The classification of abelian groups generated by time-varying automata and by Mealy automata over the binary alphabet

TL;DR

This paper classifies abelian groups generated by automata with a fixed number of states over the binary alphabet, focusing on both time-varying and Mealy automata, providing a comprehensive structural understanding.

Contribution

It provides a complete classification of abelian groups generated by n-state time-varying and Mealy automata over the binary alphabet, extending the understanding of automaton-generated groups.

Findings

Classification of abelian groups for each n

Structural characterization of automaton-generated groups

Comparison between time-varying and Mealy automata groups

Abstract

For every natural number $n$, we classify abelian groups generated by an $n$-state time-varying automaton over the binary alphabet, as well as by an $n$-state Mealy automaton over the binary alphabet.