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

  • Jarkko Kari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7410)

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.

Keywords

Cellular Automaton Product Topology Tiling Problem Universal Pattern Forward Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jarkko Kari
    • 1
  1. 1.Department of MathematicsUniversity of TurkuTurkuFinland

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