Computational Mechanics of Classical Spin Systems

David P. Feldman

Doctoral Dissertation, Department of Physics

University of California, Davis

September 1998

List of References

  1. A. R. Ammons. Corsons Inlet. In Selected Poems. Cornell University Press, 1968. Reprinted in The Norton Anthology of Poetry, Revised Shorter Edition, W. W. Norton & Co., A. W. Allison H. Barrows, C. R. Blake, A. J. Carr A. M. Eastman and H. M. English, Jr., editors.

  2. C. Anteneodo and A.R. Plastino. Some features of the Lòpez-Ruiz--Mancini--Calbet statistical measure of complexity. Physics Letters A, 223:348-354, 1996.

  3. D. Arnold. Information-theoretic analysis of phase transitions. Complex Systems, 10:143-155, 1996.

  4. H. Atmanspacher, C. Räth, and G. Weidenmann. Statistics and meta-statistics in the concept of complexity. Physica, A243:819-829, 1997.

  5. R. Badii and A. Politi. Complexity: Hierarchical structures and scaling in physics. Cambridge University Press, Cambridge, 1997.

  6. R. Badii and A. Politi. Thermodynamics and complexity of cellular automata. Phys. Rev. Lett., 78(3):444-447, 1997.

  7. R. Baierlein. Atoms and Information Theory; An Introduction to Statistical Mechanics. W. H. Freeman, San Francisco, 1971.

  8. P. Bak, C. Tang, and K. Weisenfield. Self-organized criticality: An explanation of 1/f noise. Phys. Rev. Lett., 59(4):381-384, 1987.

  9. G. A. Baker. Markov-property Monte Carlo method: One-dimensional Ising model. J. Stat. Phys., 72:621-640, 1993.

  10. Y. Bar-Yam. Dynamics of Complex Systems. Addison-Wesley, 1997.

  11. R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, 1982.

  12. C. Beck and F. Schlögl. Thermodynamics of Chaotic Systems. Cambridge University Press, 1993.

  13. C. H. Bennett. On the nature and origin of complexity in discrete, homogeneous locally-interacting systems. Found. Phys., 16:585-592, 1986.

  14. C. H. Bennett. How to define complexity in physics, and why. In W. H. Zurek, editor, Complexity, Entropy, and the Physics of Information, volume VIII of Santa Fe Institute Studies in the Sciences of Complexity, pages 137-148. Addison-Wesley, 1990.

  15. V. Berthé. Conditional entropy of some automatic sequences. Journal of Physics, A27:7993-8006, 1994.

  16. W. Bialek, C. G. Callan, and S. P. Strong. Field theories for learning probability distributions. Phys. Rev. Lett., 77:4693-4697, 1996.

  17. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman. The Theory of Critical Phenomena: An Introduction to the Renormalization Group. Oxford Science Publications, 1992.

  18. D. Blackwell and L. Koopmans. On the identifiability problem for functions of Markov chains. Ann. Math. Statist., 28:1011-1015, 1957.

  19. L. Boltzmann. Lectures on gas theory. University of California Press, Berkeley, 1964.

  20. R. Bronson. Matrix Operations. McGraw-Hill, 1989.

  21. J. G. Brookshear. Theory of Computation: Formal Languages, Automata, and Complexity. Benjamin/Cummings, 1989.

  22. A. A. Brudno. Entropy and the complexity of the trajectories of a dynamical system. Trans. Moscow Math. Soc., 44:127, 1983.

  23. S. G. Brush. History of the Lenz-Ising model. Rev. Mod. Phys., 39(1):883-893, 1967.

  24. J. A. Bucklew. Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley-Interscience, New York, 1990.

  25. G. Chaitin. On the length of programs for computing finite binary sequences. J. ACM, 13:145, 1966.

  26. G. Chaitin. Information, Randomness and Incompleteness. World Scientific, Singapore, 1987.

  27. D. Chandler. Introduction to Modern Statistical Mechanics. Oxford University Press, 1987.

  28. N. Chomsky. Language and Thought, volume Three of The Frick Collection: Anshen Transdisciplinary Lectureships in Art, Science and the Philosophy of Culture. Moyer Bell, 1993.

