Rule (f) has a period equal to the maximum of 16.

Implementation [of operators from axioms] Given an axiom system in the form {f[a, f[a, a]]  a, f[a, b]  f[b, a]} one can find rule numbers for the operators f[x, y] with k values for each variable that are consistent with the axiom system by using Module[{c, v}, c = Apply[Function, {v = Union[Level[axioms, {-1}]], Apply[And, axioms]}]; Select[Range[0, k k 2 - 1], With[{u = IntegerDigits[#, k, k 2 ]}, Block[{f}, f[x_, y_] := u 〚 -1 - k x - y 〛 ; Array[c, Table[k, {Length[v]}], 0, And]]] &]] For k = 4 this involves checking nearly 16 4 or 4 billion cases, though many of these can often be avoided, for example by using analogs of the so-called Davis–Putnam rules.

For 3-input functions, corresponding to elementary cellular automaton rules, 56 of the 256 possibilities turn out to be universal. … Other universal functions include rules 1, 45 and 202 ( If[a  1, b, c] ), but not 30, 60 or 110.

Rule 22 [with simple initial conditions] Randomness is obtained with initial conditions consisting of two black squares 4 m positions apart for any m ≥ 2 . … There is also a region of repetitive behavior on each side of the pattern; the random part in the middle expands at about 0.766 cells per step—the same speed that we found on page 949 that changes spread in this rule.

Neighbor-independent [network substitution] rules Even though the same replacement is performed at each node at each step, the networks produced are not homogeneous. … Note that there is no upper limit on the dimension that can be obtained with appropriate neighbor-independent rules.

Initial conditions [for the universe] To find the behavior of the universe one potentially needs to know not only its rule but also its initial conditions. Like the rule, I suspect that the initial conditions will turn out to be simple.

For any rule, s n for large n will behave like κ n , where κ is the largest eigenvalue of m . For rule 126 after 1 step, the characteristic polynomial for m is x 3 - 2 x 2 + x - 1 , giving κ ≃ 1.755 . … The value of this for successive t never increases; for the first 3 steps in rule 126 it is for example approximately 1, 0.811, 0.793.

And as the pictures on the facing page indicate, for any curve like Sin[x] + Sin[ α x] the relative arrangements of these crossing points turn out to be related to the output of a generalized substitution system in which the rule at each step is obtained from a term in the continued fraction representation of ( α – 1)/( α + 1) .

Examples of patterns produced by the evolution of each of the simplest possible symmetrical one-dimensional cellular automaton rules, starting from a random initial condition.

Examples of cellular automata with additive rules.