And perhaps as a result of this, it has sometimes been thought that if one could just compute frequencies of blocks of all lengths one would have a kind of universal test for randomness. … Indeed, beyond block frequencies, the only other ones that are common are those based on correlations, spectra, and occasionally run lengths—all of which we already discussed earlier in this chapter .

possible strings of a given length are eventually generated if one starts from the string representing "true". … The pictures on the next page show the fractions of strings of given lengths that are generated on successive steps in various multiway systems.

[Number representations in] practical computing Numbers used for arithmetic in practical computing are usually assumed to have a fixed length of, say, 32 bits, and thus do not need to be self-delimiting.

[Systems based on] undirected networks Networks with connections that do not have definite directions are discussed at length in Chapter 9 , mainly as potential models for space in the universe.

Implementation [of TM cellular automaton] Given a non-deterministic Turing machine with rules in the form above, the rules for a cellular automaton which emulates it can be obtained from NDTMToCA[tm_] := Flatten[{{_, h, _}  h, {s, _c, _}  e, {s, _, _}  s, {_, s, c[i_]}  s[i], {_, s, x_}  x, {a[_, _], _s, _}  s, {_, a[x_, y_], s[i_]}  a[x, y, i], {x_, _s, _}  x, {_, _, s[i_]}  s[i], Map[Table[With[{b = (# 〚 Min[Length[#], z] 〛 &)[ {x, #} /. tm]}, If[Last[b]  -1, {{a[_], a[x, #, z], e}  h, {a[ _], a[x, #, z], s}  a[x, #, z], {a[_], a[x, #, z], _}  a[b 〚 2 〛 ], {a[x, #, z], a[w_], _}  a[b 〚 1 〛 , w], {_, a[w_], a[x, #, z]}  a[w]}, {{a[_], a[x, #, z], _}  a[b 〚 2 〛 ], {a[x, #, z], a[w_], _}  a[w], {_, a[w_], a[x, #, z]}  a[b 〚 1 〛 , w]}]], {x, Max[Map[# 〚 1, 1 〛 &, tm]]}, {z, Max[Map[Length[# 〚 2 〛 ] &, tm]]}] &, Union[Map[# 〚 1, 2 〛 &, tm]]], {_, x_, _}  x}]

LFSR sequences Often referred to as pseudonoise or PN sequences, maximal length linear feedback shift register sequences have repetition period 2 n - 1 and are generated by shift registers that go through all their possible states except the one consisting of all 0's, as discussed on page 974 . … This means that every block with length up to n (except all 0's) must occur with equal frequency.

The plot below shows the lengths of the successive regions of regularity visible on the right-hand edge of the picture on page 126 over the course of the first million steps. If one works directly with a digit sequence of fixed length, dropping any carries on the left, then a repetitive pattern is typically obtained fairly quickly.

But from the network, one finds that now an infinite collection of other blocks are forbidden, beginning with the length 12 block .

On pages 363 – 369 of Chapter 8 , however, I discuss some general issues of modelling, and in Chapter 10 I consider at length not only practical but also foundational questions about perception and to some extent general thinking and consciousness.

If all 2 b possible blocks of length b occur with equal probability, then the Huffman codewords will consist of blocks equivalent to the original ones. In an opposite extreme, blocks with probabilities 1/2, 1/4, 1/8, ... will yield codewords of lengths 1, 2, 3, ...