  29. S. N. Coppersmith, T. C. Jones, L. P. Kadanoff, A. Levine, J. P. McCarten, S. R. Nagel, S. C. Venkataramani, and X. Wu. Self-organized short-term memories. Phys. Rev. Lett., 78(21):3983-3986, 1997.

  30. T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, Inc., 1991.

  31. R. T. Cox. Probability, frequency, and reasonable expectation. Am. J. Phys., 14:1-13, 1946.

  32. J. P. Crutchfield and C. Douglas. 1998. In Preparation.

  33. J. P. Crutchfield and D. P. Feldman. Statistical complexity of simple one-dimensional spin systems. Phys. Rev. E, 55(2):1239R-1243R, 1997.

  34. J. P. Crutchfield and D. P. Feldman. Regularities unseen, randomness observed: The entropy convergence hierarchy. 1998. In Preparation.

  35. J. P. Crutchfield and J. E. Hanson. Turbulent pattern bases for cellular automata. Physica D, 69:279-301, 1993.

  36. J. P. Crutchfield and N. H. Packard. Noise scaling of symbolic dynamics entropies. In H. Haken, editor, Evolution of Order and Chaos, pages 215-227, Berlin, 1982. Springer-Verlag.

  37. J. P. Crutchfield and N. H. Packard. Symbolic dynamics of one-dimensional maps: Entropies, finite precision, and noise. Intl. J. Theo. Phys., 21:433-466, 1982.

  38. J. P. Crutchfield and N. H. Packard. Symbolic dynamics of noisy chaos. Physica D, 7:201-223, 1983.

  39. J. P. Crutchfield and K. Young. Inferring statistical complexity. Phys. Rev. Lett., 63:105-108., 1989.

  40. J. P. Crutchfield and K. Young. Computation at the onset of chaos. In W. H. Zurek, editor, Complexity, Entropy and the Physics of Information, volume VIII of Santa Fe Institute Studies in the Sciences of Compexity, pages 223-269. Addison-Wesley, 1990.

  41. J. P. Crutchfield. Semantics and thermodynamics. In M. Casdagli and S. Eubank, editors, Nonlinear Modeling and Forecasting, volume XII of Santa Fe Institute Studies in the Sciences of Complexity, pages 317-359, Reading, Massachusetts, 1992. Addison-Wesley.

  42. J. P. Crutchfield. The calculi of emergence: Computation, dynamics, and induction. Physica D, 75:11-54, 1994.

  43. J. P. Crutchfield. Critical computation, phase transitions, and hierarchical learning. In M. Yamaguti, editor, Towards the Harnessing of Chaos, pages 29-46, Amsterdam, 1994. Elsevier Science.

  44. J. P. Crutchfield. Is anything ever new? Considering emergence. In G. Cowan, D. Pines, and D. Melzner, editors, Complexity: Metaphors, Models, and Reality, volume XIX of Santa Fe Institute Studies in the Sciences of Complexity, pages 479-497, Reading, MA, 1994. Addison-Wesley.

  45. J. P. Crutchfield, 1998. Personal Communication.

  46. A. Csordás and P. Szépfalusy. Singularities in Rényi information as phase transitions in chaotic states. Phys. Rev. A., 39(9):4767-4777, 1989.

  47. P. Cvitanovic. Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett., 61:2729-2732, 1988.

  48. J. Delgado and R. V. Solé. Collective-induced computation. Phys. Rev. E, 55(3):2338-2344, 1997.

  49. J. F. Dobson. Many-neighbored Ising chain. J. Math. Phys., 10(1):40-45, January 1969.

  50. K. E. Drexler. Nanosystems: molecular machinery, manufacturing, and computation. Wiley, New York, 1992.

  51. R. J. Elliot, L. Aggoun, and J. B. Moore. Hidden Markov Models: Estimation and Control, volume 29 of Applications of Mathematics. Springer, New York, 1995.

  52. K-E. Eriksson and K. Lindgren. Structural information in self-organizing systems. Physica Scripta, 1987.

  53. K.-E. Eriksson and K. Lindgren. Entropy and correlations in lattice systems. Technical report, Physical Resource Theory Group, Chalmers University of Technology, Göteborg, Sweden, 1989.

  54. 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. 1998. Submitted to the Journal of Statistical Physics.

