Advertisement

Tree-Automatic Well-Founded Trees

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

Abstract

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 ∞-rank α 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 ω ω . As a corollary, we obtain that the ordinal rank of a string-automatic (resp., tree-automatic) well-founded tree is strictly below ω 2 (resp., ω ω ). 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 ω ω . 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.

Keywords

Partial Order Decomposition Technique Computable Structure Isomorphism Problem Tree Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)zbMATHGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Delhommé, C.: Automaticité des ordinaux et des graphes homogènes. C.R. Acad. Sci. Paris Ser. I 339, 5–10 (2004)zbMATHCrossRefGoogle Scholar
  7. 7.
    Goncharov, S.S., Knight, J.F.: Computable structure and antistructure theorems. Algebra Logika 41(6), 639–681 (2002)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Huschenbett, M.: Word automaticity of tree automatic scattered linear orderings is decidable. Technical report, arXiv.org (2012), http://arxiv.org/abs/1201.5070
  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, arXiv.org (2012), http://arxiv.org/abs/1201.5495
  12. 12.
    Khoussainov, B., Minnes, M.: Model theoretic complexity of automatic structures. Ann. Pure Appl. Logic 161(3), 416–426 (2009)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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