Advertisement

International Journal of Theoretical Physics

, Volume 25, Issue 9, pp 907–938 | Cite as

Toward a quantitative theory of self-generated complexity

  • Peter Grassberger
Article

Abstract

Quantities are defined operationally which qualify as measures of complexity of patterns arising in physical situations. Their main features, distinguishing them from previously used quantities, are the following: (1) they are measuretheoretic concepts, more closely related to Shannon entropy than to computational complexity; and (2) they are observables related to ensembles of patterns, not to individual patterns. Indeed, they are essentially Shannon information needed to specify not individual patterns, but either measure-theoretic or algebraic properties of ensembles of patterns arising ina priori translationally invariant situations. Numerical estimates of these complexities are given for several examples of patterns created by maps and by cellular automata.

Keywords

Entropy Field Theory Elementary Particle Quantum Field Theory Computational Complexity 
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. Alekseev, V, M., and Yakobson, M. V. (1981).Physics Reports,75, 287.Google Scholar
  2. Allouche, J.-P., and Cosnard, M. (1984). Grenoble preprint.Google Scholar
  3. Block, L., et al. (1980). Periodic points and topological entropy of 1-dimensional maps, inLecture Notes in Mathematics, No. 819, Springer, Berlin, 1980, p. 18.Google Scholar
  4. Chaitin, G. J. (1979). Toward a mathematical definition of ‘life’, inThe Maximum Entropy Principle, R. D. Levine and M. Tribus, eds., MIT Press, Cambridge, Massachusetts.Google Scholar
  5. Christol, G., Kamae, T., Mendes France, M., and Rauzy, G. (1980).Bulletin Societé Mathematique France,108, 401.Google Scholar
  6. Collet, P., and Eckmann, J.-P. (1980).Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Boston.Google Scholar
  7. Crutchfield, J. P., and Packard, N. H. (1983).Physica,7D, 201.Google Scholar
  8. Dias de Deus, J., Dilao, R., and Noronha da Costa, A. (1984). Lissabon preprint.Google Scholar
  9. Eckmann, J. P., and Ruelle, D. (1985).Review of Modern Physics,57, 617.Google Scholar
  10. Feigenbaum, M. (1978).Journal of Statistical Physics,19, 25.Google Scholar
  11. Feigenbaum, M. (1979).Journal of Statistical Physics,21, 669.Google Scholar
  12. Grassberger, P. (1984).Physica,10D, 52.Google Scholar
  13. Grassberger, P., and Kantz, H. (1985).Physics Letters,113A, 235.Google Scholar
  14. Grossmann, S., and Thomae, S. (1977).Zeitschrift für Naturforschung,32a, 1353.Google Scholar
  15. Guckenheimer, J., and Holmes, P. (1983).Non-linear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York.Google Scholar
  16. Györgyi, G., and Szepfalusy, P. (1985).Physical Review A,31, 3477; and to be published.Google Scholar
  17. Hofstadter, D. R. (1979).Gödel, Escher, Bach, Vintage Books, New York.Google Scholar
  18. Hogg, T., and Huberman, B. A. (1985). Order, complexity, and disorder, Palo Alto preprint,Google Scholar
  19. Hopcroft, J. E. and Ullman, J. D. (1979).Introduction to Automata Theory, Lanaguages, and Computation, Addison-Wesley.Google Scholar
  20. Martin, O., Odlyzko, A., and Wolfram, S. (1984).Communication in Mathematical Physics,93, 219.Google Scholar
  21. Packard, N. (1983). Complexity of growing patterns in cellular automata, Institute of Advanced Study preprint.Google Scholar
  22. Schuster, H. G. (1984).Deterministic Chaos, Physik-Verlag, Weinheim, West Germany.Google Scholar
  23. Shannon, C. E., and Weaver, W. (1949).The Mathematical Theory of Communication, University of Illinois Press, Urbana, Illinois.Google Scholar
  24. Sinai, Ya. (1985).Commentarii Mathematici Helvetici,60, 173.Google Scholar
  25. Van Emden, M. H. (1975).An Analysis of Complexity, Mathematical Centre Tracts, Amsterdam.Google Scholar
  26. Wagoner, S. (1985). Is pi normal;,Mathematical Intelligencer,7, 65.Google Scholar
  27. Wolfram, S. (1983).Review of Modern Physics,55, 601 (1983).Google Scholar
  28. Wolfram, S. (1984a).Physica,10D, 1.Google Scholar
  29. Wolfram, S. (1984b).Communications in Mathematical Physics,96, 15.Google Scholar
  30. Wolfram, S. (1985). Random sequence generation by cellular automata, Institute for Advanced Study preprint.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • Peter Grassberger
    • 1
  1. 1.Physics DepartmentUniversity of WuppertalGauss-Strasse 20West Germany

Personalised recommendations