# Cellular Automata on Group Sets

## TL;DR

This paper generalizes cellular automata to act on various homogeneous spaces, extending classical theorems and introducing signal machines for synchronization problems.

## Contribution

It introduces cellular automata on left-homogeneous spaces and proves generalized classical theorems, also presenting signal machines for synchronization.

## Key findings

- Generalized Curtis-Hedlund-Lyndon theorem for these automata
- Extended Tarski-Følner and Garden-of-Eden theorems
- Time-optimal synchronization solution on graphs

## Abstract

We introduce and study cellular automata whose cell spaces are left-homogeneous spaces. Examples of left-homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; uniform tilings acted on by symmetries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations. For such automata and spaces, we prove, in particular, generalisations of topological and uniform variants of the Curtis-Hedlund-Lyndon theorem, of the Tarski-F{\o}lner theorem, and of the Garden-of-Eden theorem on the full shift and certain subshifts. Moreover, we introduce signal machines that can handle accumulations of events and using such machines we present a time-optimal quasi-solution of the firing mob synchronisation problem on finite and connected graphs.