Changing the Neighborhood of Cellular Automata

Abstract

In place of the traditional definition of a cellular automaton CA = (S, Q, N, f), a new definition (S, Q, f _n , ν) is given by introducing an injection called the neighborhood function ν: {0, 1,...,n − 1}→S, which provides a connection between the variables of local function f _n of arity n and neighbors of CA: image(ν) is a neighborhood of size n. The new definition allows new analysis of cellular automata. We first show that from a single local function countably many CA are induced by changing ν and then prove that equivalence problem of such CA is decidable. Then we observe what happens if we change the neighborhood. As a typical research topics, we show that reversibility of 2 states 3 neighbors CA is preserved from changing the neighborhood, but that of 3 states CA is not.