On the Complexity of Membership and Counting in Height-Deterministic Pushdown Automata

  • Nutan Limaye
  • Meena Mahajan
  • Antoine Meyer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5010)


While visibly pushdown languages properly generalise regular languages and are properly contained in deterministic context-free languages, the complexity of their membership problem is equivalent to that of regular languages. However, the corresponding counting problem could be harder than counting paths in a non-deterministic finite automaton: it is only known to be in LogDCFL.

We investigate the membership and counting problems for generalisations of visibly pushdown automata, defined using the notion of height-determinism. We show that, when the stack-height of a given PDA can be computed using a finite transducer, both problems have the same complexity as for visibly pushdown languages. We also show that when allowing pushdown transducers instead of finite-state ones, both problems become LogDCFL-complete; this uses the fact that pushdown transducers are sufficient to compute the stack heights of all real-time height-deterministic pushdown automata, and yields a candidate arithmetization of LogDCFL that is no harder than LogDCFL(our main result).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sudborough, I.H.: A note on tape-bounded complexity classes and linear context-free languages. JACM 22(4), 499–500 (1975)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Sudborough, I.: On the tape complexity of deterministic context-free language. JACM 25(3), 405–414 (1978)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ibarra, O., Jiang, T., Ravikumar, B.: Some subclasses of context-free languages in NC 1. IPL 29, 111–117 (1988)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Holzer, M., Lange, K.J.: On the complexities of linear LL(1) and LR(1) grammars. In: Ésik, Z. (ed.) FCT 1993. LNCS, vol. 710, pp. 299–308. Springer, Heidelberg (1993)Google Scholar
  5. 5.
    Barrington, D.: Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC1. JCSS 38(1), 150–164 (1989)MATHMathSciNetGoogle Scholar
  6. 6.
    Lange, K.J.: Complexity and structure in formal language theory. In: 8th CoCo, pp. 224–238. IEEE Computer Society, Los Alamitos (1993)Google Scholar
  7. 7.
    Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL recognition. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 422–432. Springer, Heidelberg (1980)Google Scholar
  8. 8.
    Braunmuhl, B.V., Verbeek, R.: Input-driven languages are recognized in log n space. In: Karpinski, M. (ed.) FCT 1983. LNCS, vol. 158, pp. 40–51. Springer, Heidelberg (1983)Google Scholar
  9. 9.
    Dymond, P.: Input-driven languages are in logn depth. IPL 26, 247–250 (1988)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: 36th STOC, pp. 202–211. ACM, New York (2004)Google Scholar
  11. 11.
    Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC 1 computation. JCSS 57(2), 200–212 (1998)MATHGoogle Scholar
  12. 12.
    Limaye, N., Mahajan, M., Rao, B.V.R.: Arithmetizing classes arround NC1 and L. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 477–488. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Limaye, N., Mahajan, M., Rao, B.V.R.: Arithmetizing classes arround NC1 and L. Technical Report ECCC TR07- (2007) submitted to TCS (spl.issue for STACS 2007) (2007)Google Scholar
  14. 14.
    Nowotka, D., Srba, J.: Height-deterministic pushdown automata. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 125–134. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Caucal, D.: Synchronization of pushdown automata. In: H. Ibarra, O., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 120–132. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. In: H. Ibarra, O., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 1–13. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Blass, A., Gurevich, Y.: A note on nested words. Technical Report MSR-TR-2006-139, Microsoft Research (October 2006)Google Scholar
  18. 18.
    Buss, S.: The Boolean formula value problem is in ALOGTIME. In: 19th STOC, pp. 123–131. ACM, New York (1987)Google Scholar
  19. 19.
    Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, Heidelberg (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Nutan Limaye
    • 1
  • Meena Mahajan
    • 1
  • Antoine Meyer
    • 2
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.LIAFAUniversité Paris Diderot – Paris 7Paris Cedex 13France

Personalised recommendations