, Volume 88, Issue 3–4, pp 193–205 | Cite as

Elementary cellular automaton Rule 110 explained as a block substitution system

Rule 110 as a block substitution system
  • Juan C. Seck-Tuoh-Mora
  • Genaro J. Martínez
  • Norberto Hernández-Romero
  • Joselito Medina-Marín


This paper presents the characterization of Rule 110 as a block substitution system of three symbols. Firstly, it is proved that the dynamics of Rule 110 is equivalent to cover the evolution space with triangles formed by the cells of the automaton. It is hence demonstrated that every finite configuration can be partitioned in several blocks of symbols and, that the dynamics of Rule 110 can be reproduced by a set of production rules applied to them. The shape of the blocks in the current configuration can be used for knowing the number of them in the next one; with this, the evolution of random configurations, ether and gliders can be modeled.


Rule 110 Elementary cellular automata Block substitution system Production rules 

Mathematics Subject Classification (2000)

68Q80 37B15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wolfram S (1986) Theory and applications of cellular automata. World Scientific Press, SingaporeMATHGoogle Scholar
  2. 2.
    Cook M (2004) Universality in elementary cellular automata. Complex Syst 15: 1–40MATHGoogle Scholar
  3. 3.
    Neary T, Woods D (2006) P-completeness of cellular automaton Rule 110. In: International colloquium on automata languages and programming (ICALP). LNCS, vol 4051, pp 132–143Google Scholar
  4. 4.
    Richard G (2008) Rule 110: universality and catenations. In: Durand B (ed) Proceedings of the first symposium on cellular automata “Journées Automates Cellulaires”, pp 141–160Google Scholar
  5. 5.
    Martínez G, McIntosh H, Seck-Tuoh-Mora J, Chapa-Vergara S (2008) Determining a regular language by glider-based structures called phases fi_1 in rule 110. J Cell Autom 3: 231–270MATHMathSciNetGoogle Scholar
  6. 6.
    McIntosh H (1999) Rule 110 as it relates to the presence of gliders. Accessed 15 Aug 2009
  7. 7.
    Martínez G, McIntosh H (2001) ATLAS: collisions of gliders like phases of ether in Rule 110. Accessed 16 Feb 2010
  8. 8.
    Wolfram S (2002) A new kind of science. Wolfram Media, ChampaignMATHGoogle Scholar
  9. 9.
    Kari J (1996) Representation of reversible cellular automata with block permutations. Math Syst Theory 29: 47–61MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Juan C. Seck-Tuoh-Mora
    • 1
  • Genaro J. Martínez
    • 2
  • Norberto Hernández-Romero
    • 1
  • Joselito Medina-Marín
    • 1
  1. 1.Centro de Investigación Avanzada en Ingeniería IndustrialUniversidad Autónoma del Estado de HidalgoPachucaMéxico
  2. 2.Ciudad Universitaria, CoyoacánMéxicoMéxico

Personalised recommendations