Local Information in One-Dimensional Cellular Automata

  • Torbjørn Helvik
  • Kristian Lindgren
  • Mats G. Nordahl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3305)

Abstract

A local information measure for a one-dimensional lattice system is introduced, and applied to describe the dynamics of one-dimensional cellular automata.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Wolfram, S. (ed.): Theory and Application of Cellular Automata. World Scientific, Singapore (1986)Google Scholar
  2. 2.
    Shannon, C.E.: A mathematical theory of communication. Bell System Tech. J. 27, 379–423, 623–656 (1948)Google Scholar
  3. 3.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev (2) 106, 620–630 (1957)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Wheeler, J.: Information, Physics, Quantum: The Search for Links, in Complexity, Entropy and the Physics of Information. Addison-Wesley, Redwood City (1988)Google Scholar
  5. 5.
    Lloyd, S.: Computational capacity of the universe. Physical Review Letters 88 (2002) 237901Google Scholar
  6. 6.
    Bekenstein, J.: Black holes and entropy. Physical Review D 7, 2333–2346 (1973)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Eriksson, K.E., Lindgren, K.: Structural information in self-organizing systems. Physica Scripta 35, 388–397 (1987)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Eriksson, K.E., Lindgren, K., Månsson, B.Å.: Structure, Context, Complexity, and Organization. World Scientific, Singapore (1987)Google Scholar
  9. 9.
    Kaneko, K.: Lyapunov analysis and information flow in coupled map lattices. Physica D 23, 436–447 (1986)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Toffoli, T.: Information transport obeying the continuity equation. IBM J. Res. Develop. 32, 29–36 (1988)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Vastano, J.A., Swinney, H.L.: Information transport in spatiotemporal systems. Physical Review Letters 60, 1773–1776 (1988)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Schreiber, T.: Spatio-temporal structure in coupled map lattices: two-point correlations versus mutual information. Journal of Physics. A. 23, L393–L398 (1990)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Schreiber, T.: Measuring information transfer. Physical Review Letters 85, 461–464 (2000)CrossRefGoogle Scholar
  14. 14.
    Lindgren, K., Eriksson, A., Eriksson, K.E.: Flows of information in spatially extended chemical dynamics. To appear in Proceedings of ALife 9 (2004)Google Scholar
  15. 15.
    Eriksson, K.E., Lindgren, K., Nordahl, M.G.: Continuity of information flow in discrete reversible systems. Chalmers preprint (1993)Google Scholar
  16. 16.
    Grassberger, P.: Toward a quantitative theory of self-generated complexity. International Journal of Theoretical Physics 25, 907–938 (1986)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hanson, J.E., Crutchfield, J.P.: Computational mechanics of cellular automata: an example. Physica D 103, 169–189 (1997)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Wolfram, S.: Universality and complexity in cellular automata. Physica D 10, 1–35 (1984)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Grassberger, P.: Chaos and diffusion in deterministic cellular automata. Physica D 10, 52–58 (1984)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Eloranta, K., Nummelina, E.: The kink of cellular automaton Rule 18 performs a random walk. Journal of Statistical Physics 69, 1131–1136 (1992)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Khinchin, A.: Mathematical Foundations of Information Theory. Dover Publications Inc., New York (1957)MATHGoogle Scholar
  22. 22.
    Cover, T.M., Thomas, J.A.: Elements of information theory. Wiley Series in Telecommunications. John Wiley & Sons Inc., New York (1991)MATHCrossRefGoogle Scholar
  23. 23.
    Toffoli, T., Margolus, N.H.: Invertible cellular automata: a review. Physica D 45, 229–253 (1990)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Frisch, Hasslacher, B., Pomeau, Y.: Lattice-gas automata for the Navier-Stokes equation. Physical Review Letters 56, 1505–1508 (1986)CrossRefGoogle Scholar
  25. 25.
    Vichniac, G.: Simulating physics with cellular automata. Physica D 10, 96–115 (1984)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Takesue, S.: Reversible cellular automata and statistical mechanics. Physical Review Letters 59, 2499–2502 (1987)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Mathematical Systems Theory 3, 320–375 (1969)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Lindgren, K.: Correlations and random information in cellular automata. Complex Systems 1, 529–543 (1987)MATHMathSciNetGoogle Scholar
  29. 29.
    Lindgren, K., Nordahl, M.G.: Complexity measures and cellular automata. Complex Systems 2, 409–440 (1988)MATHMathSciNetGoogle Scholar
  30. 30.
    Helvik, T., Lindgren, K., Nordahl, M.: Local information in discrete lattice systems (forthcoming)Google Scholar
  31. 31.
    Richardson, D.: Tesselations with local transformations. Journal of Computer and Systems Sciences 5, 373–388 (1972)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Torbjørn Helvik
    • 1
    • 2
  • Kristian Lindgren
    • 2
  • Mats G. Nordahl
    • 3
  1. 1.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway
  2. 2.Department of Physical Resource TheoryChalmers University of Technology and Göteborg UniversityGöteborgSweden
  3. 3.Innovative DesignChalmers University of TechnologyGöteborgSweden

Personalised recommendations