Attractors of D-dimensional Linear Cellular Automata

  • Giovanni Manzini
  • Luciano Margara
Automata and Formal Languages I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1373)

Abstract

In this paper we study the asymptotic behavior of D-dimensional linear cellular automata over the ring Zm (D ≥ 1, m ≥ 2). In the first part of the paper we consider non-surjective cellular automata. We prove that, after a transient phase of length at most [log2, m], the evolution of a linear non-surjective cellular automata F takes place completely within a subspace YF. This result suggests that we can get valuable information on the long term behavior of F by studying its properties when restricted to YF. We prove that such study is possible by showing that the system (YF, F) is topologically conjugated to a linear cellular automata F* defined over a different ring Zm. In the second part of the paper, we study the attractor sets of linear cellular automata. Recently, Kurka [8] has shown that CA can be partitioned into five disjoint classes according to the structure of their attractors. We present a procedure for deciding the membership in Kurka's classes for any linear cellular automata. Our procedure requires only gcd computations involving the coefficients of the local rule associated to the cellular automata.

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References

  1. 1.
    F. Blanchard, P. Kurka, and A. Maass. Topological and measure-theoretic properties of one-dimensional cellular automata. Physica D, 103:86–99, 1997.CrossRefMathSciNetGoogle Scholar
  2. 2.
    G. Cattaneo, E. Formenti, G. Manzini, and L. Margara. Ergodicity and regularity for linear cellular automata over Z m. Theoretical Computer Science. To appear.Google Scholar
  3. 3.
    G. Cattaneo, E. Formenti, G. Manzini, and L. Margara. On ergodic linear cellular automata over Z m. In 14th Annual Symposium on Theoretical Aspects of Computer Science (STACS '97), pages 427–438. LNCS n. 1200, Springer Verlag, 1997.Google Scholar
  4. 4.
    R. L. Devaney. An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Reading, MA, USA, second edition, 1989.MATHGoogle Scholar
  5. 5.
    M. Garzon. Models of Massive Parallelism. EATCS Texts in Theoretical Computer Science. Springer Verlag, 1995.Google Scholar
  6. 6.
    M. Hurley. Attractors in cellular automata. Ergodic Theory and Dynamical Systems, 10:131–140, 1990.MATHMathSciNetGoogle Scholar
  7. 7.
    M. Ito, N. Osato, and M. Nasu. Linear cellular automata over Z m. Journal of Computer and System Sciences, 27:125–140, 1983.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Kurka. Languages, equicontinuity and attractors in cellular automata. Ergodic theory and dynamical systems, 17:417–433, 1997.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    G. Manzini and L. Margara. Invertible linear cellular automata over Z m: Algorithmic and dynamical aspects. Journal of Computer and System Sciences. To appear.Google Scholar
  10. 10.
    G. Manzini and L. Margara. A complete and efficiently computable topological classification of D-dimensional linear cellular automata over Zm. In 24th International Colloquium on Automata Languages and Programming (ICALP '97). LNCS n. 1256, Springer Verlag, 1997.Google Scholar
  11. 11.
    T. Sato. Ergodicity of linear cellular automata over Zm. Information Processing Letters, 61(3):169–172, 1997.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Giovanni Manzini
    • 1
    • 2
  • Luciano Margara
    • 3
    • 4
  1. 1.Dipartimento di Scienze e Tecnologie AvanzateUniversitá di TorinoAlessandriaItaly
  2. 2.Istituto di Matematica ComputazionalePisaItaly
  3. 3.Dipartimento di Scienze dell'InformazioneUniversitá di BolognaBolognaItaly
  4. 4.International Computer Science Institute (ICSI)Berkeley CA

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