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 when 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. The former results are implemented and will be available in the GAP package FR developed by the first author.
Keywords
Cite
@article{arxiv.1803.09991,
title = {A new hierarchy for automaton semigroups},
author = {Laurent Bartholdi and Thibault Godin and Ines Klimann and Matthieu Picantin},
journal= {arXiv preprint arXiv:1803.09991},
year = {2018}
}
Comments
12 pages, accepted and presented at CIAA 2018