Repeating blocks [in cellular automata]

The discussion in the main text is mostly about repetition strictly every p steps, and no sooner. (If a system repeats for example every 3 steps, then it is inevitable that it will also repeat in the same way every 6, 9, 12, 15, etc. steps.) Finding configurations in a 1D cellular automaton that repeat with a particular period is equivalent to satisfying the kind of constraints we discussed on page 211. And as described there, if such constraints can be satisfied at all, then it must be possible to satisfy them with a configuration that consists of a repetition of identical blocks. Indeed, for period p, the length of blocks required is at most 2^{2p} (or 2^{2 p r} for range r rules).

The pictures at the bottom of the previous column summarize which periods can be obtained with various rules. Periods from 1 to 15 are represented by different rows, with period 1 at the bottom. Within each row a gray bar indicates that a particular period can be obtained with blocks of some length. The black dots indicate specific block sizes up to 25 that work.

In rule 90 (as well as other additive rules such as 60 and 150) any period can occur, but all configurations that repeat must consist of a sequence of identical blocks. For periods up to 10, examples of such blocks in rule 90 are given by the digits of

{0, 40, 24, 2176, 107904, 640, 96, 8421376, 7566031296234863392, 15561286137}

For period 1 the possible blocks are and ; for period 2 and . The total number of configurations in rule 90 that repeat with any period that divides p is always 4^{p}.

Rules 30 and 45 (as well as other one-sided additive rules) also have the property that all configurations that repeat must consist of a sequence of identical blocks. The total number of configurations in rule 30 that repeat with periods that divide 1 through 10 are {3, 3, 15, 10, 8, 99, 18, 14, 30, 163}. In general for one-sided additive rules the number of such configurations increases for large p like k^{htx p}, where h_{tx} is the spacetime entropy of page 960. (This is the analog of a standard result in dynamical systems theory about expansive homeomorphisms.)

For rules that do not show at least one-sided additivity there can be an infinite number of configurations that repeat with a given period. To find them one considers all possible blocks of length 2 p r + 1 and picks out those that after p steps evolve so that their center cell ends up the same color as it was originally. The possible configurations that repeat with period p then correspond to the finite complement language (see page 958) obtained by stringing together these blocks. For p = 2, rule 18 leaves 20 of the 32 possible length 5 blocks invariant, but these blocks can only be strung together to yield repetitions of {a, b, 0, 0}, where now a and b are not fixed, but in every case can each be either {1} or {0, 1}.

(See also page 700.)