# On groups generated by bi-reversible automata: the two-state case over a changing alphabet

## TL;DR

This paper investigates groups generated by two-state bi-reversible automata over changing alphabets, establishing conditions for free group generation and characterizing groups over binary sequences.

## Contribution

It extends bi-reversibility to changing alphabets and characterizes when such automata generate free groups or other groups, especially over binary sequences.

## Key findings

- A non-abelian free group is generated iff the alphabet sequence is unbounded.
- Characterization of groups generated by two-state bi-reversible automata over binary alphabets.
- Extension of bi-reversibility concept to automata over changing alphabets.

## Abstract

The notion of an automaton over a changing alphabet $X=(X_i)_{i\geq 1}$ is used to define and study automorphism groups of the tree $X^*$ of finite words over $X$. The concept of bi-reversibility for Mealy-type automata is extended to automata over a changing alphabet. It is proved that a non-abelian free group can be generated by a two-state bi-reversible automaton over a changing alphabet $X=(X_i)_{i\geq 1}$ if and only if $X$ is unbounded. The characterization of groups generated by a two-state bi-reversible automaton over the sequence of binary alphabets is established.