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]]

At each step in the evolution of the system, every state evolves to some new state, and this process is represented in the network by an arc that joins each node to a new node.

Given the original list s and the complete prime implicant list p the so-called Quine–McCluskey procedure can be used to find a minimal list of prime implicants, and thus a minimal DNF: QM[s_, p_] := First[Sort[Map[p 〚 # 〛 &, h[{}, Range[Length[s]], Outer[MatchQ, s, p, 1]]]]] h[i_, r_, t_] := Flatten[Map[h[Join[i, r 〚 # 〛 ], Drop[r, #], Delete[Drop[t, {}, #], Position[t 〚 All, # 〛 ], {True}]]] &, First[Sort[Position[#, True] &, t]]]], 1] h[i_, _, {}] := {i} The number of steps required in this procedure can increase exponentially with the length of p .

But if IntegerDigits[x, 2] involves no consecutive 0's then for example f[x] can be obtained from 2^(b[Join[{1, 1}, #], Length[#]] &)[IntegerDigits[x, 2]] - 1 a[{l_, _}, r_] := ({l + (5r - 3#)/2, #} &)[Mod[r, 2]] a[{l_, 0}, 0] := {l + 1, 0} a[{l_, 1}, 0] := ({(13 + #(5/2)^IntegerExponent[#, 2])/6, 0} &[6l + 2] b[list_, i_] := First[Fold[a, {Apply[Plus, Drop[list, -i]], 0}, Apply[Plus, Split[Take[list, -i], #1  #2 ≠ 0 &], 1]]] (The corresponding expression for t[x] is more complicated.)

With rules set up in this way, each step in the evolution of a network system is given by NetEvolveStep[{depth_Integer, rule_List}, list_List] := Block[ {new = {}}, Join[Table[Map[NetEvolveStep1[#, list, i] &, Replace[NeighborNumbers[list, i, depth], rule]], {i, Length[list]}], new]] NetEvolveStep1[s : {___Integer}, list_, i_] := Follow[list, i, s] NetEvolveStep1[{s1 : {___Integer}, s2 : {___Integer}}, list_, i_] := Length[list] + Length[ AppendTo[new, {Follow[list, i, s1], Follow[list, i, s2]}]] The set of nodes that can be reached from node i is given by ConnectedNodes[list_, i_] := FixedPoint[Union[Flatten[{#, list 〚 # 〛 }]] &, {i}] and disconnected nodes can be removed using RenumberNodes[list_, seq_] := Map[Position[seq, #] 〚 1, 1 〛 &, list 〚 seq 〛 , {2}] The sequence of networks obtained on successive steps by applying the rules and then removing all nodes not connected to node number 1 is given by NetEvolveList[rule_, init_, t_Integer] := NestList[(RenumberNodes[#, ConnectedNodes[#, 1]] &)[ NetEvolveStep[rule, #]] &, init, t] Note that the nodes in each network are not necessarily numbered in the order that they appear on successive lines in the pictures in the main text.

With k colors each giving a string of the same length s the recurrence relation is Thread[Map[ ϕ [#][t + 1, ω ] &, Range[k] - 1]  Apply[Plus, MapIndexed[Exp[  ω (Last[#2] - 1) s t ] ϕ [#1][t, ω ] &, Range[k] - 1 /. rules, {-1}], {1}]/ √ s ] Some specific properties of the examples shown include: (a) (Thue–Morse sequence) The spectrum is essentially Nest[Range[2 Length[#]] Join[#, Reverse[#]] &, {1}, t] .

Universal cellular automaton The rules for the universal cellular automaton are {{_, 3, 7, 18, _}  12, {_, 5, 7 | 8, 0, _}  12, {_, 3, 10, 18, _}  16, {_, 5, 10 | 11, 0, _}  16, {_, 5, 8, 18, _}  7, {_, 5, 14, 0 | 18, _}  12, {_, _, 8, 5, _}  7, {_, _, 14, 5, _}  12, {_, 5, 11, 18, _}  10, {_, 5, 17, 0 | 18, _}  16, {_, _, x : (11 | 17), 5, _}  x - 1, {_, 0 | 9 | 18, x : (7 | 10 | 16), 3, _}  x + 1, {_, 0 | 9 | 18, 12, 3, _}  14, {_, _, 0 | 9 | 18, 7 | 10 | 12 | 16, x : (3 | 5)}  8 - x, {_, _, _, 8 | 11 | 14 | 17, x : (3 | 5)}  8 - x, {_, 13, 4, _, x : (0 | 18)}  x, {18, _, 4, _, _}  18, {_, _, 18, _, 4}  18, {0, _,4, _, _}  0, {_, _, 0, _, 4}  0, {4, _, 0 | 18, 1, _}  3, {4, _, _, _, _}  4, {_, _, 4, _, _}  9, {_, 4, 12, _, _}  7, {_, 4, 16, _, _}  10, {x : (0 | 18), _, 6, _, _}  x, {_, 2, 6, 15, x : (0 | 18)}  x, {_, 12 | 16, 6, 7, _}  0, {_, 12 | 16, 6, 10, _}  18, {_, 9, 10, 6, _}  16, {_, 9, 7, 6, _}  12, {9, 15, 6, 7, 9}  0, {9, 15, 6, 10, 9}  18, {9, _, 6, _, _}  9, {_, 6, 7, 9, 12 | 16}  12, {_, 6, 10, 9, 12 | 16}  16, {12 | 16, 6, 7, 9, _}  12, {12 | 16, 6, 10, 9, _}  16, {6, 13, _, _, _}  9, {6, _, _, _, _}  6, {_, _, 9, 13, 3}  9, {_, 9, 13, 3, _}  15, {_, _, _, 15, 3}  3, {_, 3, 15, 0 | 18, _}  13, {_, 13, 3, _, 0 | 18}  6, {x : (0 | 18), 15, 9, _, _}  x, {_, 6, 13, _, _}  15, {_, 4, 15, _, _}  13, {_, _, _, 15, 6}  6, {_, _, 2, 6, 15}  1, {_, _, 1, 6, _}  2, {_, 1, 6, _, _}  9, {_, 3, 2, _, _}  1, {3, 2, _, _, _}  3, {_, _, 3, 2, _}  3, {_, 1, 9, 1, 6}  6, {_, _, 9, 1, 6}  4, {_, 4, 2, _, _}  1, {_, _, _, _, x : (3 | 5)}  x, {_, _, 3 | 5, _, x : (0 | 18)}  x, {_, _, x : (1 | 2 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17), _, _}  x, {_, _, 18, 7 | 10, 18}  18, {_, _, 0, 7 | 10, 0}  0, {_, _, 0 | 18, _, _}  9, {_, _, x_, _, _}  x} where the numbers correspond to the icons shown in the main text according to The block in the initial conditions for the universal cellular automaton corresponding to a cell with color a is given by Flatten[{Transpose[{Join[{4, 18(1 - a), 6}, Table[9, {2 2 r + 1 - 3}]], 10 - 3 rtab}], Table[{9, 1}, {r}], 9, 13}] where r is the range of the rule to be emulated ( r = 1 for elementary rules) and rtab is the list of outcomes for that rule (starting with the outcome for {1, 1, (1) ...} ).

The fossil record suggests that leaves first arose roughly 400 million years ago, probably when collections of branches which lay in a plane became joined by webbing.

So for example it is straightforward to construct a triangle in a network: one just starts from a particular node, follows geodesics to two others, then joins these with a geodesic.

This same basic scheme can be used with any associative function h — Max , GCD , And , Dot , Join or whatever—so long as suitable forms for r and s are used.