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Journal of Statistical Physics

, Volume 104, Issue 3–4, pp 817–879 | Cite as

Computational Mechanics: Pattern and Prediction, Structure and Simplicity

  • Cosma Rohilla Shalizi
  • James P. Crutchfield
Article

Abstract

Computational mechanics, an approach to structural complexity, defines a process's causal states and gives a procedure for finding them. We show that the causal-state representation—an ∈-machine—is the minimal one consistent with accurate prediction. We establish several results on ∈-machine optimality and uniqueness and on how ∈-machines compare to alternative representations. Further results relate measures of randomness and structural complexity obtained from ∈-machines to those from ergodic and information theories.

complexity computation entropy information pattern statistical mechanics causal state ∈-machine 
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REFERENCES

  1. 1.
    J.M. Yeomans, Statistical Mechanics of Phase Transitions (Clarendon Press, Oxford, 1992).Google Scholar
  2. 2.
    P. Manneville, Dissipative Structures and Weak Turbulence (Academic Press, Boston, Massachusetts, 1990).Google Scholar
  3. 3.
    P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, England, 1995).Google Scholar
  4. 4.
    M. C. Cross and P. Hohenberg, Pattern Formation Out of Equilibrium, Reviews of Modern Physics 65:851–1112 (1993).Google Scholar
  5. 5.
    J. P. Crutchfield, The calculi of emergence: Computation, dynamics, and induction, Physica D 75:11–54 (1994).Google Scholar
  6. 6.
    J. P. Crutchfield and K. Young, Inferring statistical complexity, Physical Review Letters 63:105–108 (1989).Google Scholar
  7. 7.
    J. P. Crutchfield and K. Young, Computation at the onset of chaos, In Zurek et al.,(137) pages 223–269.Google Scholar
  8. 8.
    N. Perry and P.-M. Binder, Finite statistical complexity for sofic systems, Physical Review E 60:459–463 (1999).Google Scholar
  9. 9.
    J. E. Hanson and J. P. Crutchfield, Computational mechanics of cellular automata: An example, Physica D 103:169–189 (1997).Google Scholar
  10. 10.
    D. R. Upper, Theory and Algorithms for Hidden Markov Models and Generalized Hidden Markov Models, PhD thesis (University of California, Berkeley, 1997). Online from http://www.santafe.edu/projects/CompMech/.Google Scholar
  11. 11.
    J. P. Crutchfield and M. Mitchell, The evolution of emergent computation, Proceedings of the National Academy of Sciences 92:10742–10746 (1995).Google Scholar
  12. 12.
    A. Witt, A. Neiman, and J. Kurths, Characterizing the dynamics of stochastic bistable systems by measures of complexity, Physical Review E 55:5050–5059 (1997).Google Scholar
  13. 13.
    J. Delgado and R. V. Solé, Collective-induced computation, Physical Review E 55:2338–2344 (1997).Google Scholar
  14. 14.
    W. M. Gonçalves, R. D. Pinto, J. C. Sartorelli, and M. J. de Oliveira, Inferring statistical complexity in the dripping faucet experiment, Physica A 257:385–389 (1998).Google Scholar
  15. 15.
    A. J. Palmer, C. W. Fairall, and W. A. Brewer, Complexity in the atmosphere, IEEE Transactions on Geoscience and Remote Sensing 38:2056–2063 (2000).Google Scholar
  16. 16.
    J. P. Crutchfield and C. R. Shalizi, Thermodynamic depth of causal states: Objective complexity via minimal representations, Physical Review E 59:275–283 (1999). E-print, arxiv.org, cond-mat/9808147.Google Scholar
  17. 17.
    J. L. Borges, Other Inquisitions, 1937û1952 (University of Texas Press, Austin, 1964). Trans. Ruth L. C. Simms.Google Scholar
  18. 18.
    J. P. Crutchfield, Semantics and thermodynamics, In Nonlinear Modeling and Forecasting, M. Casdagli and S. Eubank, eds., volume 12 of Santa Fe Institute Studies in the Sciences of Complexity, pages 317–359 (Addison-Wesley, Reading, Massachusetts, 1992).Google Scholar
  19. 19.
    Plato, Phaedrus.Google Scholar
  20. 20.
    A. R. Luria, The Working Brain: An Introduction to Neuropsychology (Basic Books, New York, 1973).Google Scholar
  21. 21.
