The Isomorphism Problem for ω-Automatic Trees

  • Dietrich Kuske
  • Jiamou Liu
  • Markus Lohrey
Conference paper

DOI: 10.1007/978-3-642-15205-4_31

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6247)
Cite this paper as:
Kuske D., Liu J., Lohrey M. (2010) The Isomorphism Problem for ω-Automatic Trees. In: Dawar A., Veith H. (eds) Computer Science Logic. CSL 2010. Lecture Notes in Computer Science, vol 6247. Springer, Berlin, Heidelberg

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].

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Dietrich Kuske
    • 1
  • Jiamou Liu
    • 2
  • Markus Lohrey
    • 2
  1. 1.Laboratoire Bordelais de Recherche en Informatique (LaBRI)CNRS and Université Bordeaux IBordeauxFrance
  2. 2.Institut für InformatikUniversität LeipzigGermany

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