The Isomorphism Problem for omega-Automatic Trees

The main result of this paper states that the isomorphism for ω-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, Montalbán, and Nies [HKMN08] showing that the isomorphism problem for ωautomatic structures is not in Σ 2 . Moreover, assuming the continuum hypothesis CH, we can show that the isomorphism problem for ω-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 ω-automatic trees of every finite height: (i) It is decidable (Π 1 -complete, resp,) for height 1 (2, resp.), (ii) Π 1 -hard and in Π 1 2 for height 3, and (iii) Π 1 n−3and Σ n−3-hard and in Π 1 2n−4 (assuming CH) for all n ≥ 4. All proofs are elementary and do not rely on theorems from set theory.