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

To Infinity and Beyond

We prove that if a group generated by a bireversible Mealy automaton contains an element of infinite order, then it must have exponential growth. As a direct consequence, no infinite virtually nilpotent group can be generated by a bireversible Mealy automaton

Connected Reversible Mealy Automata of Prime Size Cannot Generate Infinite Burnside Groups

The simplest example of an infinite Burnside group arises in the class of automaton groups. However there is no known example of such a group generated by a reversible Mealy automaton. It has been proved that, for a connected automaton of size at most 3, or when the automaton is not bireversible, the generated group cannot be Burnside infinite. In this paper, we extend these results to automata with bigger stateset, proving that, if a connected reversible automaton has a prime number of states, it cannot generate an infinite Burnside group

On level-transitivity and exponential growth

We prove that if the group generated by a Mealy automaton acts level-transitively on a regular rooted tree, then the semigroup generated by the dual automaton has exponential growth, hence giving a decision procedure of exponential growth for a restricted family of automaton semigroups

A Connected 3-State Reversible Mealy Automaton Cannot Generate an Infinite Burnside Group

The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, all such examples have been constructed as groups generated by non-reversible automata. Moreover, it was recently shown that 2-state reversible Mealy automata cannot generate infinite Burnside groups. Here we extend this result to connected 3-state reversible Mealy automata, using new original techniques. The results rely on a fine analysis of associated orbit trees and a new characterization of the existence of elements of infinite order

Deciding Unambiguity and Sequentiality starting from a Finitely Ambiguous Max-Plus Automaton

Finite automata with weights in the max-plus semiring are considered. The main result is: it is decidable in an effective way whether a series that is recognized by a finitely ambiguous max-plus automaton is unambiguous, or is sequential. A collection of examples is given to illustrate the hierarchy of max-plus series with respect to ambiguity.

Deciding the sequentiality of a finitely ambiguous max-plus automaton

International audienc