A Characterization of Cellular Automata Generated by Idempotents on the Full Shift

  • Ville Salo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)


In this article, we discuss the family of cellular automata generated by so-called idempotent cellular automata (CA G such that G 2 = G) on the full shift. We prove a characterization of products of idempotent CA, and show examples of CA which are not easy to directly decompose into a product of idempotents, but which are trivially seen to satisfy the conditions of the characterization. Our proof uses ideas similar to those used in the well-known Embedding Theorem and Lower Entropy Factor Theorem in symbolic dynamics. We also consider some natural decidability questions for the class of products of idempotent CA.


Cellular Automaton Periodic Point Symbolic Dynamic Spreading State Decidability Question 
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.N.: Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. Journal of Computer and System Sciences 6(5), 448–464 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Boyle, M.: Lower entropy factors of sofic systems. Ergodic Theory Dynam. Systems 3(4), 541–557 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Boyle, M., Lind, D., Rudolph, D.: The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306(1), 71–114 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Hurd, L.P., Kari, J., Culik, K.: he topological entropy of cellular automata is uncomputable. Ergodic Theory Dynam. Systems 12(2), 255–265 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Kari, J.: The nilpotency problem of one-dimensional cellular automata. SIAM J. Comput. 21(3), 571–586 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Kari, J.: Theory of cellular automata: a survey. Theor. Comput. Sci. 334, 3–33 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995)zbMATHCrossRefGoogle Scholar
  8. 8.
    Maass, R.: On the sofic limit sets of cellular automata. Ergodic Theory and Dynamical Systems 15 (1995)Google Scholar
  9. 9.
    Moore, E.F.: Machine models of self-reproduction. In: Proc. Symp. Applied Mathematics, vol. 14, p. 187–203 (1962)Google Scholar
  10. 10.
    Myhill, J.: The converse of moore’s garden-of-eden theorem. Proceedings of the American Mathematical Society 14, 685–686 (1963)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ville Salo
    • 1
  1. 1.University of TurkuFinland

Personalised recommendations