Block transformations of one-dimensional deterministic cellular automaton rules
N. Boccara, M. Röger · Journal of Physics A Mathematical and General · 1991
Given a one-dimensional cellular automaton rule f, a block transform of f is a rule T b f such that there exists between the limit sets of both rules a bijection that replaces each site value x in a configuration belonging to the limit set of f with a string x b =xx. . .x of length b in the corresponding configuration belonging to the limit set of T b f. If f is totalistic, there exists a unique totalistic block transform and a large number of nontotalistic block transforms T b f. If f is not totalistic, there are no totalistic block transforms but there still exists a large number of non-totalistic block transforms. Their number increases very rapidly with the block size b and the range of f. The range of T b f may be any integer greater than or equal to rb. Many block transforms are studied. The evolution according to rule T b f towards its limit set is discussed in terms of the annihilation of defects. These defects are often simply related to the defects characterizing the evolution according to rule f.