For nearest-neighbor rules, it suffices that each node has the same number of connections. For longer-range rules, the network must satisfy constraints of the kind discussed on page 483 . … If the connections at each node are not labelled, then only totalistic cellular automaton rules can be implemented.

[No text on this page] Network systems in which the rule depends on the number of distinct nodes reached by going up to distance two away from each node.

And as a result, the center column of rule 30 cannot be considered truly random according to such definitions. … And as discussed above, there is good evidence that the center column of rule 30 is indeed random according to all reasonable definitions of this kind. … And in Mathematica—ever since it was first released— Random[Integer] has generated 0's and 1's using exactly the rule 30 cellular automaton.

But with rules (e), (f) and (g) the networks are more complicated, and begin to seem somewhat random. … And thus, for example, even though the underlying rules for a mobile automaton always treat space and time very differently, the causal networks that emerge nevertheless often exhibit a kind of uniform randomness in which space and time somehow work in many respects the same. … And for example the pictures at the top of the facing page show the causal networks for rules (e) and (f) from the previous page —but now with each node numbered to specify the step of mobile automaton evolution from which it was derived.

One might have thought that states obtained from slices at different angles would inevitably be consistent only with different sets of underlying rules. But in fact this is not the case, and instead the exact same rules can reproduce slices at all angles. And this is a consequence of the fact that the substitution system on page 518 has the property of causal invariance—so that it gives the same causal network independent of the scheme used to apply its underlying rules.

But I suspect that this fact will be very much easier to establish for some systems than for others—with rule 110 being one of the easiest cases. … In proving the universality of rule 110, we were able to follow essentially the same basic approach. … And an example where this can be potentially done is rule 73.

Related [texture perception] models Rather than requiring particular templates to be matched, one can consider applying arbitrary cellular automaton rules. The pictures below show results from a single step of the 16 even-numbered totalistic 5-neighbor rules.

Visualization [of 2D Turing machines] The pictures below show the 2D position of the head at 500 successive steps for the rules on page 185 . … An example is the 3-state rule

Implementation [of network cellular automata] Given a network represented as a list in which element i is {a, i , b } , where a is the node reached by the above connection from node i , and b is the node reached by the below connection, each step corresponds to NetCAStep[{rule_, net_}, list_] := Map[Replace[#, rule] &, list 〚 net 〛 ]

Turing machines [from cellular automata] Given any Turing machine with rules in the form used on page 888 and k possible colors for each cell, a cellular automaton which emulates it can be constructed using TMToCA[rules_, k_:2] := Flatten[{Map[g[#, k]&, rules], {_, x_, _}  x}] g[{s_, a_}  {sp_, ap_, d_}, k_] := {If[d  1, Identity, Reverse][{k s + a, x_, _}]  k sp + x, {_, k s + a, _}  ap} If the Turing machine has s states for its head, then the cellular automaton has k (s+1) colors for each cell.