The Isomorphism Problem for omega-Automatic Trees

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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 Σ¹₂. 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 (Π⁰₁-complete, resp,) for height 1 (2, resp.), (ii) Π¹₁-hard and in Π¹₂ for height 3, and (iii) Π¹ₙ₋₃- and Σ¹ₙ₋₃-hard and in Π¹₂ₙ₋₄ (assuming CH) for all n > 3. All proofs are elementary and do not rely on theorems from set theory.