A Spectral Portrait of the Elementary Cellular Automata Rule Space

  • Eurico L. P. Ruivo
  • Pedro P. B. de Oliveira
Part of the Emergence, Complexity and Computation book series (ECC, volume 2)


Fourier spectra of cellular automata rules give a quantitative characterisation of the frequency of bit patterns present in the limit configurations generated out of their time evolution. By means of the similarity among the spectrum of each rule, the elementary cellular automata rule space is partitioned, both for periodic and non-periodic boundary conditions, thus giving rise to rule classes of spectral equivalence. The partitions generated are formally explained in terms of the dynamical behaviour of the rules, and maps of the elementary space are given in terms of graphs that represent the spectral similarity among the rules, thus displaying a spectral portrait of how the classes relate to each other.


Fourier spectrum discrete Fourier transform elementary cellular automata discrete dynamical systems dynamical behaviour of cellular automata 


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  1. 1.
    Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Mathematical Systems Theory 3(4), 320–375 (1969)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Li, W.: Power spectra of regular languages and cellular automata. Complex Systems 1, 107–130 (1987)MathSciNetMATHGoogle Scholar
  3. 3.
    Li, W.: Phenomenology of nonlocal cellular automata. Journal of Statistical Physics 68(5-6), 829–882 (1992)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Li, W., Packard, N.: The structure of the elementary cellular automata rule space. Structure 4, 281–297 (1990)MathSciNetGoogle Scholar
  5. 5.
    Ninagawa, S.: 1/f Noise in Elementary Cellular Automaton Rule 110. In: Calude, C.S., Dinneen, M.J., Păun, G., Rozenberg, G., Stepney, S. (eds.) UC 2006. LNCS, vol. 4135, pp. 207–216. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Ninagawa, S.: Evolution of One-Dimensional Cellular Automata by 1/f Noise. In: Almeida e Costa, F., Rocha, L.M., Costa, E., Harvey, I., Coutinho, A. (eds.) ECAL 2007. LNCS (LNAI), vol. 4648, pp. 905–914. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Ninagawa, S.: Power spectral analysis of elementary cellular automata. Complex Systems 17(4), 399–411 (2008)MathSciNetMATHGoogle Scholar
  8. 8.
    Oliveira, G.M., de Oliveira, P.P.B., Omar, N.: Definition and application of a five-parameter characterization of one-dimensional cellular automata rule space. Artificial Life 7(3), 277–301 (2001)CrossRefGoogle Scholar
  9. 9.
    Ruivo, E.L.P., de Oliveira, P.P.B.: Spectral similarity among elementary cellular automata. In: Formenti, E. (ed.) Proc. of AUTOMATA 2012: 18th Int. Workshop on Cellular Automata and Discrete Complex Systems, pp. 89–98 (2012)Google Scholar
  10. 10.
    Shranko, A., de Oliveira, P.P.B.: Relationships between local dynamics and global reversibility of multidimensional cellular automata with hyper-rectangular neighbourhoods (unpublished manuscript)Google Scholar
  11. 11.
    Wolfram, S.: Computation theory of cellular automata. Communications in Mathematical Physics 96(1), 15–57 (1984)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Wolfram, S.: Twenty problems in the theory of cellular automata. Physica Scripta 9(3), 170–183 (1985)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wolfram, S.: A New Kind of Science. Wolfram Media (2002)Google Scholar
  14. 14.
    Wuensche, A.: Classifying cellular automata automatically: finding gliders, filtering and relating space-time patterns, attractor basins, and the Z-parameter. Complexity 4, 73–90 (1998)MathSciNetGoogle Scholar
  15. 15.
    Xie, H.: Distinct excluded blocks and grammatical complexity of dynamical systems. Complex Systems 9, 73–90 (1995)MathSciNetMATHGoogle Scholar
  16. 16.
    Zwick, M., Shu, H.: Set-theoretic reconstructability of elementary cellular automata. General Systems 1, 1–6 (1995)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Eurico L. P. Ruivo
    • 2
  • Pedro P. B. de Oliveira
    • 1
    • 2
  1. 1.Faculdade de Computação e InformáticaUniversidade Presbiteriana MackenzieSão PauloBrazil
  2. 2.Pós-Graduação em Engenharia ElétricaUniversidade Presbiteriana MackenzieSão PauloBrazil

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