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Extensions In Reversible One-Dimensional Cellular Automata Are Equivalent With The Full Shift

Author

Listed:
  • JUAN CARLOS SECK TUOH MORA

    (Centro de Investigación Avanzada en Ingeniería Industrial, Universidad Autónoma del Estado de Hidalgo, Ciudad Universitaria, Carr. Pachuca-Tulancingo Km. 4.5, 42184 Pachuca, México)

  • MANUEL GONZÁLEZ HERNÁANDEZ

    (Centro de Investigación Avanzada en Ingeniería Industrial, Universidad Autónoma del Estado de Hidalgo, Ciudad Universitaria, Carr. Pachuca-Tulancingo Km. 4.5, 42184 Pachuca, México)

  • GENARO JUÁREZ MARTÍNEZ

    (Depto. de Ingeniería Eléctrica, Sección Computación, CINVESTAV-IPN, Av. IPN 2508, Col. San Pedro Zacatenco, 07300 DF, México)

  • SERGIO V. CHAPA VERGARA

    (Depto. de Ingeniería Eléctrica, Sección Computación, CINVESTAV-IPN, Av. IPN 2508, Col. San Pedro Zacatenco, 07300 DF, México)

Abstract

Cellular automata are discrete dynamical systems based on simple local interactions among its components, but sometimes they are able to yield quite a complex global behavior. A special kind of cellular automaton is the one where the global behavior is invertible, this type of cellular automaton is called reversible. In this paper we expose the graph representation provided by de Bruijn diagrams of reversible one-dimensional cellular automata and we define the distinct types of paths between self-loops in such diagrams. With this, we establish the way in which a reversible one-dimensional cellular automaton generates sequences composed by subsequences produced by the undefined repetition of a single state. Using this graph presentation, we define Welch diagrams which will be useful for proving that all the extensions of the ancestors in reversible one-dimensional cellular automata are equivalent to the full shift. In this way an important result of this paper is that we understand and classify the behavior of a reversible automaton analyzing the extensions of the ancestors of a given sequence by means of symbolic dynamics tools. A final example illustrates the results exposed in the paper.

Suggested Citation

  • Juan Carlos Seck Tuoh Mora & Manuel González Hernáandez & Genaro Juárez Martínez & Sergio V. Chapa Vergara, 2003. "Extensions In Reversible One-Dimensional Cellular Automata Are Equivalent With The Full Shift," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 14(08), pages 1143-1160.
    • Handle: RePEc:wsi:ijmpcx:v:14:y:2003:i:08:n:s012918310300525x
    DOI: 10.1142/S012918310300525X
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