Tree-Automatic Well-Founded Trees

  • Alexander Kartzow
  • Jiamou Liu
  • Markus Lohrey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)


We investigate tree-automatic well-founded trees. For this, we introduce a new ordinal measure for well-founded trees, called ∞-rank. The ∞-rankof a well-founded tree is always bounded from above by the ordinary (ordinal) rank of a tree. We also show that the ordinal rank of a well-founded tree of ∞-rankα is smaller than ω·(α + 1). For string-automatic well-founded trees, it follows from [16] that the ∞-rankis always finite. Here, using Delhommé’s decomposition technique for tree-automatic structures, we show that the ∞-rankof a tree-automatic well-founded tree is strictly below ωω. As a corollary, we obtain that the ordinal rank of a string-automatic (resp., tree-automatic) well-founded tree is strictly below ω2 (resp., ωω). The result for the string-automatic case nicely contrasts a result of Delhommé, saying that the ranks of string-automatic well-founded partial orders reach all ordinals below ωω. As a second application of the ∞-rankwe show that the isomorphism problem for tree-automatic well-founded trees is complete for level \(\Delta^0_{\omega^\omega}\) of the hyperarithmetical hierarchy (under Turing-reductions). Full proofs can be found in the arXiv-version [11] of this paper.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ash, C.J., Knight, J.F.: Computable structures and the hyperarithmetical hierarchy. Studies in Logic and the Foundations of Mathematics, vol. 144. North-Holland Publishing Co., Amsterdam (2000)MATHGoogle Scholar
  2. 2.
    Bárány, V., Grädel, E., Rubin, S.: Automata-based presentations of infinite structures. In: Finite and Algorithmic Model Theory. London Mathematical Society Lecture Notes Series, vol. 379, pp. 1–76. Cambridge University Press (2011)Google Scholar
  3. 3.
    Blumensath, A.: Automatic structures. Diploma thesis, RWTH Aachen (1999)Google Scholar
  4. 4.
    Blumensath, A., Grädel, E.: Finite presentations of infinite structures: Automata and interpretations. Theory Comput. Syst. 37, 642–674 (2004)Google Scholar
  5. 5.
    Calvert, W., Knight, J.F.: Classification from a computable viewpoint. Bull. Symbolic Logic 12(2), 191–218 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Delhommé, C.: Automaticité des ordinaux et des graphes homogènes. C.R. Acad. Sci. Paris Ser. I 339, 5–10 (2004)MATHCrossRefGoogle Scholar
  7. 7.
    Goncharov, S.S., Knight, J.F.: Computable structure and antistructure theorems. Algebra Logika 41(6), 639–681 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Hirschfeldt, D.R., White, W.M.: Realizing levels of the hyperarithmetic hierarchy as degree spectra of relations on computable structures. Notre Dame J. Form. Log. 43(1), 51–64 (2002)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Huschenbett, M.: Word automaticity of tree automatic scattered linear orderings is decidable. Technical report, (2012),
  10. 10.
    Kartzow, A.: First-Order Model Checking On Generalisations of Pushdown Graphs. PhD thesis, TU Darmstadt (2011)Google Scholar
  11. 11.
    Kartzow, A., Lohrey, M., Liu, J.: Tree-automatic well-founded trees. Technical report, (2012),
  12. 12.
    Khoussainov, B., Minnes, M.: Model theoretic complexity of automatic structures. Ann. Pure Appl. Logic 161(3), 416–426 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Khoussainov, B., Nerode, A.: Automatic Presentations of Structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  14. 14.
    Khoussainov, B., Nies, A., Rubin, S., Stephan, F.: Automatic structures: richness and limitations. Log. Methods Comput. Sci. 3(2):2:2, 18 (2007)MathSciNetGoogle Scholar
  15. 15.
    Khoussainov, B., Rubin, S., Stephan, F.: Automatic linear orders and trees. ACM Trans. Comput. Log. 6(4), 675–700 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kuske, D., Liu, J., Lohrey, M.: The isomorphism problem on classes of automatic structures with transitive relations. To appear in Trans. Amer. Math. Soc. (2012)Google Scholar
  17. 17.
    Kuske, D., Lohrey, M.: Automatic structures of bounded degree revisited. J. Symbolic Logic 76(4), 1352–1380 (2011)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Oliver, G.P., Thomas, R.M.: Automatic Presentations for Finitely Generated Groups. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 693–704. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill (1968)Google Scholar
  20. 20.
    Rosenstein, J.: Linear Ordering. Academic Press (1982)Google Scholar
  21. 21.
    To, A.W., Libkin, L.: Recurrent Reachability Analysis in Regular Model Checking. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 198–213. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alexander Kartzow
    • 1
  • Jiamou Liu
    • 2
  • Markus Lohrey
    • 1
  1. 1.Institut für InformatikUniversität LeipzigLeipzigGermany
  2. 2.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

Personalised recommendations