Cellular Automata for the Self-stabilisation of Colourings and Tilings

Abstract

We examine the problem of self-stabilisation, as introduced by Dijkstra in the 1970’s, in the context of cellular automata stabilising on k-colourings, that is, on infinite grids which are coloured with k distinct colours in such a way that adjacent cells have different colours. Suppose that for whatever reason (e.g., noise, previous usage, tampering by an adversary), the colours of a finite number of cells in a valid k-colouring are modified, thus introducing errors. Is it possible to reset the system into a valid k-colouring with only the help of a local rule? In other words, is there a cellular automaton which, starting from any finite perturbation of a valid k-colouring, would always reach a valid k-colouring in finitely many steps? We discuss the different cases depending on the number of colours, and propose some deterministic and probabilistic rules which solve the problem for $k\ne 3$. We also explain why the case $k=3$ is more delicate. Finally, we propose some insights on the more general setting of this problem, passing from k-colourings to other tilings (subshifts of finite type).

S. Taati—The work of ST was partially supported by NWO grant 612.001.409 of Tobias Müller.