Measures of Complexity and Chaos pp 327-338 | Cite as

# Inferring the Dynamic, Quantifying Physical Complexity

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- 1.J. P. Crutchfield, N. H. Packard, J. D. Fanner, and R. S. Shaw.
*Chaos. Sci. Am*.,**255**: 46, 1986.Google Scholar - 2.J. P. Crutchfield and B. S. McNamara. Equations of motion from a data series.
*Complex Systems*,**1**: 417, 1987.MathSciNetzbMATHGoogle Scholar - 3.J. P. Crutchfield and K. Young. Inferring statistical complexity.
*Phys. Rev. Let*,**63**:10 July, 1989.MathSciNetCrossRefGoogle Scholar - 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.J. P. Crutchfield and K. Young. Thermodynamics of minimal reconstructed machines. in preparation, 1989.Google Scholar
- 6.E. M. Gold. Language identification in the limit.
*Info. Control*,**10**: 447, 1967.zbMATHCrossRefGoogle Scholar - 7.J. P. Crutchfield and N. H. Packard. Symbolic dynamics of noisy chaos.
*Physica*,**7D**: 201, 1983.MathSciNetGoogle Scholar - 8.R. M. Wharton. Approximate language identification.
*Info. Control*,**26**: 236, 1974.MathSciNetzbMATHCrossRefGoogle Scholar - 9.E. M. Gold. Complexity of automaton identification from given
*data. Info. Control*,**37**: 302, 1978.MathSciNetzbMATHCrossRefGoogle Scholar - 10.D. Angluin. On the complexity of minimum inference of regular sets.
*Info. Control*,**39**: 337, 1978.MathSciNetzbMATHCrossRefGoogle Scholar - 11.J. P. Crutchfield.
*Noisy Chaos.*PhD thesis, University of California, Santa Cruz, 1983. published by University Microfilms Intl, Minnesota.Google Scholar - 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.J. Rissanen. Modeling by shortest data description.
*Automatica*,**14**: 462, 1978.CrossRefGoogle Scholar - 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.N. H. Packard.
*Measurements of Chaos in the Presence of Noise*. PhD thesis, University of California, Santa Cruz, 1982.Google Scholar - 16.P. Gras sberger. Toward a quantitative theory of self-generated complexity.
*Intl. J. Theo. Phys.*,**25**: 907, 1986.MathSciNetzbMATHCrossRefGoogle Scholar - 17.J. P. Crutchfield. Time is the ultrametric of causality. in preparation, 1989.Google Scholar
- 18.J. P. Crutchfield. Compressing chaos. in preparation, 1989.Google Scholar
- 19.J. Rissanen. Universal coding, information, prediction, and estimation.
*IEEE Trans. Info. Th.*, IT-**30**: 629, 1984.MathSciNetCrossRefGoogle Scholar - 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.MathSciNetzbMATHCrossRefGoogle Scholar - 21.S. Wolfram. Computation theory of cellular automata.
*Comm. Math. Phys.*,**96**: 15, 1984.MathSciNetzbMATHCrossRefGoogle Scholar - 22.R. Shaw.
*The Dripping Faucet as a Model Chaotic System*. Aerial Press, Santa Cruz, California, 1984.Google Scholar - 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.C. P. Bachas and B.A. Huberman. Complexity and relaxation of hierarchical structures.
*Phys. Rev. Let.*,**57**: 1965, 1986.MathSciNetCrossRefGoogle Scholar - 25.S. Lloyd and H. Pagels. Complexity
*as*thermodynamic depth.*Ann. Phys.*,**188**: 186, 1988.MathSciNetCrossRefGoogle Scholar