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Spectral Properties Of Reversible One-Dimensional Cellular Automata

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  • 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 Pachuca, Hidalgo, México)

  • SERGIO V. CHAPA VERGARA

    (Departamento de Ingeniería Eléctrica, Sección Computación, CINVESTAV-IPN., Av IPN 2508, Col San Pedro Zacatenco, México 07360 D.F., México)

  • GENARO JUÁREZ MARTÍNEZ

    (Departamento de Ingeniería Eléctrica, Sección Computación, CINVESTAV-IPN., Av IPN 2508, Col San Pedro Zacatenco, México 07360 D.F., México)

  • HAROLD V. McINTOSH

    (Departamento de Aplicación de Microcomputadoras, Instituto de Ciencias, UAP, México)

Abstract

Reversible cellular automata are invertible dynamical systems characterized by discreteness, determinism and local interaction. This article studies the local behavior of reversible one-dimensional cellular automata by means of the spectral properties of their connectivity matrices. We use the transformation of every one-dimensional cellular automaton to another of neighborhood size 2 to generalize the results exposed in this paper. In particular we prove that the connectivity matrices have a single positive eigenvalue equal to 1; based on this result we also prove the idempotent behavior of these matrices. The significance of this property lies in the implementation of a matrix technique for detecting whether a one-dimensional cellular automaton is reversible or not. In particular, we present a procedure using the eigenvectors of these matrices to find the inverse rule of a given reversible one-dimensional cellular automaton. Finally illustrative examples are provided.

Suggested Citation

  • JUAN CARLOS SECK TUOH MORA & SERGIO V. CHAPA VERGARA & GENARO JUÁREZ MARTÍNEZ & HAROLD V. McINTOSH, 2003. "Spectral Properties Of Reversible One-Dimensional Cellular Automata," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 14(03), pages 379-395.
    • Handle: RePEc:wsi:ijmpcx:v:14:y:2003:i:03:n:s0129183103004541
    DOI: 10.1142/S0129183103004541
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