The finiteness problem for automaton semigroups is undecidable

TL;DR

This paper proves that determining whether an automaton semigroup is finite is an undecidable problem, by linking it to the undecidability of Wang tiling problems, thus establishing a fundamental limit in automaton theory.

Contribution

It establishes the undecidability of the finiteness problem for automaton semigroups in the general case, connecting it to Wang tiling problems and cellular automata.

Findings

Finiteness problem for automaton semigroups is undecidable.

Construction of automaton from Wang tiles links tiling to semigroup finiteness.

Undecidability of plane tiling implies undecidability of automaton semigroup finiteness.

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 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.