Journal of Statistical Physics

, Volume 103, Issue 1–2, pp 45–267 | Cite as

Reliable Cellular Automata with Self-Organization

  • Peter Gács


In a probabilistic cellular automaton in which all local transitions have positive probability, the problem of keeping a bit of information indefinitely is nontrivial, even in an infinite automaton. Still, there is a solution in 2 dimensions, and this solution can be used to construct a simple 3-dimensional discrete-time universal fault-tolerant cellular automaton. This technique does not help much to solve the following problems: remembering a bit of information in 1 dimension; computing in dimensions lower than 3; computing in any dimension with non-synchronized transitions. Our more complex technique organizes the cells in blocks that perform a reliable simulation of a second (generalized) cellular automaton. The cells of the latter automaton are also organized in blocks, simulating even more reliably a third automaton, etc. Since all this (a possibly infinite hierarchy) is organized in “software,” it must be under repair all the time from damage caused by errors. A large part of the problem is essentially self-stabilization recovering from a mess of arbitrary size and content. The present paper constructs an asynchronous one-dimensional fault-tolerant cellular automaton, with the further feature of “self-organization.” The latter means that unless a large amount of input information must be given, the initial configuration can be chosen homogeneous.

probabilistic cellular automata interacting particle systems renormalization ergodicity reliability fault-tolerance error-correction simulation hierarchy self-organization 


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  1. 1.
    M. Aizenman, Translation invariance and instability of phase coexistence in the two-dimensional Ising system, Comm. Math. Phys. 73(1):83–94 (1980).Google Scholar
  2. 2.
    C. H. Bennett, G. Grinstein, Yu He, C. Jayaprakash, and D. Mukamel, Stability of temporally periodic states of classical many-body systems, Phys. Rev. A 41:1932–1935 (1990).Google Scholar
  3. 3.
    C. H. Bennett and G. Grinstein, Role of irreversibility in stabilizing complex and non-ergodic behavior in locally interacting discrete systems, Phys. Rev. Lett. 55:657–660 (1985).Google Scholar
  4. 4.
    E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays (Academic Press, New York, 1982).Google Scholar
  5. 5.
    P. Berman and J. Simon, Investigations of fault-tolerant networks of computers, in Proc. 20th Annual ACM Symp. on the Theory of Computing (1988), pp. 66–77.Google Scholar
  6. 6.
    C. Bezuidenhout and G. Grimmett, The critical contact process dies out, Ann. Probab. 18:1462–1482 (1990).Google Scholar
  7. 7.
    R. E. Blahut, Theory and Practice of Error-Control Codes (Addison-Wesley, Reading, MA, 1983).Google Scholar
  8. 8.
    P. G. de Sá and C. Maes, The Gács-Kurdyumov-Levin automaton revisited, J. Stat. Phys. 67(3/4):607–622 (1992).Google Scholar
  9. 9.
    R. L. Dobrushin and S. I. Ortyukov, Upper bound on the redundancy of self-correcting arrangements of unreliable elements, Problems of Inform. Trans. 13(3):201–208 (1977).Google Scholar
  10. 10.
    P. Gács, Reliable computation with cellular automata, J. Comput. Syst. Sci. 32(1):15–78 (1986).Google Scholar
  11. 11.
    P. Gaács, Self-correcting two-dimensional arrays, in Randomness in Computation, Silvio Micali, ed., Advances in Computing Research (a scientific annual), Vol. 5 (JAI Press, Greenwich, CT, 1989), pp. 223–326.Google Scholar
  12. 12.
    P. Gács, Deterministic parallel computations whose history is independent of the order of updating, Scholar
  13. 13.
    P.r Gács, G. L. Kurdyumov, and L. A. Levin, One-dimensional homogenuous media dissolving finite islands, Problems of Inf. Transm. 14(3):92–96 (1978).Google Scholar
  14. 14.
    P. Gács and J. Reif, A simple three-dimensional real-time reliable cellular array, J. Comput. Syst. Sci. 36(2):125–147 (1988).Google Scholar
  15. 15.
    S. Goldstein, R. Kuik, J. L. Lebowitz, and C. Maes, From PCA's to equilibrium systems and back, Commun. Math. Phys. 125:71–79 (1989).Google Scholar
  16. 16.
    L. F. Gray, The positive rates problem for attractive nearest neighbor spin systems on Z, Z. Wahrs. verw. Gebiete 61:389–404 (1982).Google Scholar
  17. 17.
    L. F. Gray, The behavior of processes with statistical mechanical properties, in Percolation Theory and Ergodic Theory of Infinite Particle Systems (Springer-Verlag, 1987), pp. 131–167.Google Scholar
  18. 18.
    G. Itkis and L. Levin, Fast and lean self-stabilizing asynchronous protocols, Proc. of the IEEE Symp. on Foundations of Computer Science (1994), pp. 226–239.Google Scholar
  19. 19.
    G. L. Kurdyumov, An example of a nonergodic homogenous one-dimensional random medium with positive transition probabilities, Sov. Math. Dokl. 19(1):211–214 (1978).Google Scholar
  20. 20.
    T. M. Liggett, Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Vol. 276 (Springer Verlag, New York, 1985).Google Scholar
  21. 21.
    J. Neveu, Bases mathematiques du calcul des probabilités (Masson et Cie, Paris, 1964).Google Scholar
  22. 22.
    K. Park, Ergodicity and mixing rate of one-dimensional cellular automata, Ph.D. thesis, Boston University, Boston, MA 02215 (1996).Google Scholar
  23. 23.
    N. Pippenger, On networks of noisy gates, Proc. of the 26th IEEE FOCS Symposium (1985), pp. 30–38.Google Scholar
  24. 24.
    C. Radin, Global order from local sources, Bull. Amer. Math. Soc. 25:335–364 (1991).Google Scholar
  25. 25.
    D. A. Spielman, Highly fault-tolerant parallel computation, Proc. of the 37th IEEE FOCS Symposium (1996), pp. 154–163.Google Scholar
  26. 26.
    T. Toffoli and N. Margolus, Cellular Automata Machines (MIT Press, Cambridge, 1987).Google Scholar
  27. 27.
    A. L. Toom, Stable and attractive trajectories in multicomponent systems, in Multicomponent Systems, R. L. Dobrushin, ed., Advances in Probability, Vol. 6 (Dekker, New York, 1980) [Translation from Russian], pp. 549–575.Google Scholar
  28. 28.
    B. S. Tsirel'son, Reliable information storage in a system of locally interacting unreliable elements, in Interacting Markov Processes in Biology, V. I. Kryukov, R. L. Dobrushin, and A. L. Toom, eds. (Scientific Centre of Biological Research, Pushchino, 1977), in Russian. Translation by Springer, pp. 24–38.Google Scholar
  29. 29.
    J. von Neumann, Probabilistic logics and the synthesis of reliable organisms from unreliable components, in Automata Studies, C. Shannon and McCarthy, eds. (Princeton University Press, Princeton, NJ, 1956).Google Scholar
  30. 30.
    W. Wang, An asynchronous two-dimensional self-correcting cellular automaton, Ph.D. thesis, Boston University, Boston, MA 02215 (1990). Short version: Proc. 32nd IEEE Symposium on the Foundations of Computer Science (1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Peter Gács
    • 1
  1. 1.Computer Science DepartmentBoston UniversityBoston

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