Skip to main content

Complexity of Maxmin-\(\omega \) Cellular Automata

  • 2252 Accesses

Part of the Springer Proceedings in Complexity book series (SPCOM)


We present an analysis of an additive cellular automaton (CA) under asynchronous dynamics. The asynchronous scheme employed is maxmin-\(\omega \), a deterministic system, introduced in previous work with a binary alphabet. Extending this work, we study the impact of a varying alphabet size, i.e., more than the binary states often employed. Far from being a simple positive correlation between complexity and alphabet size, we show that there is an optimal region of \(\omega \) and alphabet size where complexity of CA is maximal. Thus, despite employing a fixed additive CA rule, the complexity of this CA can be controlled by \(\omega \) and alphabet size. The flavour of maxmin-\(\omega \) is, therefore, best captured by a CA with a large number of states.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-96661-8_10
  • Chapter length: 10 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   169.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-96661-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   219.99
Price excludes VAT (USA)
Hardcover Book
USD   249.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.


  1. Fatès, N.A., Morvan, M.: An experimental study of robustness to asynchronism for elementary cellular automata. Complex Syst. 16, 1–27 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Granovetter, M.: Threshold models of collective behavior. Am. J. Sociol. 83(6), 1420–1443 (1978)

    CrossRef  Google Scholar 

  3. Gutowitz, H., Langton, C.: Mean field theory of the edge of chaos. In: European Conference on Artificial Life, pp. 52–64. Springer, Heidelberg (1995)

    Google Scholar 

  4. Heidergott, B., Olsder, G.J., Van der Woude, J.: Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  5. Lopez-Ruiz, R., Mancini, H.L., Calbet, X.: A statistical measure of complexity. Phys. Lett. A 209(5–6), 321–326 (1995)

    ADS  CrossRef  Google Scholar 

  6. Marr, C., Hütt, M.T.: Topology regulates pattern formation capacity of binary cellular automata on graphs. Phys. A Stat. Mech. Appl. 354, 641–662 (2005)

    CrossRef  Google Scholar 

  7. McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 5(4), 115–133 (1943)

    MathSciNet  CrossRef  Google Scholar 

  8. Patel, E.L.: Maxmin-plus models of asynchronous computation. Ph.D. thesis, University of Manchester (2012)

    Google Scholar 

  9. Patel, E.L.: Maxmin-\(\omega \): a simple deterministic asynchronous cellular automaton scheme. In: International Conference on Cellular Automata, pp. 192–198. Springer, Cham (2016)

    Google Scholar 

  10. Rényi, A.: On measures of entropy and information. Technical report, Hungarian Academy of Sciences, Budapest (1961)

    Google Scholar 

  11. Schönfisch, B., de Roos, A.: Synchronous and asynchronous updating in cellular automata. BioSystems 51(3), 123–143 (1999)

    CrossRef  Google Scholar 

  12. Watts, D.J.: A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. 99(9), 5766–5771 (2002)

    ADS  MathSciNet  CrossRef  Google Scholar 

  13. Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601–644 (1983)

    ADS  MathSciNet  CrossRef  Google Scholar 

  14. Wolfram, S.: Universality and complexity in cellular automata. Phys. D Nonlinear Phenom. 10(1–2), 1–35 (1984)

    ADS  MathSciNet  CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ebrahim L. Patel .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Patel, E.L. (2018). Complexity of Maxmin-\(\omega \) Cellular Automata. In: Morales, A., Gershenson, C., Braha, D., Minai, A., Bar-Yam, Y. (eds) Unifying Themes in Complex Systems IX. ICCS 2018. Springer Proceedings in Complexity. Springer, Cham.

Download citation