The concept of duality for automata over a changing alphabet and generation of a free group by such automata

TL;DR

This paper extends the concept of automata over a changing alphabet, adapting dual automaton methods to demonstrate how such automata can generate a free nonabelian group with just two states.

Contribution

It introduces a new framework for automata over changing alphabets and shows how they can generate free groups, expanding the understanding of automaton groups.

Findings

Automata over changing alphabets can represent free nonabelian groups.

Modified dual automaton methods are effective in this context.

A 2-state automaton suffices to generate a free group.

Abstract

In the paper, we deal with the notion of an automaton over a changing alphabet, which generalizes the concept of a Mealy-type automaton. We modify the methods based on the idea of a dual automaton and its action used by B. Steinberg et al. (2011) and M. Vorobets and Ya. Vorobets (2007,2010) and adapt them to automata over a changing alphabet. We show that this modification provides some naturally defined automaton representations of a free nonabelian group by a 2-state automaton over a changing alphabet.