A Tentative Approach for the Wadge-Wagner Hierarchy of Regular Tree Languages of Index [0, 2]

  • Jacques Duparc
  • Kevin Fournier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9118)


We provide a hierarchy of tree languages recognised by nondeterministic parity tree automata with priorities in \(\{0,1,2\}\), whose length exceeds the first fixed point of the \(\varepsilon \) operation (that itself enumerates the fixed points of \(x\mapsto \omega ^x\)). We conjecture that, up to Wadge equivalence, it exhibits all regular tree languages of index \([0,2]\).


  1. 1.
    Andretta, A., Louveau, A.: Wadge degrees and pointclasses. In: Kechris, A.S., Löwe, B., Steel, J.R. (eds.) Wadge Degrees and Projective Ordinals: The Cabal Seminar, vol. II, pp. 3–23. Cambridge University Press, Cambridge (2012)Google Scholar
  2. 2.
    Arnold, A., Duparc, J., Murlak, F., Niwiński, D.: On the topological complexity of tree languages. Logic Automata: Hist. Perspect. 2, 9–29 (2007)Google Scholar
  3. 3.
    Arnold, A., Niwiński, D.: Rudiments of \(\mu \)-Calculus. Studies in Logic and the Foundations of Mathematics. Elseiver, Amsterdam (2001)zbMATHGoogle Scholar
  4. 4.
    Duparc, J.: Wadge hierarchy and Veblen hierarchy, part I: Borel sets of finite rank. J. Symbolic Logic 66(1), 56–86 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Duparc, J., Murlak, F.: On the topological complexity of weakly recognizable tree languages. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) FCT 2007. LNCS, vol. 4639, pp. 261–273. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  6. 6.
    Finkel, O., Simonnet, P.: On recognizable tree languages beyond the Borel hierarchy. Fundam. Informaticae 95(2–3), 287–303 (2009)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. AMS 141, 1–23 (1969)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Van Wesep, R.: Wadge degrees and descriptive set theory. In: Kechris, A.S., Moschovakis, Y.N. (eds.) Cabal Seminar 76–77, pp. 151–170. Springer, Heidelberg (1978)CrossRefGoogle Scholar
  9. 9.
    Wadge, W.W.: Reducibility and determinateness on the Baire space. Ph.D. thesis, University of California, Berkeley (1984)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Information Systems Faculty of Business and EconomicsUniversity of LausanneLausanneSwitzerland
  2. 2.Équipe de Logique MathématiqueUniversité Paris DiderotParis Cedex 13France

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