Acknowledgments

The research reported here made essential use of several computer systems other than our own. We are grateful to those who made the systems available, and helped us in using them. We thank the M.I.T. Information Mechanics Group (E. Fredkin, N. Margolus, T. Toffoli, and G. Vichniac) for the use of their special-purpose two-dimensional cellular automaton simulation system, and for their hospitality and assistance. We thank R. Shaw for writing the kernel of our software simulation system for two-dimensional cellular automata in Ridge assembly language. We thank the Theoretical Division and the Center for Nonlinear Studies at Los Alamos National Laboratory for hospitality during the final stages of this work. We thank M. Prueitt at Los Alamos for making the three-dimensional illustrations, and D. Umberger for help with some Cray-1 programming. We are grateful to those mentioned and to C. Bennett, J. Crutchfield, H. Hartman, L. Hurd, J. Milnor, S. Willson, and others for discussions.

This work was supported in part by the U.S. Office of Naval Research under Contract No. N00014-80-C-0657.

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