The finiteness problem for automaton semigroups is undecidable

Pierre Gillibert (1, 2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 CAUL - Centro de Álgebra da Universidade de Lisboa

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 an Mealy automaton, such that the plane admit a valid Wang tiling if and only if the Mealy automaton generates a finite 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 semigroup is undecidable.

Mots clés: Mealy automata; automaton semigroups; cellular atomata; Wang tiles

Domaines: Mathématique discrète [cs.DM]

Pierre Gillibert. The finiteness problem for automaton semigroups is undecidable. International Journal of Algebra and Computation, 2014, 24 (1), pp.1-9. ⟨10.1142/S0218196714500015⟩. ⟨hal-00809295v2⟩