On the Dynamics of Some Exceptional Fuzzy Cellular Automata

  • Darcy Dunne
  • Angelo B. Mingarelli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4173)

Abstract

Over the past 20 years, the study of cellular automata has emerged as one of the most interesting and popular forms of “new mathematics”. The study of cellular automata has broadened into many variations of the original concepts. One such variation is the study of one-dimensional fuzzy cellular automata. The evolution and dynamics of the majority of one-dimensional fuzzy cellular automata rules can be determined analytically using techniques devised by the second author. It turns out that only 9 rules (out of 256), three of which are trivial, fail to comply with the techniques given. We give a brief overview of finite cellular automata and their fuzzification. We summarize the method used to study the majority of fuzzy rules and give some examples of its application. We analyze and uncover the dynamics of those few rules which do not conform to such techniques. Using new techniques, combined with direct analysis, we determine the long term evolution of the 4 remaining rules (since two of them were treated in detail elsewhere). We specifically analyze rules 172 and 202 and then, by deriving equivalences to the final two rules, we complete the program, initiated in 2003, of determining the long term dynamics of all 256 one-dimensional fuzzy cellular automata, thereby showing that chaotic dynamics are incompatible with this type of fuzziness, in sharp contrast with boolean cellular automata.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Flocchini, P., Geurts, F., Mingarelli, A.B., Santoro, N.: Convergence and Aperiodicity in Fuzzy Cellular Automata - Revisiting Rule 90. Physica D 42, 20–28 (2000)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Ganguly, N., Sikdar, B., Deutsch, A., Canright, G., Chaudhuri, P.: A Survey on Cellular Automata. Preprint; Center for High Performance Computing, Dresden University of Technology (2003)Google Scholar
  3. 3.
    Mingarelli, A.B., Beres, E.: The dynamics of general fuzzy cellular automata: Rule 30. WSEAS Trans. Circuits and Systems 10(3), 2211–2216 (2004)Google Scholar
  4. 4.
    Mingarelli, A.B., El-Yacoubi, S.: On the decidability of the evolution of the fuzzy cellular automaton, FCA 184. In: ICCS 2006, Reading, UK, Lecture Notes in Computer Science (to appear, 2006)Google Scholar
  5. 5.
    Mingarelli, A.B.: The global evolution of general fuzzy cellular automata. Journal of Cellular Automata (to appear, 2006)Google Scholar
  6. 6.
    Mingarelli, A.B.: The dynamics of general fuzzy cellular automata. In: Sunderam, V.S., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2005. LNCS, vol. 3515, pp. 351–359. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Mingarelli, A.B.: Fuzzy rule 110 dynamics and the golden number. WSEAS Trans. Computers 2(4), 1102–1107 (2003)Google Scholar
  8. 8.
    Reiter, C.A.: Fuzzy automata and life. Complexity 7(3), 19–29 (2002)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Von Neumann, J.: Theory of Self-Reproducing Automata. University of Illinois Press, Urbana (1966)Google Scholar
  10. 10.
    Wall, H.S.: Analytic Theory of Continued Fractions. Chelsea Publ., New York (1948)MATHGoogle Scholar
  11. 11.
    Wolfram, S.: A New Kind of Science. Wolfram Media Inc. (2002)Google Scholar
  12. 12.
    Wolfram, S.: Cellular Automata and Complexity: Collected Papers. Addison-Wesley Publishing, Reading (1994)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Darcy Dunne
    • 1
  • Angelo B. Mingarelli
    • 1
  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

Personalised recommendations