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Conjugacy of One-Dimensional One-Sided Cellular Automata is Undecidable

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SOFSEM 2018: Theory and Practice of Computer Science (SOFSEM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10706))

Abstract

Two cellular automata are strongly conjugate if there exists a shift-commuting conjugacy between them. We prove that the following two sets of pairs (F, G) of one-dimensional one-sided cellular automata over a full shift are recursively inseparable:

  1. (i)

    pairs where F has strictly larger topological entropy than G, and

  2. (ii)

    pairs that are strongly conjugate and have zero topological entropy.

Because there is no factor map from a lower entropy system to a higher entropy one, and there is no embedding of a higher entropy system into a lower entropy system, we also get as corollaries that the following decision problems are undecidable: Given two one-dimensional one-sided cellular automata F and G over a full shift: Are F and G conjugate? Is F a factor of G? Is F a subsystem of G? All of these are undecidable in both strong and weak variants (whether the homomorphism is required to commute with the shift or not, respectively). It also immediately follows that these results hold for one-dimensional two-sided cellular automata.

Research supported by the Academy of Finland Grant 296018.

J. Jalonen—Research supported by the Finnish Cultural Foundation.

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Authors and Affiliations

  1. University of Turku, Turku, Finland

    Joonatan Jalonen & Jarkko Kari

Authors
  1. Joonatan Jalonen
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  2. Jarkko Kari
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Corresponding author

Correspondence to Joonatan Jalonen .

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Editors and Affiliations

  1. Vienna University of Technology , Vienna, Austria

    A Min Tjoa

  2. ISAE-ENSMA, Chasseneuil-du-Poitou, France

    Ladjel Bellatreche

  3. Vienna University of Technology, Vienna, Austria

    Stefan Biffl

  4. Utrecht University, Utrecht, The Netherlands

    Jan van Leeuwen

  5. Academy of Sciences, Prague, Czech Republic

    Jiří Wiedermann

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Jalonen, J., Kari, J. (2018). Conjugacy of One-Dimensional One-Sided Cellular Automata is Undecidable. In: Tjoa, A., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds) SOFSEM 2018: Theory and Practice of Computer Science. SOFSEM 2018. Lecture Notes in Computer Science(), vol 10706. Edizioni della Normale, Cham. https://doi.org/10.1007/978-3-319-73117-9_16

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  • DOI: https://doi.org/10.1007/978-3-319-73117-9_16

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  • Publisher Name: Edizioni della Normale, Cham

  • Print ISBN: 978-3-319-73116-2

  • Online ISBN: 978-3-319-73117-9

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