A Tight Linear Bound on the Neighborhood of Inverse Cellular Automata

  • Eugen Czeizler
  • Jarkko Kari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3580)


Reversible cellular automata (RCA) are models of massively parallel computation that preserve information. They consist of an array of identical finite state machines that change their states synchronously according to a local update rule. By selecting the update rule properly the system has been made information preserving, which means that any computation process can be traced back step-by-step using an inverse automaton. We investigate the maximum range in the array that a cell may need to see in order to determine its previous state. We provide a tight upper bound on this inverse neighborhood size in the one-dimensional case: we prove that in a RCA with n states the inverse neighborhood is not wider than n–1, when the neighborhood in the forward direction consists of two consecutive cells. Examples are known where range n–1 is needed, so the bound is tight. If the forward neighborhood consists of m consecutive cells then the same technique provides the upper bound n m − 1–1 for the inverse direction.


Cellular Automaton Cellular Automaton Neighborhood Range Consecutive Cell Minimal Neighborhood 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amoroso, S., Patt, Y.: Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures. Journal of Computer and System Sciences 6, 448–464 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Hedlund, G.: Endomorphisms and automorphisms of shift dynamical systems. Mathematical Systems Theory 3, 320–375 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kari, J.: On the Inverse Neighborhood of Reversible Cellular Automata. In: Rosenberg, G., Salomaa, A. (eds.) Lindenmayer Systems, Impact in Theoretical Computer Science, Computer Graphics and Developmental Biology, pp. 477–495. Springer, Heidenberg (1989)Google Scholar
  4. 4.
    Kari, J.: Reversibility and surjectivity problems of cellular automata. Journal of Computer and System Sciences 48, 149–182 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Margolus, N.: Physics-like models of computation. Physica D 10, 81–95 (1984)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Moore, E.F.: Machine Models of Self-reproduction. In: Proceedings of the Symposium in Applied Mathematics, vol. 14, pp. 17–33 (1962)Google Scholar
  7. 7.
    Morita, K., Harao, M.: Computation Universality of one-dimensional reversible (injective) cellular automata. IEICE Transactions E72, 758–762 (1989)Google Scholar
  8. 8.
    Myhill, J.: The Converse to Moore’s Garden-of-Eden Theorem. In: Proceedings of the American Mathematical Society, vol. 14, pp. 685–686 (1963)Google Scholar
  9. 9.
    Richardson, D.: Tessellations with Local Transformations. Journal of Computer and System Sciences 6, 373–388 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Sutner, K.: De Bruijn graphs and linear cellular automata. Complex Systems 5, 19–31 (1991)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Toffoli, T., Margolus, N.: Invertible cellular automata: a review. Physica D 45, 229–253 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Wolfram, S.: A New Kind of Science. Wolfram Media (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Eugen Czeizler
    • 1
    • 2
  • Jarkko Kari
    • 1
    • 3
  1. 1.Department of MathematicsUniversity of TurkuFinland
  2. 2.Turku Centre for Computer ScienceTurkuFinland
  3. 3.Department of Computer ScienceUniversity of IowaIowa CityUSA

Personalised recommendations