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Computational Complexity pp 1094–1130Cite as

Firing Squad Synchronization Problem in Cellular Automata

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Article Outline

Glossary

Definition of the Subject

Introduction

Firing Squad Synchronization Problem

Variants of the Firing Squad Synchronization Problem

Firing Squad Synchronization Problem on Two-dimensional Arrays

Summary and Future Directions

Bibliography

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Abbreviations

Cellular automaton:

A cellular automaton is a discrete computational model studied in mathematics, computer science, economics, biology, physics and chemistry etc. It consists of a regular array of cells, each cell is a finite state automaton. The array can be in any finite number of dimensions. Time (step) is also discrete, and the state of a cell at time t (\( { \ge 1 } \)) is a function of the states of a finite number of cells (called its neighborhood) at time \( { t-1 } \). Each cell has a same rule set for updating its next state, based on the states in the neighborhood. At every step the rules are applied to the whole array synchronously, yielding a new configuration.

Time-space diagram:

A time-space diagram is frequently used to represent signal propagations in one‐dimensional cellular space. Usually, the time is drawn on the vertical axis and the space on the horizontal axis. The trajectories of individual signals in propagation are expressed in this diagram by sloping lines. The slope of the line represents the propagation speed of the signal. Time-space-diagrams that show the position of individual signals in time and in space are very useful for understanding cellular algorithms, signal propagations and crossings in the cellular space.

Bibliography

Primary Literature

  1. Balzer R (1967) An 8-state minimal time solution to the firing squad synchronization problem. Inf Control 10:22–42

    Article  Google Scholar 

  2. Berthiaume A, Bittner T, Perković L, Settle A, Simon J (2004) Bounding the firing synchronization problem on a ring. Theor Comput Sci 320:213–228

    Google Scholar 

  3. Beyer WT (1969) Recognition of topological invariants by iterative arrays. Ph.D. Thesis, MIT, pp 144

    Google Scholar 

  4. Burns JE, Lynch NA (1987) The Byzantine firing squad problem. In: Advances in Computing Research: Parallel and Distributed Computing. JAI press, 4:147–161

    Google Scholar 

  5. Coan BA, Dolev D, Dwork C, Stockmeyer L (1989) The distributed firing squad problem. SIAM J Comput 18(5):990–1012

    Article  MathSciNet  MATH  Google Scholar 

  6. Culik K II (1989) Variations of the firing squad synchronization problem and applications. Inf Process Lett 30:153–157

    Article  MathSciNet  MATH  Google Scholar 

  7. Culik K II, Dube S (1991) An efficient solution of the firing mob problem. Theor Comput Sci 91:57–69

    Article  MathSciNet  MATH  Google Scholar 

  8. Fischer PC (1965) Generation of primes by a one-dimensional real-time iterative array. J ACM 12(3):388–394

    Article  MATH  Google Scholar 

  9. Gerken HD (1987) über Synchronisations‐Probleme bei Zellularautomaten. Diplomarbeit, Institut für Theoretische Informatik, Technische Universität Braunschweig

    Google Scholar 

  10. Goldstein D, Kobayashi K (2004) On the complexity of network synchronization. In: Proc. of ISSAC 2004. LNCS, vol 3341. Springer, Berlin, pp 496–507

    Google Scholar 

  11. Goldstein D, Meyer N (2002) The wake up and report problem is asymptotically time-equivalent to the firing squad synchronization problem. In: Proc. of 13th Annual ACM-SIAM Symp. on Discrete Algorithms, pp 578–587

    Google Scholar 

  12. Goto E (1962) A minimal time solution of the firing squad problem. Dittoed course notes for Applied Mathematics 298, Harvard University, pp 52–59, with an illustration in color

    Google Scholar 

  13. Goto E (1966) Some puzzles on automata. In: Kitagawa T (ed) Toward computer sciences. Kyouritsu, pp 67–91 (in Japanease)

    Google Scholar 

  14. Grasselli A (1975) Synchronization of cellular arrays: The firing squad problem in two dimensions. Inf Control 28:113–124

    Article  MathSciNet  MATH  Google Scholar 

  15. Grefenstette JJ (1983) Network structure abd the firing squad synchronization problem. J Comput Syst Sci 26:139–152

    Article  MathSciNet  MATH  Google Scholar 

  16. Grigorieff S (2006) Synchronization of a bounded degree graph of cellular automata with non uniform delays in time \( { \Delta \lfloor \log_{m}(\delta) \rfloor } \). Theor Comput Sci 356(1):170–185

    Article  MathSciNet  MATH  Google Scholar 

  17. Gruska J, Torre SL, Parente M (2007) The firing squad synchronization problem on squares, toruses and rings. Int J Found Comput Sci, 18(3):637–654

