On Orbits and the Finiteness of Bounded Automaton Groups
Ievgen Bondarenko, Jan Philipp W\"achter
TL;DR
This paper presents an algorithm to determine whether a group generated by a bounded automaton is finite, using a finite-state acceptor to analyze infinite sequences and orbits.
Contribution
It introduces a novel algorithm that decides finiteness and level-transitivity of groups generated by bounded automata based on finite-state acceptors.
Findings
Algorithm effectively decides finiteness of automaton groups.
Method characterizes infinite orbits with finite-state acceptors.
Can determine level-transitivity of automaton actions.
Abstract
We devise an algorithm which, given a bounded automaton A, decides whether the group generated by A is finite. The solution comes from a description of the infinite sequences having an infinite A-orbit using a deterministic finite-state acceptor. This acceptor can also be used to decide whether the bounded automaton acts level-transitively.
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