So if one makes a list of all possible axiom systems—say starting with the simplest—where in such a list should one expect to see axiom systems that correspond to traditional areas of mathematics?

Implementation [of sound] ListPlay[data] in Mathematica generates sound output by treating the elements of data as successive samples in the waveform of the sound, typically with a default sample rate of 8000 Hz.

In general a length n list can require about n steps. But if the list contains a nested sequence, say generated using a substitution system, then about Log[n] steps should be sufficient.

In case (c), the following gives a list of the numbers of nodes generated up to step t : FoldList[Plus, 1, Join[{1, 4, 12, 10, -20, 6, 4}, Map[d, IntegerDigits[Range[4, t - 5], 2]]]] d[{___, 1}] = 1 d[{1, p : 0 .., 0}] := -Apply[Plus, 4 Range[Length[{p}]] - 1] + 6 d[{__, 1, p : 0 .., 0}] := d[{1, p, 0}] - 7 d[{___, p : 1 .., q : 0 ..., 1, 0}] := 4 Length[{p}] + 3 Length[{q}] + 2 d[{___, p : 1 .., 1, 0}] := 4 Length[{p}] + 2

Implementation [of symbolic systems] The evolution for t steps of the first symbolic system shown can be implemented simply by NestList[#/. ℯ [x_][y_]  x[x[y]]&, init, t]

Numbering scheme [for 2D constraints] The constraint numbered n allows the templates at Position[IntegerDigits[n, 2, 32], 1] in the list below.

Hashing Given data in the form of sequences of numbers between 0 and k - 1 , a very simple hashing scheme is just to compute FromDigits[Take[list, n], k] . … For a fair fraction of values of m , the hash codes obtained from this scheme change whenever any element of list is changed. If m = k s -1 then it turns out that interchanging a pair of adjacent length s blocks in list never affects the result.

Implementation [of 3/2 system] The evolution for t steps of the system at the top of the page can be computed simply by NestList[If[EvenQ[#], 3#/2, 3(# + 1)/2] &, 1, t]

Most of these operations are just done by applying ListConvolve with simple kernels. … An example originally popular in the earth and environmental sciences is so-called mathematical morphology, based on "dilation" of data consisting of 0's and 1's with a "structuring element" σ according to Sign[ListConvolve[ σ , data, 1, 0]] (as well as the dual operation of "erosion").

Iterated run-length encoding Starting say with {1} consider repeatedly replacing list by (see page 1070 ) Flatten[Map[{Length[#], First[#]} &, Split[list]]] The resulting sequences contain only the numbers 1, 2 and 3, but otherwise at first appear fairly random.