Notes

(1) Some aspects of two-dimensional cellular automata were discussed in Refs. 2 and 3, and mentioned in Ref. 4. Additive two-dimensional cellular automata were considered in Ref. 5.

(2) Several computer systems were used. The first was the special-purpose pipelined TTL machine built by the M.I.T. Information Mechanics group. This machine updates all sites on a square cellular automaton lattice 60 times per second. It is controlled by a microcomputer, with software written in FORTH. It allows for five- and nine-neighbor rules, with up to four effective values for each site. The second system was a software program running on the Ridge 32 computer. The kernel is written in assembly language; the top-level interface in the C programming language. A cellular automaton lattice is typically updated about 7 times per second. Variants of the program, with kernels written in C and FORTRAN, were used on Sun Workstations, VAX, and Cray 1 computers. One-dimensional cellular automaton simulations were carried out with our CA cellular automaton simulation package, written in C, usually running on a Sun Workstation.

(3) A cellular automaton approximation to the Euler equations is given in Ref. 16.

(4) This quantity is referred to as the ``growth rate dimension'' in Ref. 20.

(5) A few examples of class-4 behavior were however found among general rules. Requests for copies of the relevant rule tables should be directed to the authors.

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