On a series of finite automata defining free transformation groups

On Finite Monoids of Cellular Automata

For any group $G$ and set $A$, a cellular automaton over $G$ and $A$ is a transformation $τ: A^G \to A^G$ defined via a finite neighborhood $S \subseteq G$ (called a memory set of $τ$) and a local function $μ: A^S \to A$. In this paper, we assume that $G$ and $A$ are both finite and study various algebraic properties of the finite monoid $\text{CA}(G,A)$ consisting of all cellular automata over $G$ and $A$. Let $\text{ICA}(G;A)$ be the group of invertible cellular automata over $G$ and $A$. In the first part, using information on the conjugacy classes of subgroups of $G$, we give a detailed de…

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.

Freeness of automata groups vs boundary dynamics

We prove that the boundary dynamics of the (semi)group generated by the enriched dual transducer characterizes the algebraic property of being free for an automaton group. We specialize this result to the class of bireversible transducers and we show that the property of being not free is equivalent to have a finite Schreier graph in the boundary of the enriched dual pointed on some essentially non-trivial point. From these results we derive some consequences from the dynamical, algorithmic and algebraic point of view. In the last part of the paper we address the problem of finding examples of…

Kolakoski序列的群论

Group Theory of the Kolakoski Sequence

研究Kolakoski序列的迭代运行长度解码自动机的变换群及其递归结构，确定奇数次迭代下的最大轨道数。

Groups Synchronizing a Transformation of Non-Uniform Kernel

This paper concerns the general problem of classifying the finite deterministic automata that admit a synchronizing (or reset) word. (For our purposes it is irrelevant if the automata has initial or final states.) Our departure point is the study of the transition semigroup associated to the automaton, taking advantage of the enormous and very deep progresses made during the last decades on the theory of permutation groups, their geometry and their combinatorial structure. Let $X$ be a finite set. We say that a primitive group $G$ on $X$ is {\em synchronizing} if $G$ together with any non-inve…

LEF-groups and endomorphisms of symbolic varieties

Let $G$ be a group and let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic zero. Denote $A=X(k)$ the set of rational points of $X$. We investigate invertible algebraic cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by some morphism of algebraic varieties $X^M \to X$ where $M$ is a finite memory. When $G$ is locally embeddable into finite groups (LEF), then we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata. Generalizations are also obtained for finite product Hopfian…

Lamplighter groups, bireversible automata and rational series over finite rings

We realize lamplighter groups $A\wr \mathbb Z$, with $A$ a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize $A\wr \mathbb Z$ as a bireversible automaton group if and only if the $2$-Sylow subgroup of $A$ has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of $2$.

关于一类由自动机生成的多上下文无关群

On a class of poly-context-free groups generated by automata

本文研究树自动机群及其推广，通过引入可约自动机群类证明其作为多上下文无关群直极限的结构定理，并证明至少含两个生成元的树自动机群非有限生成且为顺从群。