Additive cellular automata and global injectivity

Abstract

We employ algebraic methods to investigate the global injectivity of additive cellular automata when the state alphabet is a finite commutative ring. The set of local rules of such additive cellular automata in D ≥ 1 dimensions itself forms a ring. The invertible elements in this ring correspond to the globally injective additive cellular automata. We indicate how one determines the invertible elements and thus the globally injectibe cellular automata. In particular we show that, in contrast to the general situation, the global injectivity of additive cellular automata in D > 1 dimensions is decidable. Construction of local rules for the inverse cellular automata is illustrated through examples. A strict upper bound on the radius of the inverse cellular automata is obtained. In the special case of linear cellular automata with state alphabet the ring of integers modulo m we count the globally injective cellular automata of up to a given radius. We show that in one dimension a non-globally injective additive cellular automaton is either not injective on spatially periodic configurations of any period N ϵ N + or the set of N > 0 such that the cellular automation is injective on spatially periodic configurations of period N has positive density in N +.