Groups generated by 3-state automata over a 2-letter alphabet, I

Classification of groups generated by 3-state automata over a 2-letter alphabet

This article contains most of the known results on the classification of groups generated by 3-state automata over a 2-letter alphabet, extending the previous papers 0704.3876 and math/0612178.

Two-letter group codes that preserve aperiodicity of inverse finite automata

We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite auto…

Amenability of Bounded Automata Groups on Infinite Alphabets

We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire functions.

Examples of defining groups by finite automata

We construct the groups $<A,B,C \;| \, A^2,B^2,C^2,(ABC)^2>$ and $<A,B \;| \, A^2,B^4,(AB)^4>$, using 3-state automata over the alphabets $\{1,2,3\}$ and $\{1,2,3,4\}$. In addition, we show, how to define direct powers of $G$ by automaton (when for $G$ it's given), keeping the alphabet.

An automata group of intermediate growth and exponential activity

We give a new example of an automata group of intermediate growth. It is generated by an automaton with 4 states on an alphabet with 8 letters. This automata group has exponential activity and its limit space is not simply connected.

Infinite finitely generated automata semigroups have infinite orbits

We prove that the semigroup generated by a finite state Mealy automaton $\mathcal{A}=(Q,A,τ)$ is infinite if and only if there exists some right-infinite word in the alphabet $A$ with infinite orbit.

元胞自动机的通用群

Universal groups of cellular automata

本研究证明任意字母表上的可逆元胞自动机（RCA）群存在由三个对合生成的通用子群，含所有有限生成RCA群的同构副本，并刻画了不同阶字母表下相关RCA群的结构性质。

Decomposition theorem for invertible substitutions on three-letter alphabet

We shall characterize the structure of invertible substitutions on three-letter alphabet. We show that any invertible substitution, after some cyclic operation, can be written as a finite product of permutations and Fibonacci's substitution. As a consequence, a matrix (of order 3 and with non-negative integral coefficients) is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices.

Some non-contracting automata groups

We add to the classification of groups generated by 3-state automata over a 2 letter alphabet given by Bondarenko et al., by showing that a number of the groups in the classification are non-contracting. We show that the criterion we use to prove a self-similar action is non-contracting also implies that the associated self-similarity graph introduced by Nekrashevych is non-hyperbolic.

局部有限群的一个可逆性刻画：基于元胞自动机

A Reversibility Characterization of Locally Finite Groups by Cellular Automata

证明群G是局部有限的当且仅当在每个字母表上，每个双射元胞自动机A^G→A^G都是可逆的；对于非局部有限群，存在无限字母表上的双射元胞自动机其逆不是元胞自动机。

A note on cellular automata

For an arbitrary group $G$ and arbitrary set $A$, we define a monoid structure on the set of all uniformly continuous functions $A^G\to A$ and then we show that it is naturally isomorphic to the monoid of cellular automata $\mathrm{CA}(G, A)$. This gives a new equivalent definition of a cellular automaton over the group $G$ with alphabet set $A$. We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.

On a characterization of locally finite groups in terms of linear cellular automata

We prove that a group $G$ is locally finite if and only if every surjective real (or complex) linear cellular automaton with finite-dimensional alphabet over $G$ is injective.

通过线性细胞自动机对局部有限群的闭图像刻画

Closed Image Characterizations of Locally Finite Groups via Cellular Automata

证明群G是局部有限的当且仅当每个线性细胞自动机V^G→V^G在prodiscrete拓扑下具有闭图像，解决了Ceccherini-Silberstein和Coornaert的开放问题7。

On Torsion-Free Semigroups Generated by Invertible Reversible Mealy Automata

This paper addresses the torsion problem for a class of automaton semigroups, defined as semigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet. The torsion problem is undecidable for automaton semigroups in general, but is known to be solvable within the well-studied class of (semi)groups generated by invertible bounded Mealy automata. We focus on the somehow antipodal class of invertible reversible Mealy automata and prove that for a wide subclass the generated semigroup is torsion-free.

The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable

We prove that a semigroup generated by a reversible two-state Mealy automaton is either finite or free of rank 2. This fact leads to the decidability of finiteness for groups generated by two-state or two-letter invertible-reversible Mealy automata and to the decidability of freeness for semigroups generated by two-state invertible-reversible Mealy automata.

连接拉丁方与双可逆自动机的组合结构

Combinatorial structures connecting Latin squares and bireversible automata

从组合视角探索字母变换器等自动机理论，将其转移集视为正交阵列类似物，讨论相关组合编码，用组合结构刻画各类自动机，提出字母变换器同构概念，推广对称性并保持双可逆自动机类别。