Whichever color was initially more common again eventually dominates, though with this rule it takes somewhat longer for this to occur.

The distributive laws appear at positions 2813 and 2814 in the list; it takes a long proof to obtain the second one from preceding theorems.

Pictures (f) and (g) then show how one can take the pattern of connectivity from picture (e) and lay out the updating events as nodes so as to produce an orderly network. … Each updating event, corresponding to each node in the network, can be imagined to take place at some point in spacetime.

One might however imagine that as a first approximation one could take account of underlying apparent randomness just by saying that there are certain probabilities for particles to behave in particular ways. … For in particular, if one takes two particles that have come from a single source, then the result of a measurement on one of them is found in a sense to depend too much on what measurement gets done on the other—even if there is not enough time for information travelling at the speed of light to get from one to the other.

Instead of taking each square to have a color that is chosen completely independently, one can for example take blocks of squares of some given length to have their colors chosen together. And in this case the best model is again straightforward to find: it simply takes the probabilities for different blocks to be equal to the frequencies with which these blocks occur in the data.

What does it take to find the outcome in this case? … For it implies that even if in principle one has all the information one needs to work out how some particular system will behave, it can still take an irreducible amount of computational work actually to do this.

But in the pictures on the right the results for particular cells are instead found by procedures that take much less computational effort. … But now what the cellular automata do is to take specifications of positions of cells, and then in effect compute directly from these the colors of cells.

But any replacements that take connected clusters and yield connected clusters must always maintain the connectedness of any network.

With a rule given in this form, each step in the evolution of the mobile automaton corresponds to the function MAStep[rule_, {list_List, n_Integer}] /; (1 < n < Length[list]) := Apply[{ReplacePart[list, #1, n], n + #2}&, Replace[Take[list, {n - 1, n + 1}], rule]] The complete evolution for many steps can then be obtained with MAEvolveList[rule_, init_List, t_Integer] := NestList[MAStep[rule, #]&, init, t] (The program will run more efficiently if Dispatch is applied to the rule before giving it as input.) For the mobile automaton on page 73 , the rule can be given as {{1, 1, 1}  {{0, 0, 0}, -1}, {1, 1, 0}  {{1, 0, 1}, -1}, {1, 0, 1}  {{1, 1, 1}, 1}, {1, 0, 0}  {{1, 0, 0}, 1}, {0, 1, 1}  {{0, 0, 0}, 1}, {0, 1, 0}  {{0, 1, 1}, -1}, {0, 0, 1}  {{1, 0, 1}, 1}, {0, 0, 0}  {{1, 1, 1}, 1}} and MAStep must be rewritten as MAStep[rule_, {list_List, n_Integer}] /; (1 < n < Length[list]) := Apply[{Join[Take[list, {1, n - 2}], #1, Take[list, {n + 2, -1}]], n + #2}&, Replace[Take[list, {n - 1, n + 1}], rule]]

In each case a string consisting of a single white element is eventually generated—but this takes respectively 12, 28 and 34 steps to happen.