Replication of a Binary Image on a One-Dimensional Cellular Automaton with Linear Rules

Abstract

A two-state, one-dimensional cellular automaton (1D CA) with uniform linear rules on an r + 1-neighborhood replicates any arbitrary binary image given as an initial configuration. By these linear rules, any cell gets updated by an EX-OR operation of the states of extreme (first and last) cells of its r + 1-neighborhood. These linear rules replicate the binary image in two ways on the 1D CA: one is without changing the position of the original binary image at time step t  0 and the other is by changing the position of the original binary image at time step t  0. Based on the two ways of replication, we have classified the linear rules into three types. In this paper, we have proven that the binary image of size m gets replicated exactly at time step 2k of the uniform linear rules on the r + 1-neighborhood 1D CA, where k is the least positive integer satisfying the inequality m  r ≤ 2k. We have also proved that there are exactly r * 2k -m cells between the last cell of