Tree-Automatic Well-Founded Trees

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.