Class 4 [cellular automaton] rules Other examples of class 4 totalistic rules with k = 3 colors include 357 (page 282 ), 438, 600, 792, 924, 1038, 1041, 1086, 1329 (page 282 ), 1572, 1599 (see page 70 ), 1635 (see page 67 ), 1662, 1815 (page 236 ), 2007 (page 237 ) and 2049 (see page 68 ).

The particular cellular automaton he constructed in 1952-3 had 29 possible colors for each cell, and complicated rules specifically set up to emulate the operations of components of an electronic computer and various mechanical devices. … (Ulam tried to construct a 1D analog, but ended up not with a cellular automaton, but instead with the sequences based on numbers discussed on page 908 .) … Despite the lack of investigation in science, one example of a cellular automaton did enter recreational computing in a major way in the early 1970s.

Other models [of crystal growth] There are many ways to extend the simple cellular automata shown here. … A general feature of cellular automaton rules is that they are fundamentally local. … It turns out, however, that many seemingly long-range effects can actually be captured quite easily in cellular automata.

Undecidability [of cellular automaton classes] Almost any definite procedure for determining the class of a particular rule will have the feature that in borderline cases it can take arbitrarily long, often formally showing undecidability, as discussed on page 1138 .

My cellular automaton cryptography system follows the principle of being based on a problem that is easy to state. And indeed the general problem of finding initial conditions for a cellular automaton is NP-complete (see page 767 ).

[Cellular automaton state] network properties The number of nodes and connections at step t > 1 are: rule 108: 8 , 13 ; rule 128: 2t , 2t + 2 ; rule 132: 2t + 1 , 3t + 3 ; rule 160: (t + 1) 2 , (t + 1)(t + 3) ; rule 184: 2t , 3t + 1 . For rule 126 the first few cases are {{1, 2}, {3, 5}, {13, 23}, {106, 196}, {2866, 5474}} and for rule 110 they are {{1,2}, {5, 9}, {20, 38}, {206, 403}, {1353, 2666}} The maximum size of network that can possibly be generated after t steps of cellular automaton evolution is 2 k 2 r t - 1 .

Analogous equations arise in probabilistic approximations to systems like cellular automata, as on page 953 . … (For the cellular automaton on page 339 the simple condition for equilibrium is p  p 2 (3 - 2p) , which correctly implies that 0, 1/2 and 1 are possible equilibrium densities.)

Note that the evolution rules are highly non-local, and are rather unlike those, say, in a cellular automaton.

The Threshold of Universality in Cellular Automata…With strictly repetitive initial conditions—like any cellular automaton—this must yield purely repetitive behavior.

The locality of cellular automaton rules was thought of as making them the analog for symbol sequences of continuous functions for real numbers (compare page 869 ). Of particular interest were invertible (reversible) cellular automaton rules, since systems related by these were considered conjugate or topologically equivalent. … When I started working on cellular automata in the early 1980s I wanted to see how far one could get by following ideas of statistical mechanics and dynamical systems theory and trying to find global characterizations of the possible behavior of individual cellular automata.