N. Boccara, TRANSFORMATIONS OF ONE-DIMENSIONAL CELLULAR-AUTOMATON RULES BY TRANSLATION-INVARIANT LOCAL SUBJECTIVE MAPPINGS, Physica. D, 68(3-4), 1993, pp. 416-426
If M is a noninvertible translation-invariant local surjective mapping
, it is shown that some local one-dimensional deterministic cellular a
utomaton rules F have a transform PHI by M defined by PHI . M = M . F.
When it exists PHI is local and its Wolfram's class is the same as F.
The evolution of a cellular automaton according to rule PHI is simply
related to the evolution according to rule F. In the case of class-3
rules, the evolution to the attractor may often be viewed as particle-
like structures evolving in a regular background. If the structure of
these particles and their interactions for a rule F are known, then th
e structure and interactions of the transformed particles for rule PHI
are also known. If M is a nontrivial invertible translation-invariant
local surjective mapping, PHI always exists, but it is, in general, s
ite- and time-dependent.