The Isomorphism Problem for omega-Automatic Trees

TL;DR

This paper investigates the complexity of the isomorphism problem for omega-automatic trees of finite height, establishing its high logical complexity and providing bounds for various heights, with results independent of set theory.

Contribution

It proves the isomorphism problem's complexity for omega-automatic trees of finite height is at least as hard as second-order arithmetic, and provides bounds for different tree heights.

Findings

Isomorphism for height 1 trees is decidable and $ ext{Pi}^0_1$-complete.

For height 3, the problem is $ ext{Pi}^1_1$-hard and in $ ext{Pi}^1_2$.

Bounds are given for all heights n > 3, assuming CH.

Abstract

The main result of this paper is that the isomorphism for omega-automatic trees of finite height is at least has hard as second-order arithmetic and therefore not analytical. This strengthens a recent result by Hjorth, Khoussainov, Montalban, and Nies showing that the isomorphism problem for omega-automatic structures is not $\Sigma^1_2$. Moreover, assuming the continuum hypothesis CH, we can show that the isomorphism problem for omega-automatic trees of finite height is recursively equivalent with second-order arithmetic. On the way to our main results, we show lower and upper bounds for the isomorphism problem for omega-automatic trees of every finite height: (i) It is decidable ($\Pi^0_1$-complete, resp,) for height 1 (2, resp.), (ii) $\Pi^1_1$-hard and in $\Pi^1_2$ for height 3, and (iii) $\Pi^1_{n-3}$- and $\Sigma^1_{n-3}$-hard and in $\Pi^1_{2n-4}$ (assuming CH) for all n > 3. All proofs are elementary and do not rely on theorems from set theory.