Typically the network topology of a foam continually rearranges itself through cascades of seemingly random T1 processes (rule (b) from page 511 ), with regions that reach zero size disappearing through T2 processes (reversed rule (a)).

The data that is manipulated by programs can be continuous, as can the elements of their rules.

Implementation [of causal networks] Given a list of successive positions of the active cell, as from Map[Last, MAEvolveList[rule, init, t]] (see page 887 ), the network can be generated using MAToNet[list_] := Module[{u, j, k}, u[_] = ∞ ; Reverse[ Table[j = list 〚 i 〛 ; k = {u[j - 1], u[j], u[j + 1]}; u[j - 1] = u[j] = u[j + 1] = i; i  k, {i, Length[list], 1, -1}]]] where nodes not yet found by explicit evolution are indicated by ∞ .

In general, any network that represents the evolution of a system with definite rules will have the same basic form. … The picture below shows the network obtained from a class 1 cellular automaton (rule 254) with 4 cells and thus 16 possible states. … The pictures below give corresponding results for a class 2 cellular automaton (rule 132).

And second, that in cases like rule 90 simple initial conditions led to nested or fractal patterns. … The paper also contained a small picture of rule 30 started from a single black cell. … And I do know that for example on June 1, 1984 I printed out pictures of rule 30, rule 110 and k = 2 , r = 2 totalistic code 10 (see note below ), took them with me on a flight from New York to London, and a few days later was in Sweden talking about randomness in rule 30 and its potential significance.

biological organisms are instead generated by processes whose basic rules are extremely simple—and are often chosen essentially at random.

So if two expressions are equivalent then by applying the rules of the appropriate axiom system it must be possible to get from one to the other—and in fact the picture on page 775 shows an example of how Values of expressions obtained by using operators of various forms.

I invented the rule 30 cellular automaton random number generator in 1985. … Rule 30 has no such properties. … Note that rule 45 can be used as an alternative to rule 30.

(a_  s_)  (rtab 〚 i k + a + 1 〛  k 2r (s - 1) + 1 + Mod[i k + a, k 2r ]), {i, 0, k 2r - 1}]&, net], 1] where here elementary rule 126 is specified for example by {2, 1, Reverse[IntegerDigits[126, 2, 8]]} . Starting from the set of all possible sequences, as given by AllNet[k_:2] := {Thread[(Range[k] - 1)  1]} this then yields for rule 126 the network {{0  1, 1  2}, {1  3, 1  4}, {1  1, 1  2}, {1  3, 0  4}} It is always possible to find a minimal network that represents a set of sequences. … The result from MinNet for rule 126 is {{1  3}, {0  2, 1  1}, {0  2,1  3}} .

Labelling each shape and orientation with a different color, the behavior of this system can be reproduced with equal-sized squares using the rule {3  {{1, 0}, {3, 2}}, 2  {{1}, {3}}, 1  {{3, 2}}, 0  {{3}}} starting from initial condition {{3}} .