  55. D. P. Feldman and J. P. Crutchfield. Measures of statistical complexity: Why? Phys. Lett. A, 238:244-252, 1998.

  56. R. P. Feynman. Feynman Lectures on Computation. Addison-Wesley, 1996. A. J. G. Hey and R. W. Allen, eds.

  57. K. H. Fischer and J. A. Hertz. Spin Glasses. Cambridge Studies in Magnetism. Cambridge University Press, Cambridge, 1988.

  58. J. Freund, W. Ebeling, and K. Rateitschak. Self-similar sequences and universal scaling of dynamical entropies. Phys. Rev. E, 54:5561-5566, 1996.

  59. M. Gell-Mann and S. Lloyd. Information measures, effective complexity, and total information. Complexity, 2(1):44-52, 1996.

  60. M. Gell-Mann. The quark and the jaguar: adventures in the simple and the complex. W. H. Freeman, 1994.

  61. M. Gell-Mann. What is complexity? Complexity, 1(1):16-19, 1995.

  62. W. M. Goncalves, R. D. Pinto, J. C. Sartorelli, and M. J. de Oliveira. Inferring statistical complexity in the dripping faucet experiment. Physica A, 1998. To appear.

  63. W. T. Grandy. Foundations of Statistical Mechanics. Fundamental Theories of Physics. D. Reidel, Dordrecht, 1988.

  64. P. Grassberger. Toward a quantitative theory of self-generated complexity. Intl. J. Theo. Phys., 25(9):907-938, 1986.

  65. R. M. Gray. Entropy and Information Theory. Springer-Verlag, New York, 1990.

  66. I. Guyon and P. S. P. Wang, editors. Advances in Pattern Recognition systems Using Neural Network Technologies. World Scientific, 1993.

  67. H. Haken. Synergetics. Springer-Verlag, third edition, 1983.

  68. T.C. Halsey, M. H. Jensen, L.P. Kadanoff, I. Procaccia, and B. I. Shraiman. Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A, 33:1141-1151, 1986.

  69. J. E. Hanson and J. P. Crutchfield. The attractor-basin portrait of a cellular automaton. J. Stat. Phys., 66:1415-1462, 1992.

  70. J. E. Hanson and J. P. Crutchfield. Computational mechanics of cellular automata: An example. Physica D, 103(1-4):169-189, 1997.

  71. J. E. Hanson. Computational Mechanics of Cellular Automata. PhD thesis, University of California, Berkeley, 1993.

  72. R. V. L. Hartley. Transmission of information. Bell Sys. Tech. J., July:535-563, 1928.

  73. J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading, 1979.

  74. B. A. Huberman and T. Hogg. Complexity and adaptation. Physica D, 22:376-384, 1986.

  75. E. Ising. Beitrag zur theorie des ferromagnetismus. Zeitschrift f. Physik, 31:253, 1925.

  76. E. T. Jaynes. Essays on Probability, Statistics, and Statistical Physics. Reidel, London, 1983.

  77. J. Justesen and Y. M. Shtarkov. Simple models of two-dimensional information sources and codes. To be presented at the Information Theory Symposium, Boston, MA, August 1998.

  78. A. Kato and K. Zeger. On the capacity of two-dimensional run length limited codes. To be presented at the Information Theory Symposium, Boston, MA, August 1998.

  79. S. Kauffman. At Home in the Universe: The Search for Laws of Complexity. Oxford University Press, 1995.

  80. Z. Kaufmann. Characteristic quantities of multifractals--application to the Feigenbaum attractor. Physica D, 54:75-84, 1991.

  81. A. I. Khinchin. Mathematical foundations of information theory. Dover, New York, 1957.

  82. B. Kitchens and S. Tuncel. Finitary measures for subshifts of finite type and sofic systems. Memoirs of the AMS, 58(338):1-68, 1985.

  83. W. R. Knorr. The Ancient Tradition of Geometric Problems. Birkhauser, Boston, 1986.

  84. S. Kobe. Ernst Ising--physicist and teacher. Actas: Noveno Taller Sur de Fisica del Solido, Misión, page 1, 1995. cond-mat/9605174.

  85. A. N. Kolmogorov. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk. SSSR, 119:861-864, 1958. (Russian) Math. Rev. vol. 21, no. 2035a.

