Chapter 6: Starting from Randomness

Section 7: The Notion of Attractors

Excluded blocks [in cellular automaton evolution]

As the evolution of a cellular automaton proceeds, the set of sequences that can appear typically shrinks, with progressively more blocks being excluded. In some cases the set of allowed sequences forms a so-called finite complement language (or subshift of finite type) that can be characterized completely just by saying that some finite set of blocks are excluded. But whenever the overall behavior is at all complex, there tend to be an infinite set of blocks excluded, making it necessary to use a network of the kind discussed in the main text. If there are n nodes in such a network, then if any blocks are excluded, the shortest one of them must be of length less than n. And if there are going to be an infinite number of excluded blocks, there must be additional excluded blocks with lengths between n and 2n. In rule 126, the lengths of the shortest newly excluded blocks on successive steps are 0, 3, 12, 13, 14, 14, 17, 15. It is common to see such lengths progressively increase, although in principle they can decrease by as much as 2r from one step to the next. (As an example, in rule 54 they decrease from 9 to 7 between steps 4 and 5.)

From Stephen Wolfram: A New Kind of Science [citation]