There are some numbers whose digit sequences effectively have limited length.

But as I will discuss at length in Chapter 7 one must realize that on its own this cannot explain why randomness—or complexity—should occur in any particular case.

Sometimes this set in effect includes a large fraction of the possible digit sequences of a given length—and so essentially shows nesting.

measurements, so that the amount of information needed to pick out a single arrangement is essentially the length in digits of one such number.

Redundancy can in principle be estimated by breaking text into blocks of length b , then looking for the limit of the entropy as b  ∞ (see page 1084 ).

Cyclic tag systems [emulating tag systems] From a tag system which depends only on its first element, with rules given as in the note below, the following constructs a cyclic tag system emulating it: TS1ToCT[{n_, subs_}] := With[{k = Length[subs]}, Join[Map[v[Last[#], k] &, subs], Table[{}, {k(n - 1)}]]] u[i_, k_] := Table[If[j  i + 1, 1, 0], {j, k}] v[list_, k_] := Flatten[Map[u[#, k] &, list]] The initial condition for the tag system can be converted using v[list, k] .

Nested structure of attractors Associating with each sequence of length n (and k possible colors for each element) a number Sum[a[i] k -i , {i, n}] , the set of sequences that occur in the limit n  ∞ forms a Cantor set.

The geometrical properties of a space are in general specified by its so-called metric—and this metric allows one to compute quantities based on lengths and angles from coordinates. … (This is essentially equivalent to saying that infinitesimal arc length is related to infinitesimal coordinate distances by ds 2 = g i, j dx i dx j .) … In ordinary Euclidean space, such paths are straight lines, so that the length of a path between points with lists of coordinates a and b is just the ordinary Euclidean distance Sqrt[(a - b) .

The following generates explicit lists of n -input Boolean functions requiring successively larger numbers of Nand operations: Map[FromDigits[#, 2] &, NestWhile[Append[#, Complement[Flatten[Table[Outer[1 - Times[##] &, # 〚 i 〛 , # 〚 -i 〛 , 1], {i, Length[#]}], 2], Flatten[#, 1]]] &, {1 - Transpose[IntegerDigits[Range[2 n ] - 1, 2, n]]}, Length[Flatten[#, 1]] < 2 2 n &], {2}] The results for 2-step cellular automaton evolution in the main text were found by a recursive procedure.

TMToRM[rules_] := Module[{segs, adrs}, segs = Map[TMCompile, rules] ; adrs = Thread[Map[First, rules]  Drop[FoldList[Plus, 1, Map[Length, segs]], -1]]; MapIndexed[(# /. … Note that for a Turing machine with s states, the length of the register machine program generated is between 34s and 36s .