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A Spatio-temporal Algorithmic Point of View on Firing Squad Synchronisation Problem

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7495)

Abstract

Firing Squad Synchronization Problems are well known to be solvable by voluminous transition tables describing signals traveling and colliding. In this paper, we show that it is possible to solve it by expressing directly the fact that we want a recursive division of the space into two parts of equal size, and a notification when no further division is possible. Using fields – objects associating a value to every point in space and time – as primitive objects, the solution is designed algorithmically by a semantically-intuitive decomposition of the global evolution into simpler evolutions.

The system we obtain has several interesting characteristics : it is understandable, time-optimal, tackles many initial configurations, and allows a new interpretation of the traditional signals and collisions point of view. We will quickly sketch how we can obtain a finite state automaton by reduction of the system using the Lipschitz-continuity of involved fields, and a kind of tail-recursivity property of the dependencies.

Keywords

  • Cellular Automaton
  • Initial Region
  • Recursive Schema
  • Unbounded Number
  • Gabriel Graph

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Culik, K.: Variations of the firing squad problem and applications. Information Processing Letters 30, 153–157 (1989)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Goto, E.: A minimum time solution of the firing squad synchronization problem. Courses Notes for Applied Mathematics, vol. 298. Harvard University (1962)

    Google Scholar 

  3. Grasselli, A.: Synchronization of cellular arrays: The firing squad problem in two dimensions. Information and Control 28, 113–124 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Herman, G.T., Liu, W., Rowland, S., Walker, A.: Synchronization of growing cellular automata. Information and Control 25, 103–122 (1974)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Jiang, T.: The synchronization of nonuniform networks of finite automata. Information and Control 97, 234–261 (1992)

    MATH  Google Scholar 

  6. Kobayashi, K.: The firing squad synchronisation problem for two-dimensional arrays. Information and Control 34, 177–197 (1977)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Maignan, L., Gruau, F.: Integer gradient for cellular automata: Principle and examples. In: Proceedings of the 2008 Second IEEE International Conference on Self-Adaptive and Self-Organizing Systems Workshops, pp. 321–325. IEEE Computer Society, Washington, DC (2008)

    CrossRef  Google Scholar 

  8. Maignan, L., Gruau, F.: A 1D cellular automaton that moves particles until regular spatial placement. Parallel Processing Letters 19(2), 315–331 (2009)

    CrossRef  MathSciNet  Google Scholar 

  9. Maignan, L., Gruau, F.: Convex Hulls on Cellular Automata. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds.) ACRI 2010. LNCS, vol. 6350, pp. 69–78. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  10. Maignan, L., Gruau, F.: Gabriel graphs in arbitrary metric space and their cellular automaton for many grids. ACM Trans. Auton. Adapt. Syst. 6, 12:1–12:14 (June 2011)

    Google Scholar 

  11. Mazoyer, J.: A six states minimal time solution to the firing squad synchronization problem. Theoretical Computer Science 50, 183–238 (1987)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Minsky, M.: Computation: Finite and Infinite Machines. Prentice-Hall (1967)

    Google Scholar 

  13. Moore, E.E.: Sequential machines, Selected papers. Addison-Wesley (1964)

    Google Scholar 

  14. Noguchi, K.: Simple 8-state minimal time solution to the firing squad synchronization problem. TCS 314, 303–334 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Rosenstiehl, P., Fiskel, J.R., Holliger, A.: Intelligent Graphs: Networks of Finite Automata capable of Solving Graph Problems. In: Read, R.C. (ed.) Graph Theory and Computing. Academic Press (1972)

    Google Scholar 

  16. Schmid, H., Worsch, T.: The firing squad synchronization problem with many generals for one-dimensional CA. In: Lévy, J.-J., Mayr, E.W., Mitchell, J.C. (eds.) IFIP TCS, pp. 111–124. Kluwer (2004)

    Google Scholar 

  17. Shinahr, I.: Two and three dimensional firing squad synchronization problems. Information and Control 24, 163–180 (1974)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Szwerinski, H.: Time-optimal 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)

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. Varshavsky, V.I., Marakhovsky, V.B., Peshansky, V.A.: Synchronization of interacting automata. Mathematical System Theory 4(3), 212–230 (1969)

    Google Scholar 

  20. Yunès, J.-B.: An intrinsically non minimal-time Minsky-like 6-states solution to the firing squad synchronization problem. RAIRO ITA/TIA 42(1), 55–66 (2008)

    CrossRef  MATH  Google Scholar 

  21. Yunès, J.-B.: Known CA synchronizers made insensitive to the initial state of the initiator. JCA 4(2), 147–158 (2009)

    MATH  Google Scholar 

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Maignan, L., Yunès, JB. (2012). A Spatio-temporal Algorithmic Point of View on Firing Squad Synchronisation Problem. In: Sirakoulis, G.C., Bandini, S. (eds) Cellular Automata. ACRI 2012. Lecture Notes in Computer Science, vol 7495. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33350-7_11

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  • DOI: https://doi.org/10.1007/978-3-642-33350-7_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33349-1

  • Online ISBN: 978-3-642-33350-7

  • eBook Packages: Computer ScienceComputer Science (R0)