On a free group of transformations defined by an automaton

Decidability problems in automaton semigroups

We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of $X^*$. We describe algorithms answering the word problem, and bound its complexity under some additional assumptions. We give a partial algorithm that decides in a group generated by an automaton, given $x,y$, whether an Engel identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) is satisfied. This algorithm succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, nam…

Between primitive and $2$-transitive: Synchronization and its friends

An automaton is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing $n$-state automaton has a reset word of length at most $(n-1)^2$. The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing …

Ranks of finite semigroups of one-dimensional cellular automata

Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood. In recent years, with the introduction of a generalised definition in terms of transformations of the form $τ: A^G \to A^G$ (where $G$ is any group and $A$ is any set), the the…

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 $2$.

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.

收缩分形群、Schreier图与谱

A Contracting Fractal Group, Schreier Graphs and The Spectra

本文研究了一个由5状态自动机生成的收缩分形群，分析了其自相似作用的特征，展示了对应的Schreier图，并数值近似了极限图上离散拉普拉斯算子的谱。

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…

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。

Groups and Semigroups Defined by Colorings of Synchronizing Automata

In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidable sufficient condition to have free semigroups in an automaton group. A series of examples and applications is widely discussed, in particular we show a way to color the De Bruijn automata into Mealy aut…