A 6-state Universal Semi-totalistic Cellular Automaton on Kite and Dart Penrose Tilings

In this paper we investigate certain properties of semi-totalistic cellular automata CA on the well known quasi-periodic kite and dart two dimensional tiling of the plane presented by Roger Penrose. We show that, despite the irregularity of the underlying grid, it is possible to devise a 6-state semi-totalistic CA capable of simulating any boolean circuit and any Turing machine on this aperiodic tiling.

[1]  Ferdinand Peper,et al.  An Asynchronous Cellular Automaton Implementing 2-State 2-Input 2-Output Reversed-Twin Reversible Elements , 2008, ACRI.

[2]  Katsunobu Imai,et al.  A Universal Semi-totalistic Cellular Automaton on Kite and Dart Penrose Tilings , 2012, AUTOMATA & JAC.

[3]  森田 憲一.  Constructing a reversible Turing machine by a rotary element, a reversible logic element with memory , 2014 .

[4]  Adam P. Goucher.  Gliders in Cellular Automata on Penrose Tilings , 2012, J. Cell. Autom..

[5]  Susan Stepney,et al.  The Game of Life Rules on Penrose Tilings: Still Life and Oscillators , 2010, Game of Life Cellular Automata.

[6]  Mark McClure.  A stochastic cellular automaton for three-coloring penrose tiles , 2002, Comput. Graph..

[7]  Clifford A. Reiter.  Medley of spirals from cyclic cellular automata , 2010, Comput. Graph..

[8]  Master Gardener,et al.  Mathematical games: the fantastic combinations of john conway's new solitaire game "life , 1970 .

[9]  Katsunobu Imai,et al.  A Five-State von Neumann Neighbor Universal Hyperbolic Cellular Automaton , 2006, J. Cell. Autom..

[10]  Robert Fisch,et al.  Cyclic cellular automata and related processes , 1990 .

[11]  Artiom Alhazov,et al.  Non-degenerate 2-State Reversible Logic Elements with Three or More Symbols Are All Universal , 2012, J. Multiple Valued Log. Soft Comput..

[12]  Clifford A. Reiter,et al.  A local cellular model for growth on quasicrystals , 2005 .

[13]  Susan Stepney,et al.  Investigations of Game of Life Cellular Automata Rules on Penrose Tilings: Lifetime, Ash, and Oscillator Statistics , 2010, J. Cell. Autom..