Tree-Automatic Well-Founded Trees
Martin Huschenbett (Technical University of Ilmenau), Alexander, Kartzow (University of Leipzig), Jiamou Liu (Auckland University of, Technology), Markus Lohrey (University of Leipzig)

TL;DR
This paper studies the complexity and rank bounds of tree-automatic well-founded trees and partial orders, establishing upper limits on their ordinal ranks and analyzing the complexity of their isomorphism problem.
Contribution
It proves that the ordinal rank of tree-automatic well-founded trees is below omega^omega and provides bounds for partial orders, advancing understanding of their structural complexity.
Findings
Ordinal rank of tree-automatic well-founded trees is below omega^omega
Ranks of certain tree-automatic partial orders are bounded by omega^omega^omega
Isomorphism problem for these trees is complete at level Delta^0_{omega^omega} in the hyperarithmetical hierarchy
Abstract
We investigate tree-automatic well-founded trees. Using Delhomme's decomposition technique for tree-automatic structures, we show that the (ordinal) rank of a tree-automatic well-founded tree is strictly below omega^omega. Moreover, we make a step towards proving that the ranks of tree-automatic well-founded partial orders are bounded by omega^omega^omega: we prove this bound for what we call upwards linear partial orders. As an application of our result, we show that the isomorphism problem for tree-automatic well-founded trees is complete for level Delta^0_{omega^omega} of the hyperarithmetical hierarchy with respect to Turing-reductions.
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