    N. V. S. Graham, Visual Pattern Analyzers, volume 16 of Oxford Psychology Series (Oxford University Press, Oxford, 1989).Google Scholar
  22. 22.
    S. J. Shettleworth, Cognition, Evolution and Behavior (Oxford University Press, Oxford, 1998).Google Scholar
  23. 23.
    J. T. Tou and R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Massachusetts, 1974).Google Scholar
  24. 24.
    S. P. Banks, Signal Processing, Image Processing, and Pattern Recognition (Prentice Hall, New York, 1990).Google Scholar
  25. 25.
    J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice Hall, New York, 1990).Google Scholar
  26. 26.
    Plato, Meno, In 80D Meno says: “How will you look for it, Socrates, when you do not know at all what it is? How will you aim to search for something you do not know at all? If you should meet it, how will you know that this is the thing that you did not know?” The same difficulty is raised in Theaetetus, 197 et seq.Google Scholar
  27. 27.
    A. N. Whitehead and B. Russell, Principia Mathematica, 2nd ed. (Cambridge University Press, Cambridge, England), pp. 1925–27.Google Scholar
  28. 28.
    B. Russell, Introduction to Mathematical Philosophy, The Muirhead Library of Philosophy, revised ed. (George Allen and Unwin, London, 1920). First edition, 1919. Reprinted New York: Dover Books, 1993.Google Scholar
  29. 29.
    J.P. Crutchfield, Information and its metric, In Nonlinear Structures in Physical Systemsù Pattern Formation, Chaos and Waves, L. Lam and H. C. Morris, eds. (Springer-Verlag, New York, 1990), pp. 119.Google Scholar
  30. 30.
    B. Russell, Human Knowledge: Its Scope and Limits (Simon and Schuster, New York, 1948).Google Scholar
  31. 31.
    J. Rhodes, Applications of Automata Theory and Algebra via the Mathematical Theory of Complexity to Biology, Physics, Psychology, Philosophy, Games, and Codes (University of California, Berkeley, California, 1971).Google Scholar
  32. 32.
    C. L. Nehaniv and J. L. Rhodes, KrohnûRhodes theory, hierarchies, and evolution, In Mathematical Hierarchies and Biology: DIMACS workshop, November 13û15, 1996, B. Mirkin, F. R. McMorris, F. S. Roberts, and A. Rzhetsky, eds., volume 37 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science (American Mathematical Society, Providence, Rhode Island, 1997).Google Scholar
  33. 33.
    U. Grenander, Elements of Pattern Theory, Johns Hopkins Studies in the Mathematical Sciences (Johns Hopkins University Press, Baltimore, Maryland, 1996).Google Scholar
  34. 34.
    H. Jaeger, Observable operator models for discrete stochastic time series, Neural Computation 12:1371–1398 (2000).Google Scholar
  35. 35.
    U. Grenander, Y. Chow, and D. M. Keenan, Hands: A Pattern Theoretic Study of Biological Shapes, volume 2 of Research Notes in Neural Computing (Springer-Verlag, New York, 1991).Google Scholar
  36. 36.
    U. Grenander and K. Manbeck, A stochastic shape and color model for defect detection in potatoes, American Statistical Association 12:131–151 (1993).Google Scholar
  37. 37.
    A. N. Kolmogorov, Three approaches to the quantitative definition of information, Problems of Information Transmission 1:1–7 (1965).Google Scholar
  38. 38.
    G. Chaitin, On the length of programs for computing finite binary sequences, Journal of the Association for Computing Machinery 13:547–569 (1966).Google Scholar
  39. 39.
    A. N. Kolmogorov, Combinatorial foundations of information theory and the calculus of probabilities, Russian Mathematical Surveys 38:29 (1983).Google Scholar
  40. 40.
    M. Li and P. M. B. Vitanyi, An Introduction to Kolmogorov Complexity and its Applications (Springer-Verlag, New York, 1993).Google Scholar
  41. 41.
    M. Minsky, Computation: Finite and Infinite Machines (Prentice-Hall, Englewood Cliffs, New Jersey, 1967).Google Scholar
  42. 42.
    P. Martin-Löf, The definition of random sequences, Information and Control 9:602–619 (1966).Google Scholar
  43. 43.
    L. A. Levin, Laws of information conservation (nongrowth) and aspects of the foundation of probability theory, Problemy Peredachi Informatsii 10:30–35 (1974). Translation: Problems of Information Transmission 10:206û210 (1974).Google Scholar
  44. 44.
