A Synchronization Problem in Two-Dimensional Cellular Automata

  • Hiroshi Umeo
Conference paper
Part of the Proceedings in Information and Communications Technology book series (PICT, volume 1)


The firing squad synchronization problem on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed so far. In the present paper, we give a survey on recent developments in firing squad synchronization algorithms for large-scale two-dimensional cellular automata. Several state-efficient implementations of the two-dimensional synchronization algorithms are given.


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  1. 1.
    Balzer, R.: An 8-state minimal time solution to the firing squad synchronization problem. Information and Control 10, 22–42 (1967)CrossRefGoogle Scholar
  2. 2.
    Beyer, W.T.: Recognition of topological invariants by iterative arrays. Ph.D. Thesis, MIT, p. 144 (1969)Google Scholar
  3. 3.
    Hans-D., Gerken.: Über Synchronisations - Probleme bei Zellularautomaten. Diplomarbeit, Institut für Theoretische Informatik, Technische Universität Braunschweig, p. 50 (1987)Google Scholar
  4. 4.
    Goto, E.: A minimal time solution of the firing squad problem. Dittoed course notes for Applied Mathematics 298. Harvard University, pp. 52–59 (1962)Google Scholar
  5. 5.
    Grasselli, A.: Synchronization of cellular arrays: The firing squad problem in two dimensions. Information and Control 28, 113–124 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Mazoyer, J.: A six-state minimal time solution to the firing squad synchronization problem. Theoretical Computer Science 50, 183–238 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Moore, E.F.: The firing squad synchronization problem. In: Moore, E.F. (ed.) Sequential Machines, Selected Papers, pp. 213–214. Addison-Wesley, Reading (1964)Google Scholar
  8. 8.
    Moore, F.R., Langdon, G.G.: A generalized firing squad problem. Information and Control 12, 212–220 (1968)zbMATHCrossRefGoogle Scholar
  9. 9.
    Nguyen, H.B., Hamacher, V.C.: Pattern synchronization in two-dimensional cellular space. Information and Control 26, 12–23 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Romani, F.: On the fast synchronization of tree connected networks. Information Sciences 12, 229–244 (1977)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Schmid, H.: Synchronisationsprobleme für zelluläre Automaten mit mehreren Generälen. Diplomarbeit, Universität Karsruhe (2003)Google Scholar
  12. 12.
    Settle, A., Simon, J.: Smaller solutions for the firing squad. Theoretical Computer Science 276, 83–109 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Shinahr, I.: Two- and three-dimensional firing squad synchronization problems. Information and Control 24, 163–180 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Szwerinski, H.: Time-optimum solution of the firing-squad-synchronization-problem for n-dimensional rectangles with the general at an arbitrary position. Theoretical Computer Science 19, 305–320 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Umeo, H.: A simple design of time-efficient firing squad synchronization algorithms with fault-tolerance. IEICE Trans. on Information and Systems E87-D(3), 733–739 (2004)Google Scholar
  16. 16.
    Umeo, H., Hisaoka, M., Akiguchi, S.: Twelve-state optimum-time synchronization algorithm for two-dimensional rectangular cellular arrays. In: UC 2005. LNCS, vol. 3699, pp. 214–223. Springer, Heidelberg (2005)Google Scholar
  17. 17.
    Umeo, H., Hisaoka, M., Michisaka, K., Nishioka, K., Maeda, M.: Some new generalized synchronization algorithms and their implementations for large scale cellular automata. In: Calude, C.S., Dinneen, M.J., Peper, F. (eds.) UMC 2002, vol. 2509, pp. 276–286. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    Umeo, H., Hisaoka, M., Sogabe, T.: A survey on optimum-time firing squad synchronization algorithms for one-dimensional cellular automata. Intern. J. of Unconventional Computing 1, 403–426 (2005)Google Scholar
  19. 19.
    Umeo, H., Hisaoka, M., Teraoka, M., Maeda, M.: Several new generalized linear- and optimum-time synchronization algorithms for two-dimensional rectangular arrays. In: Margenstern, M. (ed.) MCU 2004, vol. 3354, pp. 223–232. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Umeo, H., Maeda, M., Hisaoka, M., Teraoka, M.: A state-efficient mapping scheme for designing two-dimensional firing squad synchronization algorithms. Fundamenta Informaticae 74(4), 603–623 (2006)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Umeo, H., Uchino, H.: A new time-optimum synchronization algorithm for two-dimensional cellular arrays. In: Moreno Díaz, R., Pichler, F., Quesada Arencibia, A. (eds.) EUROCAST 2007. LNCS, vol. 4739, pp. 604–611. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Umeo, H., Yamawaki, T., Shimizu, N., Uchino, H.: Modeling and simulation of global synchronization processes for large-scale-of two-dimensional cellular arrays. In: Proc. of Intern. Conf. on Modeling and Simulation, AMS 2007, pp. 139–144 (2007)Google Scholar
  23. 23.
    Varshavsky, V.I., Marakhovsky, V.B., Peschansky, V.A.: Synchronization of Interacting Automata. Mathematical Systems Theory 4(3), 212–230 (1970)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Waksman, A.: An optimum solution to the firing squad synchronization problem. Information and Control 9, 66–78 (1966)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Tokyo 2009

Authors and Affiliations

  • Hiroshi Umeo
    • 1
  1. 1.Univ. of Osaka Electro-CommunicationOsakaJapan

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