A Spectral Portrait of the Elementary Cellular Automata Rule Space

Abstract

Fourier spectra of cellular automata rules give a quantitative characterisation of the frequency of bit patterns present in the limit configurations generated out of their time evolution. By means of the similarity among the spectrum of each rule, the elementary cellular automata rule space is partitioned, both for periodic and non-periodic boundary conditions, thus giving rise to rule classes of spectral equivalence. The partitions generated are formally explained in terms of the dynamical behaviour of the rules, and maps of the elementary space are given in terms of graphs that represent the spectral similarity among the rules, thus displaying a spectral portrait of how the classes relate to each other.