Iterative devices generating infinite words

  • Karel CulikII
  • Juhani Karhumäki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)


We consider various TAG-like devices that generate one-way infinite words in real time. The simplest types of these devices are equivalent to iterative morphisms (also called substitutions), automatic sequences and iterative DGSM's. We consider also a few new types. Mainly we study the comparative power of these mechanisms and develop some techniques for proving that certain devices cannot produce a particular infinite word.


Turing Machine Infinite Word Generate Tape Finite Word Inside Control 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. P. Allouche, Finite Automata in 1-D and 2-D Physics, Number Theory & Physics, ed. by J. M. Luck, P. Moussa and M. Walschmidt, in: Springer Proceedings in Physics 47, Springer-Verlag (1990).Google Scholar
  2. [2]
    J. P. Allouche, J. Betrema and J. O. Shallit, Sur des Points Fixes de Morphismes d'un Monoide Libre, R.A.I.R.O., Informatique theorique et Applications 23, 3, 235–249 (1989).Google Scholar
  3. [3]
    J. P. Allouche and M. Mendes-France, Quasi-Crystal Using Chain and Automata Theory, J. Stat. Phys 42, 5/6 (1986).Google Scholar
  4. [4]
    J. M. Autebert and J. Gabarro, Iterated GSM's and Co-CFL, Acta Informatica 26, 749–769 (1989).Google Scholar
  5. [5]
    F. Axel, J. P. Allouche, M. Kleman, M. Mendes-France and J. Peyriere, Vibrational Modes in a One Dimensional ”Quasi-Alloy”: The Morse Case, Journal de Physique, Colloque C3, Suppl. 7, Tome 47 (1986).Google Scholar
  6. [6]
    J. Berstel, Properties of Infinite Words: Recent results, in: Lecture Notes in Computer Science 349, Springer-Verlag, New York (1989).Google Scholar
  7. [7]
    A. Cobham, Uniform tag Sequences, Math. Systems Theory 6, 164–192 (1972).Google Scholar
  8. [8]
    K. Culik II, Homomorphisms: Decidability, Equality and Test Sets, in: R. Book (ed.), Formal Language Theory, Perspectives and Open Problems, Academic Press, New York (1980).Google Scholar
  9. [9]
    K. Culik II and S. Dube, Balancing Order and Chaos in Image Generation, to appear in: ICALP Proceedings, Madrid (1991).Google Scholar
  10. [10]
    K. Culik II and T. Harju, The Ω-Sequence Equivalence Problem for D0L systems is Decidable, JACM 31, 277–298 (1984).Google Scholar
  11. [11]
    K. Culik II and J. Karhumäki, Iterative Devices Generating Infinite words, Tech. Report TR 9106, Univ. of South Carolina (1991).Google Scholar
  12. [12]
    K. Culik II, J. Karhumäki and A. Lepistö, Alternating Iteration of Morphisms and the Kolakovski Sequence, in: G. Rozenberg and A. Salomaa (eds.), A memorial volume for A. Lindenmayer, Springer-Verlarg (to appear).Google Scholar
  13. [13]
    K. Culik II and A. Salomaa, On Infinite Words Obtained by Iterating Morphisms, Theoret. Comput. Sci. 19, 29–38 (1982).Google Scholar
  14. [14]
    F. M. Dekking, Regularity and Irregularity of Sequences Generated by Automata, Sem. Th. de Nombres de Bordeaux, exp. 9 (1979–1980).Google Scholar
  15. [15]
    F. M. Dekking, On the Structure of Selfgenerating Sequences, Sem. Th. de Nombres de Bordeaux, (1980–1981).Google Scholar
  16. [16]
    A. Ehrenfeucht, K. P. Lee and G. Rozenberg, Subword Complexities of Various Classes of Deterministic Developmental Languages Without Interactions, Theoret. Comput. Sci. 1, 59–76 (1975).Google Scholar
  17. [17]
    P. C. Fischer, A. A. Meyer and A. L. Rosenberg, Time-restricted Sequence Generation, J. Comput. System Sci. 4, 50–73 (1970).Google Scholar
  18. [18]
    T. Harju and M. Linna, On the Periodicity of Morphisms in Free Monoids, R.A.I.R.O., Theoret. Inform. Appl. 20, 47–54 (1986).Google Scholar
  19. [19]
    J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, MA (1979).Google Scholar
  20. [20]
    W. Kolakovski, Self Generating Runs, problem 5304, American Math. Monthly 71 (1965), solution by N. Ucoluk, same journal 73, 681–682 (1966).Google Scholar
  21. [21]
    J. Karhumäki, Two Theorems Concerning Recognizable N-subsets of δ*. Theoretical Computer Science 1, 317–323 (1976).Google Scholar
  22. [22]
    C. Kimberling, Problem 6281*, Amer. Math. Monthly 86, 793 (1979).Google Scholar
  23. [23]
    D. Knuth, Solution to Problem E 2307, Amer. Math. Monthly 79, 773–774 (1972).Google Scholar
  24. [24]
    M. L. Minsky, Computation: Finite and Infinite Machines, Prentice-Hall, Englewood Cliffs, N.J. (1967).Google Scholar
  25. [25]
    J. J. Pansiot, Decidability of Periodicity for Infinite Words, R.A.I.R.O., Theoret. Inform. Appl. 20, 43–46 (1986).Google Scholar
  26. [26]
    P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, Berlin (1990).Google Scholar
  27. [27]
    G. Rozenberg and A. Salomaa, The Mathematicai Theory of L-Systems, Academic press, New York (1980).Google Scholar
  28. [28]
    A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer-Verlag, New York (1978).Google Scholar
  29. [29]
    J. Shallit, A Generalization of Automatic Sequences, Theoretical Computer Science 61, 1–16 (1988).Google Scholar
  30. [30]
    J. Shallit, Open Problem on the Kolakovski Sequence, (private communication).Google Scholar
  31. [31]
    A. R. Smith III, Plants, Fractals, and Formal Languages, Computer Graphics 18, 1–10 (1984).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Karel CulikII
    • 1
  • Juhani Karhumäki
    • 2
  1. 1.Dept.of Computer ScienceUniversity of South CarolinaColumbiaUSA
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland

Personalised recommendations