Natural Computing

, Volume 18, Issue 4, pp 827–844 | Cite as

State-efficient realization of fault-tolerant FSSP algorithms

  • Hiroshi UmeoEmail author
  • Naoki Kamikawa
  • Masashi Maeda
  • Gen Fujita


The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) cellular arrays. It is shown that, under some constraints on the length and distribution of defective cells, any 1D cellular array of length n with p defective cell segments can be synchronized in \(2n-2+p\) steps and the algorithm is realized on a 1D cellular automaton of length \(n, 2 \le n \le 50\), having 164 states and 4792 transition rules. In addition, we give by far a smaller-state implementation of a 2D FSSP algorithm that can synchronize any 2D rectangular array of size \(m \times n\), possibly including at most O(mn) isolated defective zones, exactly in \(2(m+n)-4\) steps on a cellular automaton with only 6 states and 935 transition rules.


Cellular automaton Firing squad synchronization problem FSSP Fault-tolerant cellular automaton 



A part of this work is supported by JSPS 16K00026. The authors would like to thank reviewers for many helpful comments and suggestions to improve the paper.


  1. Balzer R (1967) An 8-state minimal time solution to the firing squad synchronization problem. Inf Control 10:22–42CrossRefGoogle Scholar
  2. Burns JE, Lynch NA (1985) The Byzantine firing squad problem. Technical report of MIT, MIT/LCS/TM-275, 1–17Google Scholar
  3. Coan BA, Dolev D, Dwork C, Stockmeyer L (1989) The distributed firing squad problem. SIAM J Comput 18:990–1012MathSciNetCrossRefGoogle Scholar
  4. Dimitriadis A, Kutrib M, Sirakoulis GCh (2016) Cutting the firing squad synchronization. In: Proceedings of ACRI 2016, LNCS 9863, pp 123–133Google Scholar
  5. Gács P (1986) Reliable computation with cellular automata. J Comput Syst Sci 32:15–78MathSciNetCrossRefGoogle Scholar
  6. Gerken HD (1987) Über Synchronisations—Probleme bei Zellularautomaten. Diplomarbeit, Institut für Theoretische Informatik, Technische Universität BraunschweigGoogle Scholar
  7. Goto E (1962) A minimal time solution of the firing squad problem. In: Dittoed course notes for applied mathematics, vol 298, Harvard University, pp 52–59Google Scholar
  8. Harao M, Noguchi S (1975) Fault tolerant cellular automata. J Comput Syst. Sci 11:171–185MathSciNetCrossRefGoogle Scholar
  9. Kutrib M, Löwe JT (2000) Fault tolerant parallel pattern recognition. In: Theoretical and practical issues on cellular automata (ACRI 2002), Springer, pp 72–80Google Scholar
  10. Kutrib M, Vollmar R (1991) Minimal time synchronization in restricted defective cellular automata. J Inform Process Cybern ELK 27:179–196zbMATHGoogle Scholar
  11. Kutrib M, Vollmar R (1995) The firing squad synchronization problem in defective cellular automata. IEICE Trans Inf Syst E78–D:895–900Google Scholar
  12. Lynch NA (1996) Distributed algorithms. Morgan Kaufmann Publishers, Califonia, ISBN 1-55860-348-4Google Scholar
  13. Mazoyer J (1987) A six-state minimal time solution to the firing squad synchronization problem. Theor Comput Sci 50:183–238MathSciNetCrossRefGoogle Scholar
  14. Moore EF (1964) The firing squad synchronization problem. In: Moore EF (ed) Sequential machines, selected papers, Addison-Wesley, Reading, MA, pp 213–214Google Scholar
  15. Nishio H, Kobuchi H (1975) Fault tolerant cellular spaces. J Comput Syst. Sci 11:150–170MathSciNetCrossRefGoogle Scholar
  16. Umeo H (2004) A simple design of time-efficient firing squad synchronization algorithms with fault-tolerance. IEICE Trans Inf Syst E87–D:733–739Google Scholar
  17. Umeo H (2009) Firing squad synchronization problem in cellular automata. In: Meyers RA (ed) Encyclopedia of complexity and system science, vol 4, Springer, pp 3537–3574Google Scholar
  18. Umeo H, Maeda M, Hisaoka M, Teraoka M (2006) A state-efficient mapping scheme for designing two-dimensional firing squad synchronization algorithms. Fund Inf 74:603–623MathSciNetzbMATHGoogle Scholar
  19. Umeo H, Kamikawa N, Maeda M, Fujita G (2018) Implementations of FSSP algorithms on fault-tolerant cellular arrays. In: Proceedings of ACRI 2018, LNCS, vol 11115, pp 274–285.
  20. Waksman A (1966) An optimum solution to the firing squad synchronization problem. Inf Control 9:66–78MathSciNetCrossRefGoogle Scholar
  21. Yunès J-B (2006) Fault tolerant solutions to the firing squad synchronization problem in linear cellular automata. J Cell Autom 1:253–268MathSciNetzbMATHGoogle Scholar
  22. Zambonelli F, Mamei M, Roli A (2002) What can cellular automata tell us about the behavior of large multi-agent systems? In: Proceedings of SELMAS 2002, Springer, pp 216–231Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Hiroshi Umeo
    • 1
    Email author
  • Naoki Kamikawa
    • 1
  • Masashi Maeda
    • 1
  • Gen Fujita
    • 1
  1. 1.University of Osaka Electro-CommunicationOsakaJapan

Personalised recommendations