Commutation of Cellular Automata Rules

This paper addresses the following problem: Given a onedimensional cellular automata (CAl defined over Z2 with a rule represented by an operat or X , determine all one-dimensional rules over Z2 which commute with X . It is shown th at the set of all such rules is given by th e solut ion set of a system of nonlinear Diophantine equations. This result is generalized to cover cellular automata whose rules obey a relat ion first studied by Ito , and to the case of idempotent rules. Connect ions are shown between the results presented in this paper and work on the commuting block map problem [24] , which is known to have significance for the study of Bernoulli shift systems. 1. Int roduction In severa l previous pap ers [5, 6] an operator form alism was int roduced to st udy cellular automata (CA) ru les defined over Zp where p is pr ime. Using thi s formalism , resul ts have been obtain ed on inversion of certain auto mata ru les [7], on ent ropic properti es of automata rules [8], and on the question of determining fixed points and shift cycles [9]. Recent ly a division algorit hm was discovered and the arit hmet ic of residues for one-d imension al auto mata rules st udied [1 0]. In this paper attent ion is restricted to one-dimensional automata defined over Z2. For this subset of cellular automata we consider t he following problem: Let X be t he operator represent ation of a given CA rule. Find the set { A I AX = X A }. Our goal is to prov ide a means of computi ng all ru les A that comm ut e with t he given rul e X . It is shown that t he set of all such A is determ ined by the solut ion set of a system of non linear Dioph antine equat ions, which exhibits an elegant st ructure. This result is significant for severa l reasons. It is of int rinsic interest for the insight it provides in the struc ture of CA rul e space [11], allowing us to study properti es of maximal commut ing subsets of t his space; it generalizes to rules sharing a relat ion of the form first st udied by Ito [1]; and , as will be discussed in section 6, it bears a direct relation to the "commuting block map" problem, and hence to commutat ion propert ies of endomorphisms of Bernoulli shift systems [24].