    Article  MATH  Google Scholar 

  18. Herman GT, Liu WH, Rowland S, Walker A (1974) Synchronization of growing cellular systems. Inf Control 25:103–122

    Article  MathSciNet  MATH  Google Scholar 

  19. Imai K, Morita K (1996) Firing squad synchronization problem in reversible cellular automata. Theor Comput Sci 165:475–482

    Article  MathSciNet  MATH  Google Scholar 

  20. Jiang T (1992) The synchronization of nonuniform networks of finite automata. Inf Comput 97:234–261

    Article  MATH  Google Scholar 

  21. Kobayashi K (1977) The firing squad synchronization problem for two-dimensional arrays. Inf Control 34:177–197

    Article  MATH  Google Scholar 

  22. Kutrib M, Vollmar R (1991) Minimal time synchronization in restricted defective cellular automata. J Inf Process Cybern ELK 27:179–196

    MATH  Google Scholar 

  23. Kutrib M, Vollmar R (1995) The firing squad synchronization problem in defective cellular automata. IEICE Trans Inf Syst E78-D(7):895–900

    Google Scholar 

  24. Mazoyer J (1986) An overview of the firing squad synchronization problem. Lecture Notes on Computer Science, vol 316. Springer, Berlin

    Google Scholar 

  25. Mazoyer J (1987) A six-state minimal time solution to the firing squad synchronization problem. Theor Comput Sci 50:183–238

    Google Scholar 

  26. Mazoyer J (1996) On optimal solutions to the firing squad synchronization problem. Theor Comput Sci 168:367–404

    Article  MathSciNet  MATH  Google Scholar 

  27. Mazoyer J (1997) A minimal-time solution to the FSSP without recursive call to itself and with bounded slope of signals. Draft, p 8

    Google Scholar 

  28. Minsky M (1967) Computation: Finite and infinite machines. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  29. Moore EF (1964) The firing squad synchronization problem. In: Moore EF (ed) Sequential Machines, Selected Papers, Addison-Wesley, Reading MA

    Google Scholar 

  30. Moore FR, Langdon GG (1968) A generalized firing squad problem. Inf Control 12:212–220

    Article  MATH  Google Scholar 

  31. Nguyen HB, Hamacher VC (1974) Pattern synchronization in two-dimensional cellular space. Inf Control 26:12–23

    Article  MathSciNet  MATH  Google Scholar 

  32. Nishimura J, Umeo H (2005) An optimum-time synchronization protocol for 1-bit-communication cellular automaton. In: Proc. of the 9th World Multi-Conference on Systemics, Cybernetics and Informatics, vol X, pp 232–237

    Google Scholar 

  33. Nishitani Y, Honda N (1981) The firing squad synchronization problem. Theor Comput Sci 14:39–61

    Article  MathSciNet  MATH  Google Scholar 

  34. Noguchi K (2004) Simple 8-state minimal time solution to the firing squad synchronization problem. Theor Comput Sci 314:303–334

    Article  MathSciNet  MATH  Google Scholar 

  35. Roka Z (1995) The firing squad synchronization problem on Cayley graphs. In: Proc. of MFCS'95. LNCS, vol 969. Springer, Berlin, pp 402–411

    Google Scholar 

  36. Romani F (1976) Cellular automata synchronization. Inform Sci 10:299–318

    MATH  Google Scholar 

  37. Romani F (1977) On the fast synchronization of tree connected networks. Inform Sci, 12:229–244

    Article  MathSciNet  MATH  Google Scholar 

  38. Romani F (1978) The parallelism principle: speeding up the cellular automata synchronization. Inf Control 36:245–255

    Article  MathSciNet  MATH  Google Scholar 

  39. Rosenstiehl P, Fiksel JR, Holliger A (1972) Intelligent graphs: networks of finite automata capable of solving graph problems. In: Graph Theory and Computing, Academic Press, New York, pp 219–265

    Google Scholar 

  40. Sanders P (1994) Massively parallel search for transition-tables of polyautomata. In: Jesshope C, Jossifov V, Wilhelmi W (eds) Proc. of the VI International Workshop on Parallel Processing by Cellular Automata and Arrays. Akademie, Berlin, pp 99–108

    Google Scholar 

  41. Schmid H, Worsch T (2004) The firing squad synchronization problem with many generals for one-dimensional. In: CA Proc. of IFIP World Congress, pp 111–124

    Google Scholar 

  42. Settle A, Simon J (1998) Improved bounds for the firing synchronization problem. In: SIROCCO 5: Proc. of the 5th International Colloquium on Structural Information and Communication Complexity. Carleton Scientific, Ontario, pp 66–81

