An 8-State Simple Reversible Triangular Cellular Automaton that Exhibits Complex Behavior

  • Kenichi MoritaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9664)


A three-neighbor triangular partitioned cellular automaton (TPCA) is a CA whose cell is triangular-shaped and divided into three parts. The next state of a cell is determined by the three adjacent parts of its neighbor cells. The framework of TPCA makes it easy to design reversible triangular CAs. Among them, isotropic 8-state (i.e., each part has two states) TPCAs, which are called elementary TPCAs (ETPCAs), are extremely simple, since each of their local transition functions is described by only four local rules. In this paper, we investigate a specific reversible ETPCA \(T_{0347}\), where 0347 is its identification number in the class of 256 ETPCAs. In spite of the simplicity of the local function and the constraint of reversibility, evolutions of configurations in \(T_{0347}\) have very rich varieties, and look like those in the Game-of-Life CA to some extent. In particular, a “glider” and “glider guns” exist in \(T_{0347}\). Furthermore, using gliders to represent signals, we can implement universal reversible logic gates in it. By this, computational universality of \(T_{0347}\) is derived.



This work was supported by JSPS KAKENHI Grant Number 15K00019.


  1. 1.
    Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berlekamp, E., Conway, J., Guy, R.: Winning Ways for Your Mathematical Plays, vol. 2. Academic Press, New York (1982)zbMATHGoogle Scholar
  3. 3.
    Fredkin, E., Toffoli, T.: Conservative logic. Int. J. Theoret. Phys. 21, 219–253 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gardner, M.: Mathematical games: The fantastic combinations of John Conway’s new solitaire game “life”. Sci. Am. 223(4), 120–123 (1970)CrossRefGoogle Scholar
  5. 5.
    Gardner, M.: Mathematical games: On cellular automata, self-reproduction, the Garden of Eden and the game “life”. Sci. Am. 224(2), 112–117 (1971)CrossRefGoogle Scholar
  6. 6.
    Imai, K., Morita, K.: A computation-universal two-dimensional 8-state triangular reversible cellular automaton. Theoret. Comput. Sci. 231, 181–191 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Margolus, N.: Physics-like model of computation. Physica D 10, 81–95 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Morita, K.: A simple construction method of a reversible finite automaton out of Fredkin gates, and its related problem. Trans. IEICE Japan E–73, 978–984 (1990)Google Scholar
  9. 9.
    Morita, K.: A simple universal logic element and cellular automata for reversible computing. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 102–113. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Morita, K.: A reversible elementary triangular partitioned cellular automaton that exhibits complex behavior (slides with simulation movies). Hiroshima University Institutional Repository (2016).
  11. 11.
    Morita, K., Harao, M.: Computation universality of one-dimensional reversible (injective) cellular automata. Trans. IEICE Japan E72, 758–762 (1989)Google Scholar
  12. 12.
    Morita, K., Ueno, S.: Computation-universal models of two-dimensional 16-state reversible cellular automata. IEICE Trans. Inf. Syst. E75–D, 141–147 (1992)Google Scholar
  13. 13.
    Wolfram, S.: Theory and Applications of Cellular Automata. World Scientific Publishing, Singapore (1986)zbMATHGoogle Scholar
  14. 14.
    Wolfram, S.: A New Kind of Science. Wolfram Media Inc, Champaign (2002)zbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  1. 1.Hiroshima UniversityHigashi-hiroshimaJapan

Personalised recommendations