On Müller Context-Free Grammars

  • Zoltán Ésik
  • Szabolcs Iván
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6224)


We define context-free grammars with Müller acceptance condition that generate languages of countable words. We establish several elementary properties of the class of Müller context-free languages including closure properties and others. We show that every Müller context-free grammar can be transformed into a normal form grammar in polynomial space without increasing the size of the grammar, and then we show that many decision problems can be solved in polynomial time for Müller context-free grammars in normal form. These problems include deciding whether the language generated by a normal form grammar contains only well-ordered, scattered, or dense words. In a further result we establish a limitedness property of Müller context-free grammars: If the language generated by a grammar contains only scattered words, then either there is an integer n such that each word of the language has Hausdorff rank at most n, or the language contains scattered words of arbitrarily large Hausdorff rank. We also show that it is decidable which of the two cases applies.


Normal Form Polynomial Time Linear Order Order Type Countable Word 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bedon, N.: Finite automata and ordinals. Theor. Comp. Sci. 156, 119–144 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bès, A., Carton, O.: A Kleene theorem for languages of words indexed by linear orderings. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 158–167. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Bloom, S.L., Ésik, Z.: Deciding whether the frontier of a regular tree is scattered. Fund. Inform. 55, 1–21 (2003)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Boasson, L.: Context-free sets of infinite words. In: Theoretical Computer Science (Fourth GI Conf., Aachen, 1979). LNCS, vol. 67, pp. 1–9. Springer, Heidelberg (1979)Google Scholar
  5. 5.
    Bruyère, V., Carton, O.: Automata on linear orderings. J. Comput. System Sci. 73, 1–24 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bruyere, V., Carton, O., Sénizergues, G.: Tree Automata and Automata on Linear Orderings. ITA 43, 321–338 (2009)zbMATHGoogle Scholar
  7. 7.
    Büchi, J.R.: The monadic second order theory of ω 1. In: Decidable theories, II. LNM, vol. 328, pp. 1–127. Springer, Heidelberg (1973)CrossRefGoogle Scholar
  8. 8.
    Choueka, Y.: Finite automata, definable sets, and regular expressions over ω n-tapes. J. Comput. System Sci. 17(1), 81–97 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cohen, R.S., Gold, A.Y.: Theory of ω-languages, parts one and two. Journal of Computer and System Science 15, 169–208 (1977)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Courcelle, B.: Frontiers of infinite trees. RAIRO Theor. Inf. 12, 319–337 (1978)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Ésik, Z., Iván, S.: Context-Free Languages of Countable Words. In: Leucker, M., Morgan, C. (eds.) Theoretical Aspects of Computing - ICTAC 2009. LNCS, vol. 5684, pp. 185–199. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Heilbrunner, S.: An algorithm for the solution of fixed-point equations for infinite words. RAIRO Theor. Inf. 14, 131–141 (1980)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Hunter, P., Dawar, A.: Complexity Bounds for Regular Games. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 495–506. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Montalbán, A.: On the equimorphism types of linear orderings. The Bulletin of Symbolic Logic 13(1), 71–99 (2007)zbMATHCrossRefGoogle Scholar
  15. 15.
    Nivat, M.: Sur les ensembles de mots infinis engendrés par une grammaire algébrique (French). RAIRO Inform. Théor. 12(3), 259–278 (1978)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Rosenstein, J.G.: Linear Orderings. Academic Press, London (1982)zbMATHGoogle Scholar
  17. 17.
    Thomas, W.: Automata on Infinite Objects. In: Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B, pp. 133–192. Elsevier/MIT Press (1990)Google Scholar
  18. 18.
    Wojciechowski, J.: Classes of transfinite sequences accepted by finite automata. Fundamenta Informaticæ 7, 191–223 (1984)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Wojciechowski, J.: Finite automata on transfinite sequences and regular expressions. Fundamenta Informaticæ 8, 379–396 (1985)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Zoltán Ésik
    • 1
  • Szabolcs Iván
    • 1
  1. 1.Department of Computer ScienceUniversity of SzegedHungary

Personalised recommendations