Advertisement

Communications in Mathematical Physics

, Volume 93, Issue 2, pp 219–258 | Cite as

Algebraic properties of cellular automata

  • Olivier Martin
  • Andrew M. Odlyzko
  • Stephen Wolfram
Article

Abstract

Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata. The complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities. The systems are usually irreversible, and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations.

Keywords

Neural Network Dynamical System Statistical Physic State Transition Complex System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys.55, 601 (1983)Google Scholar
  2. 2.
    Golomb, S.W.: Shift register sequences. San Francisco: Holden-Day 1967Google Scholar
  3. 3.
    Selmer, E.S.: Linear recurrence relations over finite fields. Dept. of Math., Univ. of Bergen, Norway (1966)Google Scholar
  4. 4.
    Miller, J.C.P.: Periodic forests of stunted trees. Philos. Trans. R. Soc. Lond. A266, 63 (1970); A293, 48 (1980)Google Scholar
  5. 4a.
    ApSimon, H.G.: Periodic forests whose largest clearings are of size 3. Philos. Trans. R. Soc. Lond. A266, 113 (1970)Google Scholar
  6. 4b.
    ApSimon, H.G.: Periodic forests whose largest clearings are of sizen≧4. Proc. R. Soc. Lond. A319, 399 (1970)Google Scholar
  7. 4c.
    Sutton, C.: Forests and numbers and thinking backwards. New Sci.90, 209 (1981)Google Scholar
  8. 5.
    Moore, E.F.: Machine models of self-reproduction. Proc. Symp. Appl. Math.14, 17 (1962) reprinted in: Essays on cellular automata, A. W. Burks. Univ. of Illinois Press (1966)Google Scholar
  9. 5a.
    Aggarwal, S.: Local and global Garden of Eden theorems. Michigan University technical rept. 147 (1973)Google Scholar
  10. 6.
    Knuth, D.: Fundamental algorithms, Reading, MA: Addison-Wesley 1968.Google Scholar
  11. 7.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford: Oxford University Press 1968Google Scholar
  12. 8.
    Mac Williams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. Amsterdam: North-Holland 1977Google Scholar
  13. 9.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978Google Scholar
  14. 10.
    Fredkin, E., Margolus, N.: Private communicationsGoogle Scholar
  15. 11.
    Ronse, C.: Non-linear shift registers: A survey. MBLE Research Lab. report, Brussels (May 1980)Google Scholar
  16. 12.
    Harao, M., Noguchi, S.: On some dynamical properties of finite cellular automaton. IEEE Trans. Comp. C-27, 42 (1978)Google Scholar
  17. 13.
    Grassberger, P.: A new mechanism for deterministic diffusion. Phys. Rev. A (to be published)Google Scholar
  18. 14.
    Guibas, L.J., Odlyzko, A.M.: String overlaps, pattern matching, and nontransitive games. J. Comb. Theory (A)30, 83 (1981)Google Scholar
  19. 15.
    Knuth, D.: Seminumerical algorithms. 2nd ed. Reading, MA: Addison-Wesley 1981Google Scholar
  20. 15a.
    Gelfand, A.E.: On the cyclic behavior of random transformations on a finite set. Tech. rept. 305, Dept. of Statistics, Stanford Univ. (August 1981)Google Scholar
  21. 16.
    Odlyzko, A.M.: UnpublishedGoogle Scholar
  22. 17.
    Lind, D.A.: Applications of ergodic theory and sofic systems to cellular automata. Physica D10 (to be published)Google Scholar
  23. 18.
    Wolfram, S.: Computation theory of cellular automata. Institute for Advanced Study preprint (January 1984)Google Scholar
  24. 19.
    Lenstra, H.W., Jr.: Private communicationGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Olivier Martin
    • 1
  • Andrew M. Odlyzko
    • 2
  • Stephen Wolfram
    • 2
    • 3
  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.Bell LaboratoriesMurray HillUSA
  3. 3.The Institute for Advanced StudyPrincetonUSA

Personalised recommendations