🤖 AI Summary
This paper resolves the long-standing open problem posed by Grigorchuk et al. concerning the undecidability of freeness for automaton semigroups and automaton monoids. The authors devise a novel encoding technique based on the Post Correspondence Problem (PCP), embedding this canonical undecidable problem into automaton algebraic structures over a fixed finite alphabet while precisely controlling the generated relations. This construction establishes the undecidability of freeness testing and, in a unified manner, derives the undecidability of several fundamental decision problems: left-cancellativity, separability, homomorphic extendability, and freeness of presentations (in the monoid case). By integrating automata theory, semigroup algebra, and computability theory, the work provides the first systematic characterization of intrinsic computational limits governing core semantic properties of automaton algebras.
📝 Abstract
We show that the freeness problems for automaton semigroups and for automaton monoids are undecidable and, thereby, solve an open problem listed by Grigorchuk, Nekrashevych and Sush-chansku{i}i. We achieve this using a new technique to encode Post's Correspondence Problem into automaton semigroups and monoids and our result even holds if we restrict the alphabet of the input automata to a constant size. The encoding allows us to precisely control the relations in the generated semigroup/monoid and the construction is quite versatile. In fact, we obtain further undecidability results on various semigroup notions (left cancellativity, equidivisibility and extending homomorphisms). Our construction can also be adapted to show that the free presentation problem for automaton monoids is undecidable (and yields a weaker statement in the semigroup case).