arXiv subjects

Simon Wacker

Publications and source records attributed to Simon Wacker.

8 recordsLinked to original sources

arXiv2017

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.

math.GR↗

arXiv2017

The Moore and the Myhill Property For Strongly Irreducible Subshifts Of Finite Type Over Group Sets

We prove the Moore and the Myhill property for strongly irreducible subshifts over right amenable and finitely right generated left homogeneous spaces with finite stabilisers. Both properties together mean that the global transition function of each big-cellular automaton with finite set of states and finite neighbourhood over such a subshift is surjective if and only if it is pre-injective. This statement is known as Garden of Eden theorem. Pre-Injectivity means that two global configurations that differ at most on a finite subset and have the same image under the global transition function must be identical.

math.GR↗

arXiv2017

Signal Machine And Cellular Automaton Time-Optimal Quasi-Solutions Of The Firing Squad/Mob Synchronisation Problem On Connected Graphs

We construct a time-optimal quasi-solution of the firing mob synchronisation problem over finite, connected, and undirected multigraphs whose maximum degrees are uniformly bounded by a constant. It is only a quasi-solution because its number of states depends on the graph or, from another perspective, does not depend on the graph but is countably infinite. To construct this quasi-solution we introduce signal machines over continuum representations of such multigraphs and construct a signal machine whose discretisation is a cellular automaton that quasi-solves the problem. This automaton uses a time-optimal solution of the firing squad synchronisation problem in dimension one with one general at one end to synchronise edges, and freezes and thaws the synchronisation of edges in such a way that all edges synchronise at the same time.

cs.FL↗

arXiv2017

Right Amenability And Growth Of Finitely Right Generated Left Group Sets

We introduce right generating sets, Cayley graphs, growth functions, types and rates, and isoperimetric constants for left homogeneous spaces equipped with coordinate systems; characterise right amenable finitely right generated left homogeneous spaces with finite stabilisers as those whose isoperimetric constant is $0$; and prove that finitely right generated left homogeneous spaces with finite stabilisers of sub-exponential growth are right amenable, in particular, quotient sets of groups of sub-exponential growth by finite subgroups are right amenable.

math.GR↗

arXiv2016

Right Amenable Left Group Sets and the Tarski-F{\o}lner Theorem

We introduce right amenability, right F{\o}lner nets, and right paradoxical decompositions for left homogeneous spaces and prove the Tarski-F{\o}lner theorem for left homogeneous spaces with finite stabilisers. It states that right amenability, the existence of right F{\o}lner nets, and the non-existence of right paradoxical decompositions are equivalent.

math.GR↗

arXiv2016

Cellular Automata on Group Sets and the Uniform Curtis-Hedlund-Lyndon Theorem

We introduce cellular automata whose cell spaces are left homogeneous spaces and prove a uniform as well as a topological variant of the Curtis-Hedlund-Lyndon theorem. Examples of left homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations.

math.GR↗

arXiv2016

The Garden of Eden Theorem for Cellular Automata on Group Sets

We prove the Garden of Eden theorem for cellular automata with finite set of states and finite neighbourhood on right amenable left homogeneous spaces with finite stabilisers. It states that the global transition function of such an automaton is surjective if and only if it is pre-injective. Pre-Injectivity means that two global configurations that differ at most on a finite subset and have the same image under the global transition function must be identical.

math.GR↗

arXiv2012

Phase Space Invertible Asynchronous Cellular Automata

While for synchronous deterministic cellular automata there is an accepted definition of reversibility, the situation is less clear for asynchronous cellular automata. We first discuss a few possibilities and then investigate what we call phase space invertible asynchronous cellular automata in more detail. We will show that for each Turing machine there is such a cellular automaton simulating it, and that it is decidable whether an asynchronous cellular automaton has this property or not, even in higher dimensions.

cs.FL↗