  86. A. N. Kolmogorov. Three approaches to the concept of the amount of information. Prob. Info. Trans., 1:1, 1965.

  87. A. N. Kolmogorov. Combinatorial foundations of information theory and the calculus of probabilities. Russ. Math. Surveys, 38:29, 1983.

  88. M. Koppel. Complexity, depth, and sophistication. Complex Systems, 1:1087-1091, 1987.

  89. H. A. Kramers and G. H. Wannier. Statistics of the two-dimensional ferromagnet: Part I. Phys. Rev., 60:252-263, 1941.

  90. R. Landauer. A simple measure of complexity. Nature, 336(6197):306-307, 1988.

  91. A. Lempel and J. Ziv. Compression of two-dimensional data. IEEE Transactions on Information Theory, 32(1):2-8, 1986.

  92. W. Lenz. Phys. Zeitschrift, 21:613, 1920.

  93. R. Lewin. Complexity: life at the edge of chaos. Macmillian Publishing Company, 1992.

  94. M. Li and P. M. B. Vitanyi. An Introduction to Kolmogorov Complexity and its Applications. Springer-Verlag, New York, 1993.

  95. W. Li. Mutual information functions versus correlation functions. J. Stat. Phys., 60(5/6):823-837, 1990.

  96. W. Li. On the relationship between complexity and entropy for Markov chains and regular languages. Complex Systems, 5(4):381-399, 1991.

  97. R. Lidl and G. Pilz. Applied Abstract Algebra. Springer, New York, 1984.

  98. D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, 1995.

  99. K. Lindgren and M. G. Nordhal. Complexity measures and cellular automata. Complex Systems, 2(4):409-440, 1988.

  100. K. Lindgren, C. Moore, and M. G. Nordahl. Complexity of two-dimensional patterns. Journal of Statistical Physics, 1998. To appear.

  101. K. Lindgren. Microscopic and macroscopic entropy. Phys. Rev. A, 38(9):4794-4798, 1988.

  102. K. Lindgren. Entropy and correlations in dynamical lattice systems. In P. Manneville, N. Boccara, G. Y. Vichniac, and R. Bidaux, editors, Cellular Automata and Modeling of Complex Systems, volume 46 of Springer Proceedings in Physics, pages 27-40, Berlin, 1990. Springer-Verlag.

  103. K. Lindgren. Entropy and correlations in discrete dynamical systems. In J. L. Casti and A. Karlqvist, editors, Beyond Belief: Randomness, Prediction and Explanation in Science, pages 88-109. CRC Press, 1991.

  104. K. Lindgren, 1998. Personal Communication.

  105. S. Lloyd and H. Pagels. Complexity as thermodynamic depth. Annals of Physics, 188:186-213, 1988.

  106. R. Lopez-Ruiz, H. L. Mancini, and X. Calbet. A statistical measure of complexity. Physics Letters A, 209:321-326, December 1995.

  107. R. Mainieri. Cycle expansion for the Lyapunov exponent of a product of random matrices. Chaos, 2(1):91-97, 1992.

  108. R. Mainieri. Thermodynamic Zeta functions for Ising models with long-range interactions. Phys. Rev. A, 45(6):3580-3591, 1992.

  109. H. Matsuda, K. Kudo, R. Nakamura, O. Yamakawa, and T. Murata. Mutual information of Ising systems. International Journal of Theoretical Physics, 35(4):839-845, 1996.

  110. B. M. McCoy and T. T. Wu. The Two-Dimensional Ising Model. Harvard, 1993.

  111. B. McMillan. The basic theorems of information theory. Ann. Math. Stat., 24:196-219, 1953.

  112. D. Michie, D. Spiegelhalter, and C. C. Taylor, editors. Machine Learning, Neural and Statistical Classification, Series in Artificial Intelligence, New York, 1994. E. Horwood.

  113. R. K. Mishra, D. Maass, and E. Zwierlein, editors. On Self-Organization: An Interdisciplinary Search for a Unifying Principle. Springer-Verlag, 1994.

  114. L. Nadel and D. Stein, editors. 1993 Lectures in Complex Systems. Addison-Wesley, 1995. See also other volumes in the series.

