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Cellular Automata, the Collatz Conjecture and Powers of 3/2

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Developments in Language Theory (DLT 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7410))

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Abstract

We discuss one-dimensional reversible cellular automata F ×3 and F ×3/2 that multiply numbers by 3 and 3/2, respectively, in base 6. They have the property that the orbits of all non-uniform 0-finite configurations contain as factors all finite words over the state alphabet {0,1,…,5}. Multiplication by 3/2 is conjectured to even have an orbit of 0-finite configurations that is dense in the usual product topology. An open problem by K. Mahler about Z-numbers has a natural interpretation in terms the automaton F ×3/2. We also remark that the automaton F ×3 that multiplies by 3 can be slightly modified to simulate the Collatz function. We state several open problems concerning pattern generation by cellular automata.

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Authors and Affiliations

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

    Jarkko Kari

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  1. Jarkko Kari
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Editors and Affiliations

  1. Deptartment of Electrical Engineering, National Taiwan University, 106, Taipei, Taiwan

    Hsu-Chun Yen

  2. Department of Computer Science, University of California, 93106, Santa Barbara, CA, USA

    Oscar H. Ibarra

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© 2012 Springer-Verlag Berlin Heidelberg

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Kari, J. (2012). Cellular Automata, the Collatz Conjecture and Powers of 3/2. In: Yen, HC., Ibarra, O.H. (eds) Developments in Language Theory. DLT 2012. Lecture Notes in Computer Science, vol 7410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31653-1_5

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  • DOI: https://doi.org/10.1007/978-3-642-31653-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31652-4

  • Online ISBN: 978-3-642-31653-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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