Advertisement

Construction of Some Nonautomatic Sequences by Cellular Automata

  • Irène Marcovici
  • Thomas Stoll
  • Pierre-Adrien Tahay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10875)

Abstract

It is known that if p is a prime number, the columns of linear CA are p-automatic sequences and all p-automatic sequences can be realized by some linear CA with memory. We give some constructions of (nonlinear) CA that realize certain nonautomatic sequences. First, we show through a recoding that from a construction with additional symbols, we can construct a CA using only the symbols occurring in the sequence. This answers a question posed by Rowland and Yassawi. Then, we propose a construction for the characteristic sequence of the integer polynomials, which are nonautomatic sequences by the Minsky–Papert criterion. We also provide a construction based on the indicator of Fibonacci numbers for the Fibonacci word, which is an emblematic nonautomatic sequence.

Keywords

Cellular automata Automatic sequences Nonautomatic sequences Computability Polynomials Fibonacci word 

Notes

Acknowledgements

The authors thank N. Fatès and E. Jeandel for fruitful discussions and the referees for valuable comments. The work has been supported by the ANR-FWF bilateral project MuDeRa “Multiplicativity: Determinism and Randomness” (France-Austria), ANR-14-CE34-0009.

References

  1. 1.
    Allouche, J.P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  2. 2.
    Blackburn, S.R.: Non-overlapping codes. IEEE Trans. Inf. Theory 61(9), 4890–4894 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. In: Mac Lane, S., Siefkes, D. (eds.) The Collected Works of J. Richard Büchi, pp. 398–424. Springer, New York (1990)CrossRefGoogle Scholar
  4. 4.
    Chee, Y.M., Kiah, H.M., Purkayastha, P., Wang, C.: Cross-Bifix-Free codes within a constant factor of optimality. IEEE Trans. Inf. Theory 59(7), 4668–4674 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Delacourt, M., Poupet, V., Sablik, M., Theyssier, G.: Directional dynamics along arbitrary curves in cellular automata. Theor. Comput. Sci. 412(30), 3800–3821 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fischer, P.C.: Generation of primes by a one-dimensional real-time iterative array. J. ACM 12(3), 388–394 (1965)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gilbert, E.N.: Synchronization of binary messages. IRE Trans. Inf. Theory 6(4), 470–477 (1960)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Korec, I.: Real-time generation of primes by a one-dimensional cellular automaton with 11 states. In: Prívara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 358–367. Springer, Heidelberg (1997).  https://doi.org/10.1007/BFb0029979CrossRefMATHGoogle Scholar
  9. 9.
    Levenshteĭn, V.I.: The maximal number of words in codes without overlap. Problemy Peredachi Informatsii 6(4), 88–90 (1970)MathSciNetMATHGoogle Scholar
  10. 10.
    Litow, B., Dumas, P.: Additive cellular automata and algebraic series. Theor. Comput. Sci. 119(2), 345–354 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mazoyer, J., Terrier, V.: Signals in one-dimensional cellular automata. Theor. Comput. Sci. 217(1), 53–80 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Minsky, M., Papert, S.: Unrecognizable sets of numbers. J. ACM 13(2), 281–286 (1966)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ritchie, R.W.: Finite automata and the set of squares. J. ACM 10(4), 528–531 (1963)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rowland, E., Yassawi, R.: A characterization of \(p\)-automatic sequences as columns of linear cellular automata. Adv. Appl. Math. 63, 68–89 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55(3), 601–644 (1983)MathSciNetCrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  • Irène Marcovici
    • 1
  • Thomas Stoll
    • 1
  • Pierre-Adrien Tahay
    • 1
  1. 1.Université de Lorraine, CNRS, Inria, IECLNancyFrance

Personalised recommendations