18 research outputs found
Cellular Automata on Group Sets
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.Comment: This is my doctoral dissertation. It consists of extended versions of
the articles arXiv:1603.07271 [math.GR], arXiv:1603.06460 [math.GR],
arXiv:1603.07272 [math.GR], arXiv:1701.02108 [math.GR], arXiv:1706.05827
[math.GR], and arXiv:1706.05893 [cs.FL
Groups, Graphs, Languages, Automata, Games and Second-order Monadic Logic
In this paper we survey some surprising connections between group theory, the
theory of automata and formal languages, the theory of ends, infinite games of
perfect information, and monadic second-order logic