Time in the Universe of Cellular Automata

  • Klaus MainzerEmail author
  • Leon Chua
Part of the SpringerBriefs in Complexity book series (BRIEFSCOMPLEXITY)


An examination of Figures (10–14) at the end of the last chapter shows that, except for period-k isle of Eden bit strings (Figs. 12, 14b), all attractors of the cellular automaton 62 have a non-empty basin of attraction with several gardens of Eden. Therefore, given any bit string on an attractor, it is impossible to retrace its dynamics in backward time to find where it had originated in the transient regime. Unlike in ordinary differential equations used in modeling dynamical systems, it is impossible, for most rules of cellular automata, to retrace its past history on the attractor. This observation leads us to exciting and deep insights in the concept of time with respect to the universe of cellular automata and physics (Chua et al. 2006; Mainzer 2002; Sachs 1987).


Cellular Automaton Modeling Dynamical System Local Rule Transient Regime Attractor Representation 
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  1. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Complex networks: structure and dynamics. Phys. Rep. 424(4–5), 175–308 (2006)MathSciNetADSCrossRefGoogle Scholar
  2. L.O. Chua, V.I. Sbitnev, S. Yoon, A nonlinear dynamics perspective of Wolfram’s new kind of science Part VI: from time-reversible attractors to the arrow of time. Int. J. Bifurcat. Chaos (IJBC) 16(5), 1097–1373 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. W. Feller, An Introduction to Probability Theory and Its Applications I (Wiley, New York, 1950)zbMATHGoogle Scholar
  4. J. Kari, Representation of reversible cellular automata with block permutation. Math. Syst. Theory 29(1), 47–61 (1996)MathSciNetzbMATHGoogle Scholar
  5. K. Mainzer, The Little Book of Time (Copernicus Books, New York, 2002)Google Scholar
  6. K. Morita, M. Harao, Computation universality of one-dimensional reversible (injective) cellular automata. Trans. IEICE E 72, 758–762 (1989)Google Scholar
  7. R.G. Sachs, The Physics of Time Reversal (University of Chicago, Chicago, 1987)Google Scholar
  8. T. Toffoli, Computation and construction universality of reversible cellular automata. J. Comput. Syst. Sci. 15, 213–231 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  9. S. Wolfram, A New Kind of Science (Wolfram Media, Champaign Il, 2002)zbMATHGoogle Scholar
  10. H.-D. Zeh, The Physical Basis of the Direction of Time, 5th edn. (Berlin, Springer, 2007)zbMATHGoogle Scholar

Copyright information

© Klaus Mainzer 2012

Authors and Affiliations

  1. 1.Technische Unviversität München, Lehrstuhl für Philosophie und WissenschaftstheorieMunichGermany
  2. 2.EECS DepartmentUniversity of CaliforniaBerkeleyUSA

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