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Some Undecidable Dynamical Properties for One-Dimensional Reversible Cellular Automata

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Algorithmic Bioprocesses

Part of the book series: Natural Computing Series ((NCS))

Abstract

Using the fact that the tiling problem of Wang tiles is undecidable even if the given tile set is deterministic by two opposite corners, it is shown that the question whether there exists a trajectory which belongs to the given open and closed set is undecidable for one-dimensional reversible cellular automata. This result holds even if the cellular automaton is mixing. Furthermore, it is shown that left expansivity of a reversible cellular automaton is an undecidable property. Also, the tile set construction gives yet another proof for the universality of one-dimensional reversible cellular automata.

J. Kari’s research supported by the Academy of Finland grant 211967.

V. Lukkarila’s research supported by the Finnish Cultural Foundation and the Fund of Vilho, Yrjö and Kalle Väisälä.

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Authors and Affiliations

  1. Department of Mathematics, University of Turku, 20014, Turku, Finland

    Jarkko Kari & Ville Lukkarila

  2. Turku Centre for Computer Science, 20520, Turku, Finland

    Jarkko Kari & Ville Lukkarila

Authors
  1. Jarkko Kari
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  2. Ville Lukkarila
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Corresponding author

Correspondence to Jarkko Kari .

Editor information

Editors and Affiliations

  1. Dept. Computer Science, University of British Columbia, Main Mall 201-2366, Vancouver, V6T 1Z4, Canada

    Anne Condon

  2. Dept. Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    David Harel

  3. Leiden Inst. Advanced Computer Science, Leiden University, Niels Bohrweg 1, Leiden, 2333 CA, Netherlands

    Joost N. Kok

  4. Turku Centre for Computer Science, Lemminkaisenkatu 14 A, Turku, 20520, Finland

    Arto Salomaa

  5. Computer Science, Computation,, California Inst. of Technology, Pasadena, 91125, U.S.A.

    Erik Winfree

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Kari, J., Lukkarila, V. (2009). Some Undecidable Dynamical Properties for One-Dimensional Reversible Cellular Automata. In: Condon, A., Harel, D., Kok, J., Salomaa, A., Winfree, E. (eds) Algorithmic Bioprocesses. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88869-7_32

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  • DOI: https://doi.org/10.1007/978-3-540-88869-7_32

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  • Print ISBN: 978-3-540-88868-0

  • Online ISBN: 978-3-540-88869-7

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