Implementation [of texture perception model] The exact matches for a template σ in data containing elements 0 and 1 can be obtained from Sign[ListCorrelate[2 σ - 1, data] - Count[ σ , 1, 2]] + 1

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]] .

Note that each step in the evolution of any additive cellular automaton can be computed as Mod[ListCorrelate[w, list, Ceiling[Length[w]/2]], k] (See page 1087 for a discussion of partial additivity.)

For 1D elementary rules the list is {{-1}, {0}, {1}} , while for 2D 5-neighbor rules it is {{-1, 0}, {0, -1}, {0, 0}, {0, 1}, {1, 0}} . … One can specify a neighborhood configuration by giving in the same order as the offset list the color of each cell in the neighborhood. … A single step in evolution of a general cellular automaton with state a and rule number num is then given by Map[IntegerDigits[num, k, k^Length[os]] 〚 -1 - # 〛 &, Apply[Plus, MapIndexed[k^(Length[os] - First[#2]) RotateLeft[a, #1] &, os]], {-1}] or equivalently by Map[IntegerDigits[num, k, k^Length[os]] 〚 -# - 1 〛 &, ListCorrelate[Fold[ReplacePart[k #1, 1, #2 + r + 1] &, Array[0 &, Table[2r + 1, {d}]], os], a, r + 1], {d}]

More colors [in additive cellular automata] The pictures below show generalizations of rule 90 to k possible colors using the rule CAStep[k_Integer, a_List] := Mod[RotateLeft[a] + RotateRight[a], k] or equivalently Mod[ListCorrelate[{1, 0, 1}, a, 2], k] .

Flatten[{1, 0, CTList[{{1, 0, 0, 1}, {0, 1, 1, 0}}, {0, 1}, t]}] gives for example the Thue–Morse substitution system {1  {1, 0}, 0  {0, 1}} . In example (a), the elements are correlated, so that slower growth occurs than in the estimate above. In example (c), the elements are again correlated: the growth is by an average of ( √ 5 - 1)/2 ≃ 0.618 elements at each step, and the first elements on alternate steps form the same nested sequence as obtained from the substitution system {1  {1, 0}, 0  {1}} .

Fractal dimensions [of additive cellular automata] The total number of nonzero cells in the first t rows of the pattern generated by the evolution of an additive cellular automaton with k colors and weights w (see page 952 ) from a single initial 1 can be found using g[w_, k_, t_] := Apply[Plus, Sign[NestList[Mod[ ListCorrelate[w, #, {-1, 1}, 0], k] &, {1}, t - 1]], {0, 1}] The fractal dimension of this pattern is then given by the large m limit of Log[k,g[w, k,k m + 1 ]/g[w, k, k m ]] When k is prime it turns out that this can be computed as d[w_, k_:2] := Log[k,Max[Abs[Eigenvalues[With[ {s = Length[w] - 1}, Map[Function[u, Map[Count[u, #] &, #1]], Map[Flatten[Map[Partition[Take[#, k + s - 1], s, 1] &, NestList[Mod[ListConvolve[w, #], k] &, #, k - 1]], 1] &, Map[Flatten[Map[{Table[0, {k - 1}], #} &, Append[#, 0]]] &, #]]] &[Array[IntegerDigits[#, k, s] &, k s - 1]]]]]]] For rule 90 one gets d[{1, 0, 1}] = Log[2, 3] ≃ 1.58 .