Tree-Automatic Well-Founded Trees

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.

💡 Research Summary

The paper investigates the expressive power of tree‑automatic structures when restricted to well‑founded trees and certain well‑founded partial orders. Using the decomposition technique originally introduced by Delhommé for tree‑automatic structures, the authors show that any well‑founded tree that can be presented by a finite‑state tree automaton has an ordinal rank strictly below ω^ω. The proof proceeds by recursively splitting the tree into a bounded‑depth front part and a collection of sub‑trees, each of which is again tree‑automatic. By inductively bounding the rank of each sub‑tree and carefully accounting for the increase caused by the combination step, they demonstrate that the overall rank never reaches the first fixed point of the ω‑exponential hierarchy, i.e., ω^ω.

Beyond trees, the authors introduce the class of “upwards linear partial orders” (ULPOs). A partial order is upwards linear if any two elements that share a common upper bound are comparable. This structural restriction allows ULPOs to be encoded as tree‑automatic structures whose levels are linearly ordered. By adapting the tree‑decomposition argument and adding an extra layer of ordinal analysis, they prove that the rank of any tree‑automatic ULPO is bounded by ω^{ω^{ω}}. This result constitutes a step toward the conjectured bound ω^{ω^{ω}} for all tree‑automatic well‑founded partial orders.

The paper then applies these rank bounds to the isomorphism problem for tree‑automatic well‑founded trees. The authors construct a uniform reduction from any decision problem in the hyperarithmetical class Δ⁰_{ω^ω} to the isomorphism problem for such trees, showing Δ⁰_{ω^ω}‑hardness. Conversely, they present a decision procedure that, given two tree‑automatic presentations of well‑founded trees, determines isomorphism by traversing the rank hierarchy up to ω^ω, establishing Δ⁰_{ω^ω}‑membership. Hence the isomorphism problem is Δ⁰_{ω^ω}‑complete under Turing reductions.

The paper is organized as follows. Section 1 reviews background on automatic structures, ordinal ranks, and previous results on string‑automatic and tree‑automatic presentations. Section 2 details Delhommé’s decomposition method, reformulates it for the well‑founded setting, and proves the ω^ω rank bound for trees. Section 3 defines ULPOs, proves the ω^{ω^{ω}} bound, and discusses why the linearity condition is essential for the argument. Section 4 contains the complexity analysis of the isomorphism problem, providing both hardness and membership proofs. Section 5 concludes with a discussion of open problems, notably whether the ω^{ω^{ω}} bound can be extended to all tree‑automatic well‑founded partial orders and how the techniques might apply to other automatic structures such as graph‑automatic or higher‑dimensional automatic presentations.

In summary, the authors establish new ordinal rank limits for tree‑automatic well‑founded trees (below ω^ω) and for a significant subclass of well‑founded partial orders (below ω^{ω^{ω}}). They leverage these limits to give a precise hyperarithmetical classification of the isomorphism problem, showing it is complete for Δ⁰_{ω^ω}. The results deepen our understanding of the interplay between automatic presentations, ordinal combinatorics, and descriptive set‑theoretic complexity.