Cellular automata, the collatz conjecture and powers of 3/2

  • Authors:

  • Jarkko Kari

  • Affiliations:

  • Department of Mathematics, University of Turku, Turku, Finland

  • Venue:
  • DLT'12 Proceedings of the 16th international conference on Developments in Language Theory
  • Year:
  • 2012

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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.