The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable

TL;DR

This paper proves that the finiteness of groups generated by two-letter invertible-reversible Mealy automata is decidable, establishing a clear dichotomy and enabling algorithmic determination of their structure.

Contribution

It establishes that semigroups from reversible two-state Mealy automata are either finite or free of rank 2, leading to decidability results for related groups and semigroups.

Findings

Semigroup generated by reversible two-state Mealy automaton is either finite or free of rank 2

Decidability of finiteness for groups from two-state or two-letter invertible-reversible Mealy automata

Decidability of freeness for semigroups from two-state invertible-reversible Mealy automata

Abstract

We prove that a semigroup generated by a reversible two-state Mealy automaton is either finite or free of rank 2. This fact leads to the decidability of finiteness for groups generated by two-state or two-letter invertible-reversible Mealy automata and to the decidability of freeness for semigroups generated by two-state invertible-reversible Mealy automata.