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

The notion of an automaton over a changing alphabet X=(Xi)i≥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=(Xi)i≥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.