    V. G. Gurzadyan, Kolmogorov complexity as a descriptor of cosmic microwave background maps, Europhysics Letters 46:114–117 (1999).Google Scholar
  45. 45.
    R. J. Solomonoff, A formal theory of inductive inference, Information and Control 7:1–22 and 224û254 (1964).Google Scholar
  46. 46.
    P. M. B. Vitányi and M. Li, Minimum description length induction, Bayesianism, and Kolmogorov complexity, E-print, arxiv.org, cs.LG/9901014, 1999.Google Scholar
  47. 47.
    G. W. Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems and Adaptation (MIT Press, Cambridge, Massachusetts, 1998).Google Scholar
  48. 48.
    J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, Singapore, 1989).Google Scholar
  49. 49.
    C. H. Bennett, How to define complexity in physics, and why, In Zurek et al.,(137) pages 137–148.Google Scholar
  50. 50.
    M. Koppel, Complexity, depth, and sophistication, Complex Systems 1:1087–1091 (1987).Google Scholar
  51. 51.
    M. Koppel and H. Atlan, An almost machine-independent theory of program-length complexity, sophistication and induction, Information Sciences 56:23–44 (1991).Google Scholar
  52. 52.
    D. C. Dennett, Real patterns, Journal of Philosophy 88:27–51 (1991). Reprinted in ref. 138.Google Scholar
  53. 53.
    J. P. Crutchfield, Is anything ever new? Considering emergence, In Complexity: Metaphors, Models, and Reality, G. Cowan, D. Pines, and D. Melzner, eds., volume 19 of Santa Fe Institute Studies in the Sciences of Complexity (Addison-Wesley, Reading, Massachusetts, 1994), pp. 479–497.Google Scholar
  54. 54.
    J. H. Holland, Emergence: From Chaos to Order (Addison-Wesley, Reading, Massachusetts, 1998).Google Scholar
  55. 55.
    L. Boltzmann, Lectures on Gas Theory (University of California Press, Berkeley, 1964).Google Scholar
  56. 56.
    H. Cramér, Mathematical Methods of Statistics (Almqvist and Wiksells, Uppsala, 1945). Republished by Princeton University Press, 1946, as vol. 9 in the Princeton Mathematics Series, and as a paperback, in the Princeton Landmarks in Mathematics and Physics series, 1999.Google Scholar
  57. 57.
    C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal 27:379–423 (1948). Reprinted in ref. 139.Google Scholar
  58. 58.
    D. Hume, A Treatise of Human Nature: Being an Attempt to Introduce the Experimental Method of Reasoning into Moral Subjects (John Noon, London, 1739). Reprint (Oxford: Clarendon Press, 1951) of original edition, with notes and analytical index.Google Scholar
  59. 59.
    M. Bunge, Causality: The Place of the Causal Princple in Modern Science (Harvard University Press, Cambridge, Massachusetts, 1959). Reprinted as Causality and Modern Science, NY: Dover Books, 1979.Google Scholar
  60. 60.
    W. C. Salmon, Scientific Explanation and the Causal Structure of the World (Princeton University Press, Princeton, 1984).Google Scholar
  61. 61.
    P. Billingsley, Ergodic Theory and Information, Tracts on Probablity and Mathematical Statistics (Wiley, New York, 1965).Google Scholar
  62. 62.
    R. M. Gray, Entropy and Information Theory (Springer-Verlag, New York, 1990).Google Scholar
  63. 63.
    T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).Google Scholar
  64. 64.
    W. O. Ockham, Philosophical Writings: A Selection, Translated, with an Introduction, by Philotheus Boehner, O.F.M., Late Professor of Philosophy, The Franciscan Institute (Bobbs-Merrill, Indianapolis, 1964). First pub. various European cities, early 1300s.Google Scholar
  65. 65.
    Anonymous, Kuan Yin Tzu, T'ang Dynasty, Written in China during the T'ang dynasty. Partial translation in Joseph Needham, Science and Civilisation in China, vol. II (Cambridge University Press, 1956), p. 73.Google Scholar
  66. 66.
    D. P. Feldman and J. P. Crutchfield, Discovering non-critical organization: Statistical mechanical, information theoretic, and computational views of patterns in simple one-dimensional spin systems, Journal of Statistical Physics submitted (1998), Santa Fe Institute Working Paper 98-04-026, http://www.santafe.edu/projects/CompMech/papers/ DNCO.html.Google Scholar
  67. 67.
