For it takes roughly an hour to make the picture on page 27 by hand, and it would take a few weeks to make the picture on page 29 . … And while it takes only rather basic

It turns out that often there is, and it is that even though a string may be short it may nevertheless take a great many steps to reach. … And often one finds that even a short string can take a rather large number of steps to produce. … Certainly one can take the rules for any multiway system and add transformations that immediately generate particular short strings.

LFSR cryptanalysis Given a sequence obtained from a length n LFSR (see page 975 ) Nest[Mod[Append[#, Take[#, -n] . vec], 2] &, list, t] the vector of taps vec can be deduced from LinearSolve[Table[Take[seq, {i, i + n - 1}], {i, n}], Take[seq, {n + 1, 2n}], Modulus  2] (An iterative algorithm in n taking about n 2 rather than n 3 steps was given by Elwyn Berlekamp and James Massey in 1968.)

Doubling rules [cellular automata] Rule (a) is {{0, 2, _}  5, {5, 3, _}  5, {5, _, _}  1, {_, 5, _}  1, {_, 2, _}  3, {_, 3, 2}  2, {_, 1, 2}  4, {_, 4, _}  3, {4, 3, _}  4, {4, 0, _}  2, {_, x_, _}  x} and takes 2n 2 + n steps to yield Table[1, {2n}] given input Append[Table[1, {n - 1}], 2] . Rule (b) is {{_, 2, _}  3, {_, 1, 2}  2, {3, 0, _}  1, {3, _, _}  3, {_, 3, _}  1, {_, x_, _}  x} and takes 3n steps. Rule (c) is k = 3 , r = 1 rule 5407067979 and takes 3n - 1 steps.

In this chapter what I will do is to take what we have learned, and look at a sequence of fairly specific kinds of systems in nature and elsewhere, and in each case discuss how the most obvious features of their behavior arise. … And in fact, to do this for even just one kind of system would most likely take at least another whole book, if not much more.

The networks are laid out in analogy to the space networks on page 479 , with nodes being placed on successive rows if they take progressively more connections to reach from the top node.

If both of these were white on the previous step, then take the new color of the cell to be whatever the previous color of its left-hand neighbor was. Otherwise, take the new color to be the opposite of that.

The pictures below show what happens if one takes various patterns, arranges their rows one after another in a long line, and then applies pointer-based encoding to the resulting sequences. … The basic answer is that one needs to take account of the two-dimensional nature of the patterns.

And as the pictures on the facing page show, different Turing machines can take very different numbers of steps to do the computations they do. … But while this means that for a given input each of them yields the same output, the pictures demonstrate that they usually take a different number of steps to do so.

Pattern (a) dies out after 36 steps; pattern (b) takes 1017 steps.