The isomorphism problem for ω-automatic trees

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The main result of this paper states that the isomorphism problem 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 (2008) [12] showing that the isomorphism problem for ω -automatic structures is not in Σ 2 1 . 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 0 -complete, resp.), for height 1 (2, resp.), (ii) Π 1 1 -hard and in Π 2 1 for height 3, and (iii) Π n − 3 1 - and Σ n − 3 1 -hard and in Π 2 n − 4 1 (assuming CH ) for height n ⩾ 4 . All proofs are elementary and do not rely on theorems from set theory.