    J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation (Addison-Wesley, Reading, 1979), 2nd edition of Formal Languages and Their Relation to Automata, 1969.Google Scholar
  68. 68.
    H. R. Lewis and C. H. Papadimitriou, Elements of the Theory of Computation, 2nd ed. (Prentice-Hall, Upper Saddle River, New Jersey, 1998).Google Scholar
  69. 69.
    J. G. Kemeny and J. L. Snell, Finite Markov Chains (Springer-Verlag, New York, 1976).Google Scholar
  70. 70.
    J. G. Kemeny, J. L. Snell, and A. W. Knapp, Denumerable Markov Chains, 2nd ed. (Springer-Verlag, New York, 1976).Google Scholar
  71. 71.
    J. E. Hanson, Computational Mechanics of Cellular Automata, PhD thesis (University of California, Berkeley, 1993).Google Scholar
  72. 72.
    G. Bateson, Mind and Nature: A Necessary Unity (E. P. Dutton, New York, 1979).Google Scholar
  73. 73.
    S. Kullback, Information Theory and Statistics, 2nd ed. (Dover Books, New York, 1968). First edition New York: Wiley, 1959.Google Scholar
  74. 74.
    K. L. Klinkner, C. R. Shalizi, and J. P. Crutchfield, Extensive state estimation: Reconstructing causal states by splitting, Manuscript in preparation, 2001.Google Scholar
  75. 75.
    J. P. Crutchfield and N. H. Packard, Symbolic dynamics of noisy chaos, Physica D 7:201–223 (1983).Google Scholar
  76. 76.
    R. Shaw, The Dripping Faucet as a Model Chaotic System (Aerial Press, Santa Cruz, California, 1984).Google Scholar
  77. 77.
    P. Grassberger, Toward a quantitative theory of self-generated complexity, International Journal of Theoretical Physics 25:907–938 (1986).Google Scholar
  78. 78.
    K. Lindgren and M. G. Nordahl, Complexity measures and cellular automata, Complex Systems 2:409–440 (1988).Google Scholar
  79. 79.
    W. Li, On the relationship between complexity and entropy for Markov chains and regular languages, Complex Systems 5:381–399 (1991).Google Scholar
  80. 80.
    D. Arnold, Information-theoretic analysis of phase transitions, Complex Systems 10:143–155 (1996).Google Scholar
  81. 81.
    W. Bialek and N. Tishby, Predictive information, E-print, arxiv.org, cond-mat/ 9902341, 1999.Google Scholar
  82. 82.
    J. P. Crutchfield and D. P. Feldman, Statistical complexity of simple one-dimensional spin systems, Physical Review E 55:1239R–1243R (1997).Google Scholar
  83. 83.
    W. R. Ashby, An Introduction to Cybernetics (Chapman and Hall, London, 1956).Google Scholar
  84. 84.
    H. Touchette and S. Lloyd, Information-theoretic limits of control, Physical Review Letters 84:1156–1159 (1999).Google Scholar
  85. 85.
    A. Lempel and J. Ziv, Compression of two-dimensional data, IEEE Transactions in Information Theory IT-32:2–8 (1986).Google Scholar
  86. 86.
    D. P. Feldman, Computational Mechanics of Classical Spin Systems, PhD thesis (University of California, Davis, 1998). Online at http://hornacek.coa.edu/dave/ Thesis/thesis.html.Google Scholar
  87. 87.
    D. G. Mayo, Error and the Growth of Experimental Knowledge (Science and Its Conceptual Foundations, University of Chicago Press, Chicago, 1996).Google Scholar
  88. 88.
    J. P. Crutchfield and C. Douglas, Imagined complexity: Learning a random process. Manuscript in preparation, 1999.Google Scholar
  89. 89.
    R. Lidl and G. Pilz, Applied Abstract Algebra (Springer, New York, 1984).Google Scholar
  90. 90.
    E. S. Ljapin, Semigroups, volume 3 of Translations of Mathematical Monographs (American Mathematical Society, Providence, Rhode Island, 1963).Google Scholar
  91. 91.
    K. Young, The Grammar and Statistical Mechanics of Complex Physical Systems, PhD thesis (University of California, Santa Cruz, 1991).Google Scholar
  92. 92.
    P. Billingsley, Probability and Measure, Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1979).Google Scholar
  93. 93.
