Universality of Reversible Hexagonal Cellular Automata

Issue		RAIRO-Theor. Inf. Appl. Volume 33, Number 6, November-December 1999


Page(s)		535 - 550
DOI		10.1051/ita:1999131

DOI: 10.1051/ita:1999131

Theoret. Informatics Appl. 33 (1999) 535-550

Universality of Reversible Hexagonal Cellular Automata

Kenichi Morita
Hiroshima University, Faculty of Engineering, Higashi-Hiroshima 739-8527, Japan; (morita@ke.sys.hiroshima-u.ac.jp)

Maurice Margenstern
G.I.F.M., Université de Metz, I.U.T. de Metz, Départment d'Informatique, Île du Saulcy, 57045 Metz Cedex 1, France; (margens@iut.univ-metz.fr)

Katsunobu Imai
Hiroshima University, Faculty of Engineering, Higashi-Hiroshima 739-8527, Japan; (imai@ke.sys.hiroshima-u.ac.jp)

Received March 18, 1999. Accepted November 12, 1999

Abstract: We define a kind of cellular automaton called a hexagonal partitioned cellular automaton (HPCA), and study logical universality of a reversible HPCA. We give a specific 64-state reversible HPCA H₁, and show that a Fredkin gate can be embedded in this cellular space. Since a Fredkin gate is known to be a universal logic element, logical universality of H₁ is concluded. Although the number of states of H₁ is greater than those of the previous models of reversible CAs having universality, the size of the configuration realizing a Fredkin gate is greatly reduced, and its local transition function is still simple. Comparison with the previous models, and open problems related to these model are also discussed.

Keywords and phrases: Cellular automata, reversibility, universality.

AMS Subject Classification: 68Q80, 68Q05

Communicated by: Ch. Choffrut

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