Inferring the Dynamic, Quantifying Physical Complexity

  • James P. Crutchfield
Part of the NATO ASI Series book series (NSSB, volume 208)

Summary

Through its formalization of inductive inference, computational learning theory provides a foundation for the inverse problem of chaotic data analysis: inferring the deterministic equations of motion underlying observed random behavior in physical systems. Integrating the geometric and statistical techniques of dynamical systems with learning theory provides a framework for consistently, although not absolutely, distinguishing between deterministic chaos and extrinsic fluctuations at a given level of computational resources. Two approaches to the inverse problem, estimating symbolic equations of motion and reconstructing minimal automata from chaotic data series, are reviewed from this point of view. With an inferred model dynamic the dynamical entropies and dimensions can be estimated. More interestingly, its structural properties give a measure of the intrinsic computational complexity of the underlying process.

Keywords

Inverse Problem Inference Method Inductive Inference Dynamical System Theory Deterministic Chaos 
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.

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References

  1. 1.
    J. P. Crutchfield, N. H. Packard, J. D. Fanner, and R. S. Shaw. Chaos. Sci. Am., 255: 46, 1986.Google Scholar
  2. 2.
    J. P. Crutchfield and B. S. McNamara. Equations of motion from a data series. Complex Systems, 1: 417, 1987.MathSciNetMATHGoogle Scholar
  3. 3.
    J. P. Crutchfield and K. Young. Inferring statistical complexity. Phys. Rev. Let, 63:10 July, 1989.MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. P. Crutchfield and K. Young. Computation at the onset of chaos. In W. Zurek, editor, Entropy, Complexity, and Physics of Information. Addison-Wesley, 1989. to appear.Google Scholar
  5. 5.
    J. P. Crutchfield and K. Young. Thermodynamics of minimal reconstructed machines. in preparation, 1989.Google Scholar
  6. 6.
    E. M. Gold. Language identification in the limit. Info. Control, 10: 447, 1967.MATHCrossRefGoogle Scholar
  7. 7.
    J. P. Crutchfield and N. H. Packard. Symbolic dynamics of noisy chaos. Physica, 7D: 201, 1983.MathSciNetGoogle Scholar
  8. 8.
    R. M. Wharton. Approximate language identification. Info. Control, 26: 236, 1974.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    E. M. Gold. Complexity of automaton identification from given data. Info. Control, 37: 302, 1978.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    D. Angluin. On the complexity of minimum inference of regular sets. Info. Control, 39: 337, 1978.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    J. P. Crutchfield. Noisy Chaos. PhD thesis, University of California, Santa Cruz, 1983. published by University Microfilms Intl, Minnesota.Google Scholar
  12. 12.
    N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw. Geometry from a time series. Phys. Rev. Let., 45: 712, 1980.CrossRefGoogle Scholar
  13. 13.
    J. Rissanen. Modeling by shortest data description. Automatica, 14: 462, 1978.CrossRefGoogle Scholar
  14. 14.
    A. W. Biermann and J. A. Feldman. On the synthesis of finite-state machines from samples of their behavior. IEEE Trans. Comp., C-21: 592, 1972.Google Scholar
  15. 15.
    N. H. Packard. Measurements of Chaos in the Presence of Noise. PhD thesis, University of California, Santa Cruz, 1982.Google Scholar
  16. 16.
    P. Gras sberger. Toward a quantitative theory of self-generated complexity. Intl. J. Theo. Phys., 25: 907, 1986.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    J. P. Crutchfield. Time is the ultrametric of causality. in preparation, 1989.Google Scholar
  18. 18.
    J. P. Crutchfield. Compressing chaos. in preparation, 1989.Google Scholar
  19. 19.
    J. Rissanen. Universal coding, information, prediction, and estimation. IEEE Trans. Info. Th., IT-30: 629, 1984.MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. P. Crutchfield and N. H. Packard. Symbolic dynamics of one-dimensional maps: Entropies, finite precision, and noise. Intl. J. Theo. Phys., 21: 433, 1982.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    S. Wolfram. Computation theory of cellular automata. Comm. Math. Phys., 96: 15, 1984.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    R. Shaw. The Dripping Faucet as a Model Chaotic System. Aerial Press, Santa Cruz, California, 1984.Google Scholar
  23. 23.
    C. H. Bennett. On the nature and origin of complexity in discrete, homogeneous locally-interacting systems. Found. Phys., 16: 585, 1986.MathSciNetCrossRefGoogle Scholar
  24. 24.
    C. P. Bachas and B.A. Huberman. Complexity and relaxation of hierarchical structures. Phys. Rev. Let., 57: 1965, 1986.MathSciNetCrossRefGoogle Scholar
  25. 25.
    S. Lloyd and H. Pagels. Complexity as thermodynamic depth. Ann. Phys., 188: 186, 1988.MathSciNetCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • James P. Crutchfield
    • 1
  1. 1.Physics DepartmentUniversity of CaliforniaBerkeleyUSA

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