    J. L. Doob, Stochastic Processes, Wiley Publications in Statistics (Wiley, New York, 1953).Google Scholar
  94. 94.
    M. Loéve, Probability Theory, 1st ed. (D. Van Nostrand Company, New York, 1955).Google Scholar
  95. 95.
    M. M. Rao, Conditional Measures and Applications, volume 177 of Monographs and Textbooks in Pure and Applied Mathematics (Marcel Dekker, New York, 1993).Google Scholar
  96. 96.
    N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, Geometry from a time series, Physical Review Letters 45:712–716 (1980).Google Scholar
  97. 97.
    F. Takens, Detecting strange attractors in fluid turbulence, In D. A. Rand and L. S. Young, editors, Symposium on Dynamical Systems and Turbulence, volume 898 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981), pp. 366.Google Scholar
  98. 98.
    J. P. Crutchfield and B. S. McNamara, Equations of motion from a data series, Complex Systems 1:417–452 (1987).Google Scholar
  99. 99.
    J. Neyman, First Course in Probability and Statistics (Henry Holt, New York, 1950).Google Scholar
  100. 100.
    D. Blackwell and M. A. Girshick, Theory of Games and Statistical Decisions (Wiley, New York, 1954). Reprinted New York: Dover Books, 1979.Google Scholar
  101. 101.
    R. D. Luce and H. Raiffa, Games and Decisions: Introduction and Critical Survey (Wiley, New York, 1957).Google Scholar
  102. 102.
    I. M. Gel'fand and A. M. Yaglom, Calculation of the amount of information about a random function contained in another such function, Uspekhi Matematicheski Nauk 12:3–52 (1956). Trans. in American Mathematical Society Translations 12(2):199û246 (1959).Google Scholar
  103. 103.
    P. E. Caines, Linear Stochastic Systems (Wiley, New York, 1988).Google Scholar
  104. 104.
    D. Blackwell and L. Koopmans, On the identifiability problem for functions of finite Markov chains, Annals of Mathematical Statistics 28:1011–1015 (1957).Google Scholar
  105. 105.
    H. Ito, S.-I. Amari, and K. Kobayashi, Identifiability of hidden Markov information sources and their minimum degrees of freedom, IEEE Transactions on Information Theory 38:324–333 (1992).Google Scholar
  106. 106.
    P. Algoet, Universal schemes for prediction, gambling and portfolio selection, The Annals of Probability 20:901–941 (1992). See also an important Correction, The Annals of Probability 23:474û478 (1995).Google Scholar
  107. 107.
    A. N. Kolmogorov, Interpolation und extrapolation von stationären zufälligen folgen, Bull. Acad. Sci. U.S.S.R., Math. 3:3–14 (1941). In Russian with German summary.Google Scholar
  108. 108.
    N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series: With Engineering Applications, The Technology Press of the Massachusetts Institute of Technology, Cambridge, Massachusetts, 1949, “First published during the war as a classifed report to Section D 2, National Defense Research Council”.Google Scholar
  109. 109.
    N. Wiener, Nonlinear Problems in Random Theory (The Technology Press of the Massachusetts Institute of Technology, Cambridge, Massachusetts, 1958).Google Scholar
  110. 110.
    M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, 2nd ed. (Robert E. Krieger Publishing Company, Malabar, Florida, 1989). Reprint, with additions, of the first edition, New York: John Wiley, 1980.Google Scholar
  111. 111.
    N. Wiener, Cybernetics: Or, Control and Communication in the Animal and the Machine, 2nd ed. (MIT Press, Cambridge, Massachusetts, 1961). First edition New York: Wiley, 1948.Google Scholar
  112. 112.
    N. Chomsky, Three models for the description of language, IRE Transactions on Information Theory 2:113 (1956).Google Scholar
  113. 113.
    N. Chomsky, Syntactic Structures, volume 4 of Janua linguarum, series minor (Mouton, The Hauge, 1957).Google Scholar
  114. 114.
    B. A. Trakhtenbrot and Y. M. Barzdin, Finite Automata (North-Holland, Amsterdam, 1973).Google Scholar
  115. 115.
    E. Charniak, Statistical Language Learning, Language, Speech and Communication (MIT Press, Cambridge, Massachusetts, 1993).Google Scholar
  116. 116.
    M. J. Kearns and U. V. Vazirani, An Introduction to Computational Learning Theory (MIT Press, Cambridge, Massachusetts, 1994).Google Scholar
  117. 117.
    V. N. Vapnik, The Nature of Statistical Learning Theory, 2nd ed. (Springer-Verlag, Berlin, 2000).Google Scholar
  118. 118.
    L. G. Valiant, A theory of the learnable, Communications of the Association for Computing Machinery 27:1134–1142 (1984).Google Scholar
  119. 119.
    M. A. Boden, Precis of “The Creative Mind: Myths and Mechanisms',” Behaviorial and Brain Sciences 17:519–531 (1994).Google Scholar
  120. 120.
    C. Thornton, Truth from Trash: How Learning Makes Sense, Complex Adaptive Systems (MIT Press, Cambridge, Massachusetts, 2000).Google Scholar
  121. 121.
    J. Pearl, Causality: Models, Reasoning, and Inference (Cambridge University Press, Cambridge, England, 2000).Google Scholar
  122. 122.
    M. I. Jordan, ed., Learning in Graphical Models, volume 89 of NATO Science Series D: Behavioral and Social Sciences (Dordrecht, 1998).Google Scholar
  123. 123.
    P. Spirtes, C. Glymour, and R. Scheines, Causation, Prediction, and Search, Adaptive Computation and Machine Learning (MIT Press, Cambridge, Massachusetts, 2000).Google Scholar
  124. 124.
    D. V. Lindley, Bayesian Statistics, a Review (Society for Industrial and Applied Mathematics, Philadelphia, 1972).Google Scholar
  125. 125.
    J. Rissanen, A universal data compression system, IEEE Transactions in Information Theory IT-29:656–664 (1983).Google Scholar
  126. 126.
    P. Bühlmann and A. J. Wyner, Variable length Markov chains, Technical Report 497 (UC Berkeley Statistics Department, 1997). Online from http://www.stat. berkeley.edu/tech-reports/.Google Scholar
  127. 127.
    A. Lempel and J. Ziv, On the complexity of finite sequences, IEEE Transactions in Information Theory IT-22:75–81 (1976).Google Scholar
  128. 128.
    J. Ziv and A. Lempel, A universal algorithm for sequential data compression, IEEE Transactions in Information Theory IT-23:337–343 (1977).Google Scholar
  129. 129.
    R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics, volume 6 of Cambridge Nonlinear Science Series (Cambridge University Press, Cambridge, 1997).Google Scholar
  130. 130.
    B. Weiss, Subshifts of finite type and sofic systems, Monatshefte für Mathematik 77:462–474 (1973).Google Scholar
  131. 131.
    C. Moore, Recursion theory on the reals and continuous-time computation, Theoretical Computer Science 162:23–44 (1996).Google Scholar
  132. 132.
    C. Moore, Dynamical recognizers: Real-time language recognition by analog computers, Theoretical Computer Science 201:99–136 (1998).Google Scholar
  133. 133.
    P. Orponen, A survey of continuous-time computation theory, In D.-Z. Du and K.-I. Ko, editors, Advances in Algorithms, Languages, and Complexity (Kluwer Academic, Dordrecht, 1997), pp. 209–224.Google Scholar
  134. 134.
    L. Blum, M. Shub, and S. Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bulletin of the American Mathematical Society 21:1–46 (1989).Google Scholar
  135. 135.
    C. Moore, Unpredictability and undecidability in dynamical systems, Physical Review Letters 64:2354–2357 (1990).Google Scholar
  136. 136.
    S. Sinha and W. L. Ditto, Dynamics based computation, Physical Review Letters 81:2156–2159 (1998).Google Scholar
  137. 137.
    W. H. Zurek, editor, Complexity, Entropy, and the Physics of Information, volume 8 of Santa Fe Institute Studies in the Sciences of Complexity 0 (Addison-Wesley, Reading, Massachusetts, 1990).Google Scholar
  138. 138.
    D. C. Dennett, Brainchildren: Essays on Designing Minds, Representation and Mind (MIT Press, Cambridge, Massachusetts, 1997).Google Scholar
  139. 139.
    C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbana, Illinois, 1963).Google Scholar
  140. 140.
    B. F. Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, Cambridge, England, 1980).Google Scholar

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© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Cosma Rohilla Shalizi
    • 1
    • 2
  • James P. Crutchfield
    • 1
  1. 1.Santa Fe InstituteSanta Fe
  2. 2.Physics DepartmentUniversity of Wisconsin-MadisonMadison

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