THE FINITENESS PROBLEM FOR AUTOMATON SEMIGROUPS IS UNDECIDABLE

Abstract

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 NW-deterministic Wang tile set. It follows that the finiteness problem for automaton semigroups is undecidable.