  115. K. Nagel and M. Paczuski. Emergent traffic jams. Phys. Rev. E, 51:2909-2918, 1995.

  116. K. Nagel. Particle hopping models and traffic flow theory. Phys. Rev. E, 53:4655-4672, 1996.

  117. M. G. Nordahl. Cellular automata probability measures. In P. Manneville, N. Boccara, G. Y. Vichniac, and R. Bidaux, editors, Cellular Automata and Modeling of Complex Systems, volume 46 of Springer Proceedings in Physics. Springer-Verlag, Berlin, 1990.

  118. Hero of Alexandria. Opera, volume III: Metrica. B. G. Teubner, Leipzig, 1903.

  119. Y. Oono. Large deviation and statistical physics. Prog. Theo. Phys., 99:165-205, 1989.

  120. E. Ordentlich and R. M. Roth. On the redundancy of two-dimensional balanced codes. To be presented at the Information Theory Symposium, Boston, MA, August 1998.

  121. E. Ott. Chaos in Dynamical Systems. Cambridge University Press, 1993.

  122. N. H. Packard. Measurements of Chaos in the Presence of Noise. PhD thesis, University of California, Santa Cruz, 1982.

  123. C. H. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, Massachusetts, 1994.

  124. G. Parisi. Statistical Field Theory, volume 66 of Frontiers in Physics. Addison-Wesley, 1988.

  125. A. Paz. Introduction to Probabilistic Automata. Academic Press, New York, 1971.

  126. R. Peierls. Proc. Cambridge Phil. Soc., 32:477, 1936.

  127. H.-O. Peitgen, H. Jürgens, and D. Saupe. Chaos and Fractals: New Frontiers of Science. Springer-Verlag, 1992.

  128. L. Peliti and A. Vulpiani, editors. Measures of Complexity. Springer-Verlag, 1988.

  129. P. Perner, P. Wang, and A. Rosenfeld, editors. Advances in Structural and Syntactical Pattern Recognition. Springer, 1996.

  130. M. Plischke and B. Bergensen. Equilibrium Statistical Physics. Prentice Hall, 1989.

  131. M. Raijmakers. Epigensis in Neural Network Models of Cognitive Development: Bifurcations, More Powerful Structures, and Cognitive Concepts. PhD thesis, Universiteit van Amsterdam, 1996.

  132. J. Rhodes. Applications of Automata Theory and Algebra via the mathematical theory of complexity to: Biology, Physics, Psychology, Philosophy, Games, Codes. 1971.

  133. J. Rissanen. Modeling by shortest data description. Automatica, 14:465-471, 1978.

  134. J. Rissanen. Universal coding, information, prediction, and estimation. IEEE Trans. Info. Th., IT-30:629-636, 1984.

  135. J. Rissanen. Stochastic Complexity in Statistical Inquiry. World Scientific Publisher, Singapore, 1989.

  136. Harry S. Robertson. Statistical Theromphysics. Prentice Hall, 1993.

  137. J. Rothstein. Generalized entropy, boundary conditions, and biology. In R.D. Levine and M. Tribus, editors, The Maximum Entropy Formalism. MIT Press, Cambridge, Massachusetts, 1979.

  138. D. Ruelle. Statistical Mechanics: Rigorous Results. Addison Wesley, 1989.

  139. D. Saad and S. A. Solla. On-line learning in soft committee machines. Phys. Rev. E, 52:4225-4243, 1995.

  140. T. D. Schultz, D. C. Mattis, and E. H. Lieb. Two-dimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys., 36:856-871, 1964.

  141. J. Schurmann. Pattern Classification: A Unified View of Statistical and Neural Approaches. Wiley, New York, 1996.

  142. C. E. Shannon and W. Weaver. The Mathematical Theory of Communication. University of Illinois Press, 1963.

  143. C. E. Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379-423, 1948. as reprinted in ``The Mathematical Theory of Communication'', C. E. Shannon and W. Weaver, University of Illinois Press, Champaign-Urbana (1963).

  144. R. Shaw. The Dripping Faucet as a Model Chaotic System. Aerial Press, Santa Cruz, California, 1984.

  145. D. Sheinwald, A. Lempel, and J. Ziv. Two-dimensional encoding by finite-state encoders. IEEE Transactions on Communications, 38(3):341-347, 1990.

