Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

If one starts from a given initial string, then typically one will generate different strings by applying different replacements. But if one is going to get the same causal network, then it must always be the case that there are replacements one can apply to the strings one has generated that yield the same final string.

Given only an output list NestList[Mod[a #, m]&, x, n] parameters {a, m} that generate the list can be found for sufficiently large n from With[{ α = Apply[(#2 . Rest[list]/#1) &, Apply[ ExtendedGCD, Drop[list, -1]]]}, {Mod[ α , #], #} &[ Fold[GCD[#1, If[#1  0, #2, Mod[#2, #1]]] &, 0, ListCorrelate[{ α , -1}, list]]]] With slightly more effort both x and {a, m} can be found just from First[IntegerDigits[list, 2, p]] .

Order of replacements [in sequential substitution systems] For many sequential substitution systems the evolution effectively stops because a string is produced to which none of the replacements given apply. In most sequential substitution systems there is more than one possible replacement that can in principle apply at a particular step, so the order in which the replacements are tried matters.

[Generating] arbitrary transformations [between networks] By applying the string transformation rules on page 1035 at appropriate locations, it is possible to transform any string of A 's and B 's to any other. And the analog of this for networks is that by applying the rules shown below at appropriate locations it is possible to transform any network into any other.

For the generalization of rule 90, the values of the left and right cells are added together, and the value of the cell on the next step is then found by applying the continuous generalization of the modulo 2 function shown at the right.

[No text on this page] Examples of applying various rules for cellular automaton evolution to the sequences from page 594 .

 [i, k], {i, 0, t - 1}, {j, stot}, {k, j + 1, stot}], Table[Apply[Or, Table[  [i, j], {j, n + i, Max[0, n - i], -2}]], {i, 0, t}], Table[! …  [i, k], {i, 0, t}, {j, n + i, Max[0, n - i], -2}, {k, j + 2, n + i}], Table[Apply[Or, Table[  [i, j, k], {k, 0, ktot - 1}]], {i, 0, t - 1}, {j, Max[1, n - i], n + i}], Table[! …  [i, j, z 〚 1, 2 〛 ] || ## &, Apply[Sequence, Map[If[i < t - 1, {  [i + 1, # 〚 1 〛 ],  [ i + 1, j - # 〚 3 〛 ],  [i + 1, j, # 〚 2 〛 ]}, {  [i + 1, j - # 〚 3 〛 ]}]&, z 〚 2 〛 ]]]], rules], {i, 0, t - 1}, {j, n + i, Max[1, n - i], -2}], Apply[Or, Table[  [i, 0], {i, n, t, 2}]]} /.