The Isomorphism Problem for-Automatic Trees

Dietrich Kuske, Jiamou Liu, Markus Lohrey

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].

External IDs:dblp:conf/csl/KuskeLL10