[Periods in] rules 30 and 45 Maximum periods are often achieved with initial conditions consisting of a single black cell. Particularly for rule 30, however, there are quite a few exceptions. … For rule 45, the maximum possible period discussed above is achieved for n = 9 , but does not appear to be achieved for any larger n .

Maintaining simple rules [for networks] An important reason for considering models based solely on trivalent networks is that they allow simpler evolution rules to be maintained (see page 508 ). If nodes can have more than three connections, then they will often be able to evolve to have any number of connections—in which case one must give what is in effect an infinite set of rules to specify what to do for each number of connections.

The pictures below show what happens if the programs operate by applying elementary cellular automaton rules t times to 2t + 1 inputs. The plots on the left show cumulative scores in the Evens and Odds game; the array on the right indicates for each of the 256 possible rules the average number of wins it gets against each of the 256 rules. … But the rules that win most often typically seem to do so in rather simple ways.

[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. The rules can be implemented using GeneralCARule as given on page 867 .

Order dependence [in symbolic systems] The operation expr /. lhs  rhs in Mathematica has the effect of scanning the functional representation of expr from left to right, and applying rules whenever possible while avoiding overlaps. (Standard evaluation in Mathematica is equivalent to expr //. rules and uses the same ordering, while Map uses a different order.) One can have a rule be applied only once using Module[{i = 1}, expr /. lhs  rhs /; i++  1] Many symbolic systems (including the one on page 103 ) have the so-called Church–Rosser property (see page 1036 ) which implies that if a fixed point is reached in the evolution of the system, this fixed point will be the same regardless of the order in which rules are applied.

Rule 30 cryptography Rule 30 is known to have many of the properties desirable for practical cryptography. … In 1985 and soon thereafter a number of people (notably Richard and Carl Feynman ) tried to cryptanalyze rule 30, but without success. … Rule 30 has been widely used for random sequence generation, but for a variety of reasons I have not in the past much emphasized its applications in cryptography.

When m itself divides k , the cellular automaton rule is {_, b_, c_}  m Mod[b, k/m] + Quotient[c, k/m] ; in other cases the rule can be obtained by composition. … In all cases the cellular automaton rule, like the original operation on numbers, is invertible. The inverse rule, corresponding to multiplication by 1/m , can be obtained by applying the rule for multiplication by the integer k q /m , then shifting right by q positions.

The next 8 are specific to rule 110. … The universality of rule 110 presumably implies that the axiom system given is universal. … The axioms as they are stated apply to any rule 110 evolution, regardless of initial conditions.

Examples of fractal patterns produced by repeatedly applying the geometrical rules shown for a total of 12 steps. … The presence of this nested structure is an inevitable consequence of the fact that the rule for replacing an element at a particular position does not depend in any way on other elements.

The basic idea is to have rules that specify how the connections coming out of each node should be rerouted on the basis of the local structure of the network around that node. But to see the effect of any such rules, one must first find a uniform way of displaying the networks that can be produced.