This list can then be updated using CCAEvolveStep[f_, list_List] := Map[f, (RotateLeft[list] + list + RotateRight[list])/3] CCAEvolveList[f_, init_List, t_Integer] := NestList[CCAEvolveStep[f, #] &, init, t] where for the rule on page 157 f is FractionalPart[3#/2] & while for the rule on page 158 it is FractionalPart[# + 1/4] & .

Multicolor Turing machines [from 2-color TMs] Given rules in the form on page 888 for a Turing machine with s states and k colors the following yields an equivalent Turing machine with With[{c = Ceiling[Log[2, k]]}, (3 2 c + 2c - 7) s] states (always less than 6.03 k s ) and 2 colors: TMToTM2[rule_, s_, k_] := # /. MapIndexed[ #1  First[#2] &, Union[Map[# 〚 1, 1 〛 &, #]]] &[ With[{b = Ceiling[Log[2, k]] - 1}, Flatten[Table[ {Table[{Table[{{m, i, n, d}, c}  {{m, Mod[i, 2 n - 1 ], n - 1, d}, Quotient[i, 2 n - 1 ], 1}, {n, 2, b}, {i, 0, 2 n - 1}], Table[{ {m, i, 1, d}, c}  {{m, -1, 1, d}, i, d}, {i, 0, 1}], Table[ {{m, -1, n, d}, c}  {{m, -1, n + 1, d}, c, d}, {n, b - 1}], {{m, -1, b, d}, c}  {{0, 0, m}, c, d}}, {d, -1, 1, 2}], Table[{{i, n, m}, c}  {{ i + 2 n c, n + 1, m}, c, -1}, {n, 0, b - 1}, {i, 0, 2 n - 1}], With[{r = 2 b }, Table[ If[i + r c ≥ k, {}, Cases[rule, ({m, i + r c}  {x_, y_, z_})  {{i, b, m}, c}  {{x, Mod[y, r], b, z}, Quotient[y, r], 1})]], {i, 0, r - 1}]]}, {m, s}, {c, 0, 1}]]]] Some of these states are usually unnecessary, and in the main text such states have been pruned.

As discussed on page 1121 Schönfinkel introduced certain specific rules that he suggested could be used to build up functions defined in logic. … For the most part, however, only Schönfinkel's specific rules were ever used, and only rather specific forms of behavior were investigated. … (One can always set up the analog of types by having rules only for expressions whose heads have particular structures.)

The basic mistake is usually to make the implicit assumption that computation must be done in some rather specific way—that does not happen to be consistent with the way we have for example seen that it can be done in rule 110.

Probably this reflects not so much a similarity in underlying rules, but rather similarity in features that are most noticeable to the human visual system.

And one feature of the simple branching processes I describe is that for purely mathematical reasons, their rules always produce structures that are of limited size.

Distribution of chaotic behavior For iterated maps, unlike for discrete systems such as cellular automata, one can get continuous ranges of rules by varying parameters.

Different runs [of initially random cellular automata] The qualitative behavior seen with a given cellular automaton rule will normally look exactly the same for essentially all different large random initial conditions—just as it does for different parts of a single initial condition.

The sequential limit [in generalized substitution systems] Even when the order of applying rules does not matter, using the scheme of a sequential substitution system will often give different results.

(Any cellular automaton rule with an n -cell neighborhood corresponds to such a function; digit sequences in rule numbers correspond to explicit tables of values.)