The Isomorphism Problem for omega -Automatic Trees

Abstract The main result of this paper is that the isomorphism problem for ω -automatic trees of finite height is at least as hard as second-order arithmetic and therefore not analytical. This strengthens a recent result by Hjorth, Khoussainov, Montalbán, and Nies [9] showing that the isomorphism problem for ω -automatic structures is not \(\Sigma^1_2\). 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 ( \(\Pi^0_1\) -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 ≥ 4. All proofs are elementary and do not rely on theorems from set theory. Complete proofs can be found in [18].

Authors

• Dietrich Kuske
• Jiamou Liu
• Markus Lohrey

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