An asynchronous solution to the synchronisation problem for binary one-dimensional cellular automata

Abstract

Because cellular automata rely on totally local and synchronous processing at the neighbourhood of each cell, the issue of global synchronisation of the cell states has a long tradition in cellular automata literature. Here, the following benchmark synchronisation problem in the context of one-dimensional binary cellular automata rule is addressed: in periodic boundary condition, a rule needs to be conceived so as to lead any initial configuration to converge, at some time in future, to a cyclic regime of period 2 characterised by all cells alternating between fully homogeneous configurations of 0s and 1s. In spite of its simplicity, the problem has recently been shown not to have a solution by synchronously updating the cell states. In contrast, we provide a perfect solution to the problem, with a 4-neighbour rule, by replacing the synchronous update of the cells of the lattice with a deterministic, block sequential asynchronous update schedule. Achieving synchrony by asynchronous means represents an example that makes it evident the fact that asynchrony can be explored to give more freedom to the cells of a CA, in terms of the local information they exchange with each other. Since synchronisation phenomena and problems are pervasive in many contexts in science and engineering, it is a valuable resource to understand how synchronisation can be achieved and controlled in simple computational models, such as cellular automata.