  146. P. H. Siegel and J. K. Wolf. Bit-stuffing bounds on the capacity of two-dimensional constrained arrays. To be presented at the Information Theory Symposium, Boston, MA, August 1998.

  147. Ja. G. Sinai. On the concept of entropy of a dynamical system. Dokl. Akad. Nauk. SSSR, 124:768-771, 1959.

  148. R. V. Solé, S. C. Manrubia, B. Luque, J. Delgado, and J. Bascompte. Phase transitions and complex systems. Complexity, 1(4):13-26, 1996.

  149. R. V. Solé, C. Blanc, and B. Luque. Statistical measures of complexity for strongly interacting systems. 1998. SFI Working Paper 97-11-083, Submitted to Physics Letters A.

  150. M. R. Spiegel. Mathematical Handbook of Formulas and Tables. McGraw-Hill, 1968.

  151. K. W. Sulston, B. L. Burrows, and A. N. Chishti. Recursive procedures for measuring disorder in non-periodic sequences. Physica A, 217:146-160, 1995.

  152. P. Szépfalusy and G. Györgyi. Entropy decay as a measure of stochasticity in chaotic systems. Phys. Rev. A, 33(4):2852-2855, 1986.

  153. B. A. Trakhtenbrot and Ya. M. Barzdin. Finite Automata. North-Holland, Amsterdam, 1973.

  154. D. R. Upper. Theory and Algorithms for Hidden Markov Models and Generalized Hidden Markov Models. PhD thesis, University of California, Berkeley, 1997.

  155. M. N. van Emden. An Analysis of Complexity, volume 35 of Mathematical Centre Tracts. Mathematisch Centrum Amsterdam, 1971.

  156. E. van Nimwegen, J. P. Crutchfield, and M. Mitchell. Finite populations induce metastability in evolutionary search. Physics Letters A, 229(3):144-150, 1997.

  157. E. van Nimwegen, J. P. Crutchfield, and M. Mitchell. Statistical dynamics of the Royal Road genetic algorithm. Theoret. Comp. Sci., 1998. In press.

  158. B. Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, and H. Scheingraber. A comparative classification of complexity measures. Chaos, Solitons & Fractals, 4(1):133-173, 1994.

  159. D. Walgraef. Spatio-Temporal Pattern Formation, with Examples from Physics, Chemistry, and Materials Science. Springer-Verlag, 1997.

  160. C. S. Wallace and D. M. Boulton. An information measure for classification. Comput. J., 11:185, 1968.

  161. S. Watanabe. Knowing and Guessing; A Quantitative Study of Inference and Information. Wiley, New York, 1969.

  162. T. L. H. Watkin, A. Rau, and M. Biehl. The statistical mechanics of learning a rule. Rev. Mod. Phys., 65:499-556, 1993.

  163. W. Weeks and Richard E. Blahut. The capacity and coding gain of certain checkerboard codes. IEEE Transactions on Information Theory, 44(3):1193-1203, 1998.

  164. K. Wilson. Problems in physics with many scales of length. Scientific American, 241(2):158-179, 1979.

  165. A. Witt, A. Neiman, and J. Kurths. Characterizing the dynamics of stochastic bistable systems by measures of complexity. Physical Review, E55(5):5050-5059, 1997.

  166. S. Wolfram. Statistical mechanics of cellular automata. Reviews of Modern Physics, 55(3):601-644, 1983.

  167. S. Wolfram. Computation theory of cellular automata. Communications in Mathematical Physics, 96(15-57), 1984.

  168. S. Wolfram. Universality and complexity in cellular automata. Physica, 10D:1-35, 1984.

  169. S. Wolfram, editor. Theory and Applications of Cellular Automata. World Scientific, 1986.

  170. J. M. Yeomans. Statistical Mechanics of Phase Transitions. Clarendon Press, Oxford, 1992.

  171. K. Young and J. P. Crutchfield. Fluctuation spectroscopy. Chaos, Solitons, and Fractals, 4:5-39, 1993.

  172. K. Young. The Grammar and Statistical Mechanics of Complex Physical Systems. PhD thesis, University of California, Santa Cruz, 1991.