Turing machines [emulating cellular automata] Given the rules for an elementary cellular automaton in the form used on page 867 , the following will construct a Turing machine which emulates it: CAToTM[rules_] := {{q[a_, b_], c : (0 | 1)}  {q[b, c], {a, b, c} /. rules, 1}, {q[_, _], x}  {p[0], 0, -1}, {p[a_], b : (0 | 1)}  {p[b], a, -1}, {p[_], x}  {q[0, 0], 0, 1}} Given a list of initial cell colors for the cellular automaton, the initial tape for the Turing machine consists of Join[{0, 0}, list, {0, 0}] surrounded by x 's, with the head of the Turing machine on the first 0 in state q[0, 0] . … By combining identical cases in rules and writing rules as compositions of ones with smaller neighborhoods one can for example readily construct Turing machines with 4 states and 3 colors that emulate 166 of the elementary cellular automata.

Domains [in code 976] Some of the effective rules for interfaces between black and white domains are easy to state. Given a flat interface, the layer of cells immediately on either side of this interface behaves like the rule 150 1D cellular automaton. … One example is totalistic code 52, which is a direct analog in the 4-neighbor case of the rule illustrated here.

Probabilistic rules [for cellular automata] There appears to be a discrete transition as a function of the size of the perturbations, similar to phase transitions seen in the phenomenon of directed percolation. Note that if one just uses the original cellular automata rules, then with any nonzero probability of reversing the colors of cells, the patterns will be essentially destroyed. With more complicated cellular automaton rules, one can get behavior closer to the continuous cellular automata shown here.

The rules shown are of the same kind as on the facing page , and include most of the 64 possibilities that leave a state that contains only white cells unchanged.

Implementation [of finite automata for nested patterns] Given the rules for a substitution system in the form used on page 931 a finite automaton (as on page 957 ) which yields the color of each cell from the digit sequences of its position is Map[Flatten[MapIndexed[#2 - 1  Position[rules, #1  _] 〚 1, 1 〛 &, Last[#], {-1}]] &, rules] This works in any number of dimensions so long as each replacement yields a block of the same cuboidal form.

Implementation [of 2D Turing machines] With rules represented as a list of elements of the form {s, a}  {sp, ap, {dx, dy}} ( s is the state of the head and a the color of the cell under the head) each step in the evolution of a 2D Turing machine is given by TM2DStep[rule_, {s_, tape_, r : {x_, y_}}] := Apply[{#1, ReplacePart[tape, #2, {r}], r + #3} &, {s, tape 〚 x, y 〛 } /. rule]

Rule 225 [with simple initial conditions] With initial conditions consisting of a single black cell, this class 3 rule yields a regular nested pattern, as shown on page 58 . … Rule 225 can be expressed as ¬ p ⊻ (q ∨ r) .

Rule 94 [with simple initial conditions] With appropriate initial conditions this class 2 rule can yield both nested and random behavior, as shown below.

Responsibility [and free will] It is often assumed that if there are definite underlying rules for our brains then it cannot be meaningful to say that we have any ultimate moral or legal responsibility for our actions. … The theory popular from the behavioral sciences tends to assume that human actions can be understood from underlying rules for the brain, and that people should be dealt with according to the rules they have—which can perhaps be modified by some form of treatment. But computational irreducibility can make it essentially impossible to find what general behavior will arise from particular rules—making it difficult to apply this theory.

[No text on this page] More steps in the evolution of the reversible cellular automaton with rule 37R.