    Google Scholar 

  43. Settle A, Simon J (2002) Smaller solutions for the firing squad. Theor Comput Sci 276:83–109

    Article  MathSciNet  MATH  Google Scholar 

  44. Shinahr I (1974) Two- and three-dimensional firing squad synchronization problems. Inf Control 24:163–180

    Article  MathSciNet  MATH  Google Scholar 

  45. Szwerinski H (1982) Time-optimum solution of the firing-squad-synchronization-problem for n‑dimensional rectangles with the general at an arbitrary position. Theor Comput Sci 19:305–320

    Article  MathSciNet  MATH  Google Scholar 

  46. Torre SL, Napoli M, Parente D (1996) Synchronization of one-way connected processors. Complex Syst 10:239–255

    MATH  Google Scholar 

  47. Torre SL, Napoli M, Parente D (1998) Synchronization of a line of identical processors at a given time. Fundamenta Informaticae 34:103–128

    MathSciNet  MATH  Google Scholar 

  48. Torre SL, Napoli M, Parente M (2000) A compositional approach to synchronize two dimensional networks of processors. Theor Inf Appl 34:549–564

    Article  MATH  Google Scholar 

  49. Torre SL, Napoli M, Parente M (2001) Firing squad synchronization problem on bidimensional cellular automata with communication constraints. In: Proc. of MCU 2001. LNCS,vol 2055. Springer, Berlin, pp 264–275

    Google Scholar 

  50. 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) Proc. of Cellular Automata Workshop, pp 65

    Google Scholar 

  51. Umeo H (2001) Linear-time recognition of connectivity of binary images on 1-bit inter-cell communication cellular automaton. Parallel Computing, 27:587–599

    Google Scholar 

  52. Umeo H (2004) A simple design of time-efficient firing squad synchronization algorithms with fault-tolerance. IEICE Trans Inf Syst E87-D(3):733–739

    Google Scholar 

  53. Umeo H, Hisaoka M, Akiguchi S (2005) A twelve‐state optimum‐time synchronization algorithm for two‐dimensional rectangular arrays. In: Proc. of the 4th International Conference on Unconventional Computation: UC 2005. LNCS, vol 3699. Springer, Berlin, pp 214–223

    Google Scholar 

  54. Umeo H, Hisaoka M, Sogabe T (2005) A survey on optimum-time firing squad synchronization algorithms for one-dimensional cellular automata. Int J Unconv Comput 1:403–426

    Google Scholar 

  55. Umeo H, Hisaoka M, Teraoka M, Maeda M (2005) Several new generalized linear- and optimum-time synchronization algorithms for two-dimensional rectangular arrays. In: Margenstern M (ed) Proc. of the 4th International Conference on Machines, Computations and Universality: MCU 2004. LNCS, vol 3354. Springer, Berlin, pp 223–232

    Google Scholar 

  56. Umeo H, Hisaoka M, Michisaka K, Nishioka K, Maeda M (2002) Some new generalized synchronization algorithms and their implementations for large scale cellular automata. In: Proc. of the 3rd International Conference on Unconventional Models of Computation: UMC 2002. LNCS, vol 2509. Springer, Berlin, pp 276–286

    Google Scholar 

  57. Umeo H, Kamikawa N (2003) Real-time generation of primes by a 1-bit-communication cellular cutomaton. Fundamenta Informaticae 58:421–435

    MathSciNet  MATH  Google Scholar 

  58. Umeo H, Kamikawa N, Yunès JB (2008) A family of smallest symmetrical four-state firing squad synchronization protocols for one-dimensional ring celluar automata. Luniver Press, Beckington, pp 174–186

    Google Scholar 

  59. Umeo H, Maeda M, Fujiwara N (2002) An efficient mapping scheme for embedding any one-dimensional firing squad synchronization algorithm onto two-dimensional arrays. In: Proc. of the 5th International Conference on Cellular Automata for Research and Industry. LNCS, vol 2493. Springer, Berlin, pp 69–81

    Google Scholar 

  60. Umeo H, Maeda M, Hisaoka M, Teraoka M (2006) A state-efficient mapping scheme for designing two-dimensional firing squad synchronization algorithms. Fundamenta Informaticae 74(4):603–623

    MathSciNet  MATH  Google Scholar 

  61. Umeo H, Maeda M, Hongyo K (2006) A design of symmetrical six-state 3n‑step firing squad synchronization algorithms and their implementations. In: Proc. of 7th International Conference on Cellular Automata for Research and Industry, ACRI2006. LNCS, vol 4173. Springer, Berlin, pp 157–168

    Google Scholar 

  62. Umeo H, Michisaka K, Kamikawa N (2003) A synchronization problem on 1-bit-communication cellular automata. In: Proc. of International Conference on Computational Science-ICCS2003. LNCS, vol 2657. Springer, Berlin, pp 492–500

