), pages 77–91, 2002.Google Scholar
  4. [4]
    R. A. Bosch. Maximum density stable patterns in variants of Conway’s game of Life. Operations Research Letters, 27:7–11, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    R. Dechter and J. Pearl. Tree clustering for constraint networks. Artificial Intelligence, 38:353–366, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    M. Gardner. The fantastic combinations of John Conway’s new solitaire game. Scientific American, 223:120–123, 1970.CrossRefGoogle Scholar
  7. [7]
    I. P. Gent and B. M. Smith. Symmetry Breaking During Search in Constraint Programming. In W. Horn, editor, Proceedings ECAI’2000, pages 599–603, 2000.Google Scholar
  8. [8]
    K. Stergiou and T. Walsh. Encodings of Non-Binary Constraint Satisfaction Problems. In Proceedings AAAI’99, pages 163–168, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Barbara M. Smith
    • 1
  1. 1.University of HuddersfieldHuddersfieldUK

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