Rules versus constraints See page 940 .

Rule 73 The pattern has a few definite regularities. … (Any block in rule 73 consisting of an even number of black cells will evolve to a structure that remains fixed forever, as mentioned on page 954 .)

The rules are of the same kind as in the previous picture, except that in the third case shown here, the gray level of each neighboring cell is multiplied by 1.13 before the average is done.

Substitution systems [from cellular automata] Given a substitution system with rules in the form such as {1  {0}, 0  {0, 1}} used on page 889 , the rules for a cellular automaton which emulates it are obtained from SSToCA[rules_] := {{b, b, p[x_, _]}  s[x], {_, s[v : (0 | 1)], p[x_, _]}  p[v, x], {_, p[_, y_], _}  s[y], {_, s[v : (0 | 1)], _m}  m[v], {s[0 | 1], m[v : (0 | 1)], _}  s[v], {b, m[v : (0 | 1)], _}  r[v], {_, r[v : (0 | 1)], _}  (Replace[v, rules] /.

But out of a million randomly chosen rules, there will typically be a few that show complex behavior. … The rule specifies in which of the four possible directions the head should move at each step.

In general, however, one can set up network systems that have rules in which different operations are performed at different nodes, depending on the local structure of the network near each node. … But as soon as one allows dependence on slightly longer-range features of the network, much more complicated behavior immediately Evolution of network systems whose rules involve the addition of new nodes.

In a system like a cellular automaton that is based on explicit rules, it is always straightforward to take the rule and apply it to see Examples of patterns produced by systems in which not only must the arrangement of colors in each neighborhood match one of a fixed set of templates, but also a certain template from this set must occur at least once in the pattern.

In the first mechanism, randomness is explicitly introduced into the underlying rules for the system, so that a random color is chosen for every cell at each step. … The initial conditions for the system are chosen randomly, but then the subsequent evolution of the system is assumed to follow definite rules that involve no randomness.

of the causal networks for rules (e) and (f) requires following the underlying mobile automaton evolution for 2447 and 731 steps respectively. … For example, one might think that the fact that all the networks we have seen so far grow at most linearly with time must be an inevitable consequence of the one-dimensional character of the mobile Causal networks corresponding to rules (e) and (f) from page 493 , with each node explicitly labelled to specify from which step of mobile automaton evolution it is derived.

The pictures below show some simple examples of rules with this property. … The first rule shown has the effect of sorting the elements in the string.