    Google Scholar 

  63. Umeo H, Michisaka K, Kamikawa N, Kanazawa M (2007) State-efficient one-bit-communication solutions for some classical cellular automata problems. Fundamenta Informaticae 78:449–465

    MathSciNet  MATH  Google Scholar 

  64. Umeo H, Sogabe T, Nomura Y (2000) Correction, optimization and verification of transition rule set for Waksman's firing squad synchronization algorithm. In: Proc. of the Fourth Intern. Conference on Cellular Automata for Research and Industry. Springer, London, pp 152–160

    Google Scholar 

  65. Umeo H, Uchino H (2007) A new time-optimum synchronization algorithm for two-dimensional cellular arrays. In: Quesada-Arenciba A, Rodriguez JC, Moreno-Diaz R Jr, R Moreno-Diaz (eds) Proc. of Intern. Conf. on Computer Aided Systems Theory, EUROCAST 2007. LNCS, vol 4739. Springer, Berlin, pp 604–611

    Chapter  Google Scholar 

  66. Umeo H, Yamawaki T, Shimizu N, Uchino H (2007) 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

    Google Scholar 

  67. Umeo H, Yanagihara T (2007) A smallest five-state solution to the firing squad synchronization problem. In: Proc. of 5th Intern. Conf. on Machines, Computations, and Universality, MCU 2007. LNCS, vol 4664. Springer, Berlin, pp 292–302

    Google Scholar 

  68. Umeo H, Yanagihara T, Kanazawa M (2006) State-efficient firing squad synchronization protocols for communication-restricted cellular automata. In: Proc. of 7th International Conference on Cellular Automata for Research and Industry, ACRI2006. LNCS, vol 4173. Springer, Berlin, pp 169–181

    Google Scholar 

  69. Varshavsky VI, Marakhovsky VB, Peschansky VA (1970) Synchronization of Interacting Automata. Math Syst Theor 4(3):212–230

    Google Scholar 

  70. Vivien H (1996) A quasi-optimal time for synchronization two interacting finite automata. J Algebra Comput 6(2):261–267

    Article  MathSciNet  MATH  Google Scholar 

  71. Vivien H (2005) Cellular automata: a geometrical approach. Draft, pp 412

    Google Scholar 

  72. Vollmar R (1979) Algorithmen in Zellularautomaten, Teubner, Stuttgart, pp 192

    Book  MATH  Google Scholar 

  73. Vollmar R (1982) Some remarks about the “Efficiency” of polyautomata. Int J Theor Phys 21(12):1007–1015

    Article  MathSciNet  MATH  Google Scholar 

  74. Waksman A (1966) An optimum solution to the firing squad synchronization problem. Inf Control 9:66–78

    Article  MathSciNet  MATH  Google Scholar 

  75. Worsch T (2000) Linear time language recognition on cellular automata with restricted communication. In: Gonnet GH, Panario D, Viola A (eds) Proc. of LATIN 2000:Theoretical Informatics. LNCS, vol 1776. Springer, Berlin, pp 417–426

    Google Scholar 

  76. Yunès JB (1994) Seven-state solution to the firing squad synchronization problem. Theor Comput Sci 127:313–332

    Google Scholar 

  77. Yunès JB (2006) Fault tolerant solutions to the firing squad synchronization problem in linear cellular automata. J Cell Autom 1:253–268

    Google Scholar 

  78. Yunès JB (2007) Simple new algorithms which solve the firing squad synchronization problem: A 7-states 4n‑steps solution. In: Proc. of 5th Intern. Conf. on Machines, Computations, and Universality, MCU 2007. LNCS, vol 4664. Springer, Berlin, pp 316–324

    Google Scholar 

  79. Yunès JB (2008) An intrinsically non minimal-time Minsky-like 6-states solution to the firing squad synchronization problem. Theor Inf Appl 42(1):55–68

    Google Scholar 

  80. Yunès JB (2007) A 4-states algebraic solution to linear cellular automata synchronization. submitted to Information Processing Letters

    Google Scholar 

Books and Reviews

  1. Delorme M, Mozoyer J (eds) (1999) Cellular Automata. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  2. Ilachinski A (2001) Cellular Automata – A Discrete Universe –. World Scientific, Singapore

    Google Scholar 

  3. Wolfram S (1994) Cellular Automata and Complexity. Addison-Wesley Publishing Company, Reading

    MATH  Google Scholar 

  4. Wolfram S (2002) A New Kind of Science. Wolfram Media, Champaign

    MATH  Google Scholar 

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Authors and Affiliations

  1. Electro-Communication, University of Osaka, Osaka, Japan

    Hiroshi Umeo

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  1. Hiroshi Umeo
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Umeo, H. (2012). Firing Squad Synchronization Problem in Cellular Automata. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_70

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