Regular Languages of Nested Words: Fixed Points, Automata, and Synchronization

  • Marcelo Arenas
  • Pablo Barceló
  • Leonid Libkin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)


Nested words are a restriction of the class of visibly pushdown languages that provide a natural model of runs of programs with recursive procedure calls. The usual connection between monadic second-order logic (MSO) and automata extends from words to nested words and gives us a natural notion of regular languages of nested words.

In this paper we look at some well-known aspects of regular languages – their characterization via fixed points, deterministic and alternating automata for them, and synchronization for defining regular relations – and extend them to nested words. We show that mu-calculus is as expressive as MSO over finite and infinite nested words, and the equivalence holds, more generally, for mu-calculus with past modalities evaluated in arbitrary positions in a word, not only in the first position. We introduce the notion of alternating automata for nested words, show that they are as expressive as the usual automata, and also prove that Muller automata can be determinized (unlike in the case of visibly pushdown languages). Finally we look at synchronization over nested words. We show that the usual letter-to-letter synchronization is completely incompatible with nested words (in the sense that even the weakest form of it leads to an undecidable formalism) and present an alternative form of synchronization that gives us decidable notions of regular relations.


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  1. 1.
    Alur, R., Arenas, M., Barceló, P., Etessami, K., Immerman, N., Libkin, L.: First-order and temporal logics for nested words. In: LICS 2007Google Scholar
  2. 2.
    Alur, R., Chaudhuri, S., Madhusudan, P.: A fixpoint calculus for local and global program flows. In: POPL 2006, pp. 153–165Google Scholar
  3. 3.
    Alur, R., Chaudhuri, S., Madhusudan, P.: Languages of nested trees. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 329–342. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Alur, R., Etessami, K., Madhusudan, P.: A temporal logic of nested calls and returns. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 467–481. Springer, Heidelberg (2004)Google Scholar
  5. 5.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: STOC 2004, pp. 202–211Google Scholar
  6. 6.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 1–13. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Arnold, A., Niwinski, D.: Rudiments of μ-calculus. North-Holland, Amsterdam (2001)CrossRefGoogle Scholar
  8. 8.
    Bárány, V., Löding, C., Serre, O.: Regularity problems for visibly pushdown languages. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 420–431. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Barceló, P., Libkin, L.: Temporal logics over unranked trees. In: LICS 2005, pp. 31–40.Google Scholar
  10. 10.
    Benedikt, M., Libkin, L., Schwentick, T., Segoufin, L.: Definable relations and first-order query languages over strings. J. ACM 50(5), 694–751 (2003)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Benedikt, M., Libkin, L., Neven, F.: Logical definability and query languages over ranked and unranked trees. In: ACM TOCL. Extended abstract in LICS 2002 and LICS 2003, vol. 8(2), ACM Press, New York (2007)Google Scholar
  12. 12.
    Blumensath, A., Grädel, E.: Automatic structures. In: LICS 2000, pp. 51–62.Google Scholar
  13. 13.
    Bruyère, V., Hansel, G., Michaux, C., Villemaire, R.: Logic and p-recognizable sets of integers. Bull. Belg. Math. Soc. 1, 191–238 (1994)MATHGoogle Scholar
  14. 14.
    Cleaveland, R., Steffen, B.: A linear-time model-checking algorithm for the alternation-free modal mu-calculus. In: Larsen, K.G., Skou, A. (eds.) CAV 1991. LNCS, vol. 575, pp. 48–58. Springer, Heidelberg (1992)Google Scholar
  15. 15.
    Elgot, C., Mezei, J.: On relations defined by generalized finite automata. IBM J. Res. Develop. 9, 47–68 (1965)MATHMathSciNetGoogle Scholar
  16. 16.
    Janin, D., Walukiewicz, I.: On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 263–277. Springer, Heidelberg (1996)Google Scholar
  17. 17.
    Lautemann, C., Schwentick, T., Thérien, D.: Logics for context-free languages. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 205–216. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  18. 18.
    Löding, C., Madhusudan, P., Serre, O.: Visibly pushdown games. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 408–420. Springer, Heidelberg (2004)Google Scholar
  19. 19.
    Neven, F., Schwentick, Th: Query automata over finite trees. TCS 275, 633–674 (2002)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Niwinski, D.: Fixed points vs. infinite generation. In: LICS 1988, pp. 402–409Google Scholar
  21. 21.
    Peng, F., Chawathe, S.: Xpath queries on streaming data. In: SIGMOD 2003, pp. 431–442.Google Scholar
  22. 22.
    Safra, S.: On the complexity of omega-automata. In: FOCS 1988, pp. 319–327Google Scholar
  23. 23.
    Segoufin, L., Vianu, V.: Validating streaming XML documents. In: PODS 2002, pp. 53–64.Google Scholar
  24. 24.
    Seidl, H.: Deciding equivalence of finite tree automata. SICOMP 19(3), 424–437 (1990)MATHMathSciNetGoogle Scholar
  25. 25.
    Thomas, W.: Languages, automata, and logic. Handbook of Formal Languages, vol. 3 (1997)Google Scholar
  26. 26.
    Thomas, W.: Infinite trees and automaton-definable relations over ω-words. TCS 103, 143–159 (1992)MATHCrossRefGoogle Scholar
  27. 27.
    Vardi, M.Y.: An automata-theoretic approach to linear temporal logic. Banff Higher Order Workshop, pp. 238-266 (1995)Google Scholar
  28. 28.
    Vardi, M.Y.: Reasoning about the past with two-way automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Marcelo Arenas
    • 1
  • Pablo Barceló
    • 2
  • Leonid Libkin
    • 3
  1. 1.Pontificia Universidad Católica deChile
  2. 2.Universidad deChile
  3. 3.University of Edinburgh 

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