Unfolding Complexity and Modelling Asymptotic Scaling Behavior

  • R. Badii
  • M. Finardi
  • G. Broggi
Part of the NATO ASI Series book series (NSSB, volume 280)


We discuss the problem of quantifying complexity in the framework of a hierarchical modelling of physical systems. The analysis is first performed on the set of all symbolic sequences which label non-empty regions in phase-space. A dynamical process represented by a shift map is associated with each admissible doubly-infinite sequence. The “grammatical” rules governing it are unfolded by using variable-length prefix-free codewords and described by means of allowed transitions on a “logic” tree. The derived model is employed to make predictions about the scaling behaviour of the system’s observables at each level of resolution. The complexity of the system, relative to the unfolding scheme, is evaluated through a generalization of the information gain by comparing prediction and observation. Rapidly converging estimates of thermodynamic averages can be obtained from the logic tree in the general incomplete-folding case using a transfer-matrix technique, related to the theory of scaling functions.


Nonlinear Dynamical System Topological Entropy Regular Language Logic Tree Symbolic Sequence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    (a) “Chaos and Complexity”, R. Livi, S. Ruffo, S. Ciliberto and M. Buiatti Eds., World Scientific, Singapore (1988); (b) “Measures of Complexity and Chaos”, N.B. Abraham, A. Albano, T. Passamante and P. Rapp Eds., Plenum, New York (1990).Google Scholar
  2. [2]
    J. von Neumann, “Theory of Self-Reproducing Automata”, A. Burks ed., University of Illinois Press, Urbana, Illinois (1966).Google Scholar
  3. [3]
    K. Binder and A.P. Young, Rev.Mod.Phys. 58, 801 (1986).ADSCrossRefGoogle Scholar
  4. [4]
    J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields”, Springer, New York (1986).Google Scholar
  5. [5]
    R. Badii, in [lb]; PSI Report 61, 1, Villigen, Switzerland (dy1990); Weizmann preprint, Rehovot, Israel (1988); R. Badii, M. Finardi and G. Broggi, in “Information Dynamics”, H. Atmanspacher Ed., Plenum, New York (1990).Google Scholar
  6. [6]
    V.M. Alekseev and M.V. Yakobson, Phys.Rep. 75, 287 (1981).MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    M.J. Feigenbaum, J. Stat. Phys. 52, 527 (1988).MathSciNetADSMATHCrossRefGoogle Scholar
  8. [8]
    D. Ruelle, “Thermodynamic Formalism”, Vol. 5 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, MA (1978).Google Scholar
  9. [9]
    R.J. Solomonoff, Inf. Control 7, 1 (1964); A.N. Kolmogorov, Probl. Inform. Transm. 1, 1 (1965); G. Chaitin, J. Assoc.Comp.Math. 13, 547 (1966).MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    J.E. Hopcroft and J.D. Ullman, “Introduction to Automata Theory, Languages and Computation”, Addison-Wesley, Reading, MA (1979).MATHGoogle Scholar
  11. [11]
    G. Rozenberg and A. Salomaa, “The Mathematical Theory of L Systems”, Academic Press, London (1980).MATHGoogle Scholar
  12. [12]
    J.P. Crutchfield, this issue.Google Scholar
  13. [13]
    R.M. Wharton, Inform. Contr. 26, 236 (1974).MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    B. Weiss, Monatshefte für Mathematik 77, 462 (1973).MATHCrossRefGoogle Scholar
  15. [15]
    P. Collet and J.P. Eckmann, “Iterated Maps of the Interval as Dynamical Systems”, Birkhauser, Cambridge, MA, (1980).Google Scholar
  16. [16]
    I. Procaccia, S. Thomae and C. Tresser, Phys.Rev. A35, 1884 (1987).MathSciNetADSGoogle Scholar
  17. [17]
    J.P. Crutchfield and K. Young in “Complexity, Entropy and Physics of Information”, W. Zurek Ed., Addison-Wesley, Reading, MA, (1989).Google Scholar
  18. [18]
    P. Grassberger, Wuppertal preprint B 89-26, (1989).Google Scholar
  19. [19]
    J.P. Crutchfield and K. Young, Phys.Rev.Lett. 63, 105 (1989).MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    D. Auerbach and I. Procaccia, Phys.Rev. A41, 6602 (1990).MathSciNetADSGoogle Scholar
  21. [21]
    S. Lloyd and H. Pagels, Ann. of Phys. 188, 186 (1988).MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    If system A is described by a number of rules which grows, for increasing resolution, more rapidly than that of system B, the overall descriptive effort is dominated by the properties of A. As a consequence, complexity can be meaningfully defined only in the limit of infinitely extended patterns and characterizes the scaling behaviour of the physical process: it must equal zero, in particular, for systems specified by a finite number of dynamical rules, in agreement with point 1.Google Scholar
  23. [23]
    G. D’Alessandro and A. Politi, Phys.Rev.Lett. 64, 1609 (1989).MathSciNetCrossRefGoogle Scholar
  24. [24]
    R. Hamming, “Coding and Information Theory”, Prentice-Hall, Englewood Cliffs, NJ (1986).MATHGoogle Scholar
  25. [25]
    D. Auerbach, P. Cvitanović, J.P. Eckmann, G. Gunaratne and I. Procaccia, Phys. Rev.Lett. 58, 2387 (1987); P. Cvitanović, Phys.Rev.Lett. 61, 2729 (1988); C. Grebogi, E. Ott and J.A. Yorke, Phys.Rev. A36, 3522 (1988) and Phys.Rev. A37, 1711 (1988).MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    R. Badii, Riv. Nuovo Cim. 12, N° 3, 1 (1989).MathSciNetCrossRefGoogle Scholar
  27. [27]
    R. Artuso, E. Aurell and P. Cvitanović, Niels Bohr Institute preprints NBI-89-41 and NBI-89-42.Google Scholar
  28. [28]
    P. Cvitanovič, in Proceedings of the Workshop in Condensed Matter, Atomic and Molecular Physics, Trieste, Italy (1986); D. Katzen and I. Procaccia, Phys.Rev.Lett. 58, 1169 (1987); P. Grassberger, R. Badii and A. Politi, J.Stat.Phys. 51, 135 (1988); G. Broggi and R. Badii, Phys.Rev. 39A, 434 (1989).Google Scholar
  29. [29]
    P. Paoli, A. Politi, G. Broggi, M. Ravani and R. Badii, Phys.Rev.Lett. 62, 2429 (1989); P. Paoli, A. Politi and R. Badii, Physica D36, 263 (1989).MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    P. Szépfalusy, T. Tel and G. Vattay, Eötvös preprint, Budapest (1990).Google Scholar
  31. [31]
    R. Badii and A. Politi, Phys. Rev. 35A, 1288 (1987); R. Badii and G. Broggi, Phys.Rev. 41A, 1165 (1990).MathSciNetADSGoogle Scholar
  32. [32]
    M.J. Feigenbaum, M.H. Jensen and I. Procaccia, Phys.Rev.Lett. 57, 1503 (1986).MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    M.A. Sepúveda and R. Badii in [1b]; R. Badii, M. Finardi and G. Broggi, PSI preprint, PSI-LUS-05 (1990).Google Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • R. Badii
    • 1
  • M. Finardi
    • 1
  • G. Broggi
    • 2
  1. 1.Paul Scherrer InstitutVilligenSwitzerland
  2. 2.Physik-Institut der UniversitätZurichSwitzerland

Personalised recommendations