The isomorphism problem for ω-automatic trees

Abstract

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 Σ21. 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 (Π10-complete, resp.), for height 1 (2, resp.), (ii) Π11-hard and in Π21 for height 3, and (iii) Πn−31- and Σn−31-hard and in Π2n−41 (assuming CH) for height n⩾4. All proofs are elementary and do not rely on theorems from set theory.