Computing by Temporal Order: Asynchronous Cellular Automata

TL;DR

This paper investigates the computational capabilities of elementary cellular automata under asynchronous updates on finite rings, demonstrating how certain rules can generate all functions or bijections on binary states for small sizes.

Contribution

It shows that ECA rule 57 can compute all functions and bijections on small finite rings through asynchronous updates, expanding understanding of cellular automata computational power.

Findings

ECA57 can map any binary configuration to any other for n <= 11.

ECA57 can compute all even bijections on configurations for n <= 10.

Characterization of non-bijective functions computable by asynchronous rules.

Abstract

Our concern is the behaviour of the elementary cellular automata with state set 0,1 over the cell set Z/nZ (one-dimensional finite wrap-around case), under all possible update rules (asynchronicity). Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule. We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules.