The point is that of the 32 5-cell neighborhoods involved in the 2D cellular automaton rule, only some subset will have the property that the center cell remains unchanged after applying the rule.

Intrinsic synchronization in cellular automata Taking the rules for an ordinary cellular automaton and applying them sequentially will normally yield very different results.

Implementation [of conserved quantity test] Whether a k -color cellular automaton with range r conserves total cell value can be determined from Catch[Do[ (If[Apply[Plus, CAStep[rule, #] - #] ≠ 0, Throw[False]] &)[ IntegerDigits[i, k, m]], {m, w}, {i, 0, k m - 1}]; True] where w can be taken to be k 2r , and perhaps smaller.

With a rule given in this form, a single step in the evolution of the Turing machine can be implemented with the function TMStep[rule_List, {s_, a_List, n_}] /; (1 ≤ n ≤ Length[a]) := Apply[{#1, ReplacePart[a, #2, n], n + #3}&, Replace[{s, a 〚 n 〛 }, rule]] The evolution for many steps can then be obtained using TMEvolveList[rule_, init_List, t_Integer] := NestList[TMStep[rule, #]&, init, t] An alternative approach is to represent the complete state of the Turing machine by MapAt[{s, #}&, list, n] , and then to use TMStep[rule_, c_] := Replace[c, {a___, x_, h_List, y_, b___}  Apply[{{a, x, #2, {#1, y}, b}, {a, {#1, x}, #2, y, b}} 〚 #3 〛 &, h /. rule]] The result of t steps of evolution from a blank tape can also be obtained from (see also page 1143 ) s = 1; a[_] = 0; n = 0; Do[{s, a[n], d} = {s, a[n]} /. rule; n += d, {t}]

Ulam systems Having formulated the system around 1960, Stanislaw Ulam and collaborators (see page 877 ) in 1967 simulated 120 steps of the process shown below, with black cells after t steps occurring at positions Map[First, First[Nest[UStep[p[q[r[#1], #2]] &, {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}, #] &, ({#, #} &)[{{{0, 0}, {0, 0}}}], t]]] UStep[f_, os_, {a_, b_}] := {Join[a, #], #} &[f[Flatten[ Outer[{#1 + #2, #1} &, Map[First, b], os, 1], 1], a]] r[c_]:= Map[First, Select[Split[Sort[c], First[#1]  First[#2] &], Length[#]  1 &]] q[c_, a_] := Select[c, Apply[And, Map[Function[u, qq[#1, u, a]], a]] &] p[c_]:= Select[c, Apply[And, Map[Function[u, pp[#1, u]], c]] &] pp[{x_, u_}, {y_, v_}] := Max[Abs[x - y]] > 1 || u  v qq[{x_, u_}, {y_, v_}, a_] := x  y || Max[Abs[x - y]] > 1 || u  y || First[Cases[a, {u, z_}  z]]  y These rules are fairly complicated, and involve more history than ordinary cellular automata.

With rules in this form the network update is simply NetStep[rule_, net_]:= Block[{new}, net /. rule /. new[n_]  n + Apply[Max, Map[First, net]]] Note that just as we discussed for strings on page 1033 the direct use of /. here corresponds to a particular scheme for applying the update rule.

The pictures below show what happens if the programs operate by applying elementary cellular automaton rules t times to 2t + 1 inputs.

And by applying various forms of thresholding one can often pick out at least approximately the stored vector closest to the piece of data given.

Nesting is also seen in curves obtained by applying bitwise functions to n and 2n for successive n .

Order dependence [in symbolic systems] The operation expr /. lhs  rhs in Mathematica has the effect of scanning the functional representation of expr from left to right, and applying rules whenever possible while avoiding overlaps.