Universality and Complexity in Cellular Automata


Universality and Complexity in Cellular Automata
S. Wolfram / Universality and complexity in cellular automata
then this dependence may be arbitrarily complex,
and the behaviour of the system can be found by
1,10 procedure significantly simpler than direct sim~Iation. No meaningful prediction is therefore
possible for such systems.
I am grateful to many people for discussions,
including C. Bennett, J. Crutchfield, D. Friedan, P.
Gacz, E. Jen, D. Lind, O. Martin, A. Odlyzko, N .
Packard, S2. Shenker, W. Thurston, T. Toffoli and
S. Willson. I am particularly grateful to J. Milnor
for extensive discussions and suggestions.
[I] S. Wolfram, 'Statistical mechanics of cellular automata",
Rev . Mod. Phys. 55 (1983) 601.
[2] O. Martin, A.M. Odlyzko and S. Wolfram, " Algebraic
properties of cellular automata", Bell Laboratories report
(January 1983); Comm. Math. Phys., to be published.
[3] D. Lind, " Applications of ergodic theory and sofic systems
to cellular automata", University of Washington preprint
(April 1983); Physica 100 (1984) 36 (these proceedings).
[4] S. Wolfram, "CA: an interactive cellular automaton simulator for the Sun Workstation and VAX", presented and
demonstrated at the Interdisciplinary Workshop on Cellular Automata, Los Alamos (March 1983).
[5] T. Toffoli, N. Margolus and G . Vishniac, private demonstrations.
[6] P. Billingsley, Ergodic Theory and Information (Wiley,
New York, 1965).
[7] D . Knuth, Semi numerical Algorithms, 2nd. ed. (AddisonWesley, New York, 1981), section 3.5.
[8] R.G. Gallager, Information Theory and Reliable Commu• _ nications (Wiley, New York, 1968).
[9] J.D. Farmer, "Dimension, fractal measures and the probabilistic structure of chaos", in : Evolution of Order and
Chaos in Physics, Chemistry and Biology, H. Haken, ed .
(Springer, Berlin, 1982).
[10] J.D. Farmer, private communication.
[II] B. Mandelbrot, The Fractal Geometry of nature (Freeman, San Francisco, 1982).
[12] J.D. Farmer, " Information dimension and the probabilistic structure of chaos", Z. Naturforsch. 37a (1982)
[13] P. Grassberger, to be published.
[l4] P. Diaconis, private communication; C. Stein, unpublished
[15] F.S. Beckman, " Mathematical Foundations of Programming (Addison-Wesley, New York , 1980).
[16] J.E. Hopcroft and J.D. Ullman, Introduction to Automata
Theory, Languages, and Computation (Addison-Wesley,
New York, 1979).
[17] Z. Manna, Mathematical Theory of Computation
(McGraw-Hill, New York, 1974).
[18] M. Minsky, Computation: Finite and Infinite Machines
(Prentice-Hall, London, 1967).
[19] B. Weiss, " Subshifts of finite type and sofic systems",
Monat. Math. 17 (1973) 462. E.M. Coven and M.E. Paul,
" Sofic systems", Israel J. Math. 20 (1975) 165.
[20] P. Grassberger, " A new mechanism for deterministic
diffusion", Wuppertal preprint WU B 82- 18 (1982).
[2I] J. Milnor, unpublished notes.
[22] R.W. Gosper, unpublished; R. Wainwright, "Life is universa!!", Proc. Winter Simul. Conf., Washington D .C.,
ACM (1974). E.R. Berlekamp, J.H . Conway and R.K.
Guy, Winning Ways, for Your Mathematical Plays, vol. 2
(Academic Press, New York, 1982), chap. 25.
[23] R.W. Gosper, "Exploiting regularities in large cellular
spaces", Physica 100 (1984) 75 (these proceedings).
[24] G. Chaitin, "Algorithmic information theory" , IBM J.
Res. & Dev. , 21 (1977) 350; "Toward a mathematical
theory of life", in : The Maximum Entropy Formalism,
R.D. Levine and M. Tribus, ed. (MIT press, Cambridge,
MA, 1979).
[25] C. Bennett, "On the logical "depth" of sequences and their
reducibilities to random sequences", IBM report (April
1982) (to be published in Info. & Control) .

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