A Spectral Outlook on the Elementary Cellular Automata with Cyclic Configurations and Block-Sequential Asynchronous Updates

Abstract

Spectral analysis has been used previously in the literature to analyse the space-time diagrams of the elementary cellular automata in the frequency domain, as it allows for a distinct perspective on the dynamics and limit behaviour of the rules. Asynchronous cellular automata are variants of cellular automata whose cells have their states updated at different time steps, either stochastically or deterministically. Here, by relying on the latter – the block-sequential update scheme – the entire elementary rule space is computationally probed, over cyclic configurations, according to their discrete Fourier spectra. The rule space is accounted for in its compact description, which became possible after we introduced a characterisation of dynamically equivalent rules under asynchronous updates. Since the number of possible update schemes depends on the configuration size, a reasoned choice had to be made in order to define an appropriate set of updates, which led us to three distinct families, each one with ten updates. Analysis of the spectra obtained was then carried out by means of a proposed measure of asynchronism of the updates, and by grouping the rules into similarity classes, according to the Fourier spectra entailed by each update.