A design of generalized minimum-state-change FSSP algorithms and their implementations

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Abstract

The firing squad synchronization problem (FSSP) on cellular automata has been studied extensively for more than 50 years, and a rich variety of FSSP algorithms has been proposed. Here we consider the FSSP from a view point of state-change complexity that models the energy consumption of SRAM-type storage with which cellular automata might be built. In the present paper, we propose minimum-state-change generalized FSSP (GFSSP) algorithms for synchronizing any one-dimensional (1D) cellular automaton, where the initial synchronization operation is started by any cell in the array. First, we construct two minimum-time, minimum-state-change GFSSP implementations on finite state automata: one is based on Goto’s algorithm, known as the first minimum-time FSSP algorithm that was reconstructed again recently in Umeo et al. (A new reconstruction and the first implementation of Goto’s FSSP algorithm, 2017), and the other is based on Gerken’s (Diplomarbeit, Institut für Theoretische Informatik, Technische Universität Braunschweig, pp 1–50, 1987) one. These implementations are optimal not only in time but also in the state-change complexity. The implementations of the minimum-time GFSSP algorithms are the first ones having the minimum-state-change complexity. In addition, we also present a six-state 145-rule non-minimum-time, minimum-state-change GFSSP implementation. The implemented GFSSP algorithm is the smallest one, known at present, in number of states of the finite state automaton.

Keywords

Cellular automaton Firing squad synchronization problem FSSP State-change complexity 
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Notes

Acknowledgements

A part of this work is supported by JSPS 16K00026.

References

  1. Balzer R (1967) An 8-state minimal time solution to the firing squad synchronization problem. Inf Control 10:22–42CrossRefMATHGoogle Scholar
  2. Gerken H-D (1987) Über Synchronisationsprobleme bei Zellularautomaten. Diplomarbeit, Institut für Theoretische Informatik. Technische Universität BraunschweigGoogle Scholar
  3. Goto E (1962) A minimal time solution of the firing squad problem. Dittoed course notes for Applied Mathematics 298 (with an illustration in color). Harvard University, HarvardGoogle Scholar
  4. Mazoyer J (1987) A six-state minimal time solution to the firing squad synchronization problem. Theoret Comput Sci 50:183–238MathSciNetCrossRefMATHGoogle Scholar
  5. Minsky M (1967) Computation: finite and infinite machines. Prentice-Hall, Upper Saddle RiverMATHGoogle Scholar
  6. Moore EF (1964) The firing squad synchronization problem. In: Moore EF (ed) Sequential machines, selected papers. Addison-Wesley, Reading, pp 213–214Google Scholar
  7. Moore FR, Langdon GG (1968) A generalized firing squad problem. Inf Control 12:212–220CrossRefMATHGoogle Scholar
  8. Schmid H, Worsch T (2004) The firing squad synchronization problem with many generals for one-dimensional CA. In: Proceedings of IFIP World Congress, pp 111–124Google Scholar
  9. Szwerinski H (1982) Time-optimum solution of the firing-squad-synchronization-problem for \(n\)-dimensional rectangles with the general at an arbitrary position. Theoret Comput Sci 19:305–320MathSciNetCrossRefMATHGoogle Scholar
  10. Umeo H (1996) A note on firing squad synchronization algorithms–a reconstruction of Goto’s first-in-the-world optimum-time firing squad synchronization algorithm. In: Kutrib M, Worsch T (eds) Proceedings of IFIP cellular automata workshop 1996. Schloss Rauischholzhausen, Giessen, p 65Google Scholar
  11. Umeo H (2004) A simple design of time-efficient firing squad synchronization algorithms with fault-tolerance. IEICE Trans Inf Syst E87–D(3):733–739Google Scholar
  12. Umeo H (2009) Firing squad synchronization problem in cellular automata. In: Meyers RA (ed) Encyclopedia of complexity and system science, vol 4. Springer, New York, pp 3537–3574CrossRefGoogle Scholar
  13. Umeo H, Imai K (2016) A class of minimum-time, minimum-state-change generalized FSSP algorithms. In: El Yacoubi S et al (eds) Proceedings of ACRI 2016, LNCS, vol 9863. Springer, Heidelberg, pp 1–11Google Scholar
  14. Umeo H, Kamikawa N, Nishioka K, Akiguchi S (2010) Generalized firing squad synchronization protocols for one-dimensional cellular automata—a survey. Acta Phys Pol B Proc Suppl 3:267–289Google Scholar
  15. Umeo H, Maeda M, Sousa A, Taguchi K (2015a) A class of non-optimum-time \(3n\)-step FSSP algorithms—a survey. In: Proceedings of the 13th international conference on parallel computing technologies, PaCT 2015, LNCS, vol 9251. Springer, Heidelberg, pp 231–245Google Scholar
  16. Umeo H, Imai K, Sousa A (2015b) A generalized minimum-time, minimum-state-change FSSP algorithm. In: Proceedings of the 4th international conference on the theory and practise of natural computing, TPNC 2015, LNCS, vol 9477. Springer, Heidelberg, pp 161–173Google Scholar
  17. Umeo H, Hirota M, Nozaki Y, Imai K, Sogabe T (2017) A new reconstruction and the first implementation  of Goto’s FSSP algorithm. Appl Math Comput. doi: 10.1016/j.amc.2017.05.015
  18. Vollmar R (1982) Some remarks about the efficiency of polyautomata. Int J Theore Phys 21(12):1007–1015MathSciNetCrossRefMATHGoogle Scholar
  19. Waksman A (1966) An optimum solution to the firing squad synchronization problem. Inf Control 9:66–78MathSciNetCrossRefMATHGoogle Scholar
  20. Yunès J-B (2008) An intrinsically non minimal-time Minsky-like 6-states solution to the firing squad synchronization problem. Theor Inf Appl 42(1):55–68MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.University of Osaka Electro-CommunicationOsakaJapan

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