A new hierarchy for automaton semigroups
: Laurent Bartholdi, Thibault Godin, Ines Klimann, Matthieu Picantin
: Cezar Câmpeanu
: International Conference on Implementation and Application of Automata (CIAA)
Publisher: Springer Verlag
: 2018
: Lecture Notes in Computer Science
: Implementation and Application of Automata. 23rd International Conference, CIAA 2018, Charlottetown, PE, Canada, July 30 – August 2, 2018, Proceedings
: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
: Lecture Notes in Computer Science
: 10977
: 71
: 83
: 13
: 978-3-319-94811-9
: 978-3-319-94812-6
: 0302-9743
DOI: https://doi.org/10.1007/978-3-319-94812-6_7
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.
