Hao Wang in 1963 constructed an example that deletes just 2 elements at each step and adds at most 3 elements—but has a large number of colors.

Random causal networks If one assumes that there are events at random positions in continuous spacetime, then one can construct an effective causal network for them by setting up connections between each event and all events in its future light cone—then deleting connections that are redundant in the sense that they just provide shortcuts to events that could otherwise be reached by following multiple connections.

Given an original DNF list s , this can be done using PI[s, n] : PI[s_, n_] := Union[Flatten[ FixedPointList[f[Last[#], n] &, {{}, s}] 〚 All, 1 〛 , 1]] g[a_, b_] := With[{i = Position[Transpose[{a, b}], {0,1}]}, If[Length[i]  1 && Delete[a, i] === Delete[b, i], {ReplacePart[a, _, i]}, {}]] f[s_, n_] := With[ {w = Flatten[Apply[Outer[g, #1, #2, 1] &, Partition[Table[ Select[s, Count[#, 1]  i &], {i, 0, n}], 2, 1], {1}], 3]}, {Complement[s, w, SameTest  MatchQ], w}] The minimal DNF then consists of a collection of these prime implicants. … 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 .

[Redundancy in] text As the picture below illustrates, English text typically remains intelligible until about half its characters have been deleted, indicating that it has a redundancy of around 0.5.

Note that although the Turing machine can emulate any number of colors in the tag system, it can only emulate directly rules that delete exactly 2 elements at each step.

One can find the sequences of length n that work by using Nest[DeleteCases[Flatten[Map[Table[Append[#, i - 1], {i, k}] &, #], 1], {___, x__, x__, ___}] &, {{}}, n] and the number of these grows roughly like 3 n/4 .

Another approach is to consider reducing whole networks to so-called minors by deleting connections or merging connected nodes, and in this case Wagner's theorem shows that any non-planar network must be exactly reducible to either K 5 or K 3,3 .

Hump m in the picture of sequence (c) shown is given by FoldList[Plus, 0, Flatten[Nest[Delete[NestList[Rest, #, Length[#] - 1], 2]&, Append[Table[1, {m}], 0], m]] - 1/2] The first 2 m elements in the sequence can also be generated in terms of reordered base 2 digit sequences by FoldList[Plus, 1, Map[Last[Last[#]]&, Sort[Table[{Length[#], Apply[Plus, #], 1 - #}& [ IntegerDigits[i, 2]], {i, 2 m }]]]] Note that the positive and negative fluctuations in sequence (f) are not completely random: although the probability for individual fluctuations in each direction seems to be the same, the probability for two positive fluctuations in a row is smaller than for two negative fluctuations in a row.

= {}, AllNet[k], q = ISets[b = Map[Table[ Position[d, NetStep[net, #, a]] 〚 1, 1 〛 , {a, 0, k - 1}]&, d]]; DeleteCases[MapIndexed[#2 〚 2 〛 - 1  #1 &, Rest[ Map[Position[q, #] 〚 1, 1 〛 &, Transpose[Map[Part[#, Map[ First, q]]&, Transpose[b]]], {2}]] - 1, {2}], _  0, {2}]]] DSets[net_, k_:2] := FixedPoint[Union[Flatten[Map[Table[NetStep[net, #, a], {a, 0, k - 1}]&, #], 1]]&, {Range[Length[net]]}] ISets[list_] := FixedPoint[Function[g, Flatten[Map[ Map[Last, Split[Sort[Part[Transpose[{Map[Position[g, #] 〚 1, 1 〛 &, list, {2}], Range[Length[list]]}], #]], First[#1]  First[#2]&], {2}]&, g], 1]], {{1}, Range[2, Length[list]]}] If net has q nodes, then in general MinNet[net] can have as many as 2 q -1 nodes.