The finiteness problem for automaton semigroups is undecidable

The finiteness problem for automaton groups and semigroups has been widely studied, several partial positive results are known. However we prove that, in the most general case, the problem is undecidable. We study the case of automaton semigroups. Given a NW-deterministic Wang tile set, we construct a Mealy automaton, such that the plane admits a valid Wang tiling if and only if the Mealy automaton generates a infinite semigroup. The construction is similar to a construction by Kari for proving that the nilpotency problem for cellular automata is unsolvable. Moreover Kari proves that the tiling of the plane is undecidable for NWdeterministic Wang tile set. It follows that the finiteness problem for automaton semigroups is undecidable.