Note (c) for Computations in Cellular Automata…[Rules for the] squaring cellular automaton The rules are {{0, _, 3}  0, {_, 2, 3}  3, {1, 1, 3}  4, {_, 1, 4}  4, {1 | 2, 3, _}  5,{p : (0 | 1), 4, _}  7 - p, {7, 2, 6}  3, {7, _, _}  7, {_, 7, p : (1 | 2)}  p, {_, p : (5 | 6), _}  7 - p, {5 | 6, p : (1 | 2), _}  7 - p, {5 | 6, 0, 0}  1, {_, p : (1 | 2), _}  p, {_, _, _}  0} and the initial conditions consist of Append[Table[1, {n}], 3] surrounded by 0 's.

(As discussed above, any cellular automaton rule can be represented as an appropriate combination of bitwise functions.)

History [of cyclic tag systems] Cyclic tag systems were studied by Matthew Cook in 1994 in connection with working on the rule 110 cellular automaton for this book.

Probabilistic estimates [of cellular automaton properties] One way to get estimates for density and other properties of class 3 cellular automata is to make the assumption that the color of each cell at each step is completely random. … And in general, the probabilities for all 8 possible combinations of 3 cells are given by probs = Apply[Times, Table[IntegerDigits[8 - i, 2, 3], {i, 8}] /. {1  p, 0  1 - p}, {1}] In terms of these probabilities the density at the next step in the evolution of cellular automaton with rule number m is then given by Simplify[probs . … (At least for the 256 elementary cellular automata this iterated map is never chaotic.)

Backtracking [in cellular automata] If one wants to find out which of the 2 n possible initial conditions of width n evolve to yield a specific column of colors in a system like an elementary cellular automaton one can usually do somewhat better than just testing all possibilities. … The picture below shows trees obtained for the column in various elementary cellular automata.

Note (f) for The Threshold of Universality in Cellular Automata…Encodings [of cellular automaton rules] Generalizing the setup in the main text one can say that a cellular automaton i can emulate j if there is some encoding function ϕ that encodes the initial conditions a j for j as initial conditions for i , and which has an inverse that decodes the outcome for i to give the outcome for j . … Convolutional codes (related to sequential cellular automata) are the other major class of codes studied in coding theory, but in their usual form these do not seem especially useful for our present purposes.

Discrete quantum mechanics While there are many issues in finding a complete underlying discrete model for quantum phenomena, it is quite straightforward to set up continuous cellular automata whose limiting behavior reproduces the evolution of probability amplitudes in standard quantum mechanics. … From the discussion of page 1024 one can reproduce the 1D diffusion equation with a continuous block cellular automaton in which the new value of each block is given by {{1 - ξ , ξ }, { ξ , 1 - ξ }} . … One might hope to be able to get an ordinary cellular automaton with a limited set of possible values by choosing a suitable θ .

But as discussed on page 786 , nothing similar is true for equations involving only integers, and in this case finding solutions can in effect require following the evolution of a system like a cellular automaton for infinitely many steps.

(Iterating the related function BitXor[i, 2i] yields numbers whose digit sequences correspond to the rule 60 cellular automaton).

An example suggested by Stanislaw Ulam around 1960 (in a peculiar attempt to get a 1D analog of a 2D cellular automaton; see pages 877 and 928 ) starts with {1, 2} , then successively appends the smallest number that is the sum of two previous numbers in just one way, yielding {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, …} With this initial condition, the sequence is known to go on forever.