A New Hierarchy for Automaton Semigroups

Abstract

We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata. We prove that the Order Problem is decidable whenever the state-activity is bounded. The Order Problem remains open for the next level of this hierarchy, that is, when the state-activity is linear. Gillibert showed that it is undecidable in the whole family. We extend the aforementioned hierarchy via a semi-norm making it more coarse but somehow more robust and we prove that the Order Problem is still decidable for the first two levels of this alternative hierarchy.