The finiteness of a group generated by a 2-letter invertible-reversible   Mealy automaton is decidable
  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. 
Ines Klimann (LIAFA)
cs.FL math.GR