Advertisement

Counting Paths in VPA Is Complete for #NC1

  • Andreas Krebs
  • Nutan Limaye
  • Meena Mahajan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6196)

Abstract

We give a #NC 1 upper bound for the problem of counting accepting paths in any fixed visibly pushdown automaton. Our algorithm involves a non-trivial adaptation of the arithmetic formula evaluation algorithm of Buss, Cook, Gupta, Ramachandran ([9]). We also show that the problem is #NC 1 hard. Our results show that the difference between #BWBP and #NC 1 is captured exactly by the addition of a visible stack to a nondeterministic finite-state automata.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allender, E.: Arithmetic circuits and counting complexity classes. In: Krajicek, J. (ed.) Complexity of Computations and Proofs. Quaderni di Matematica, vol. 13, pp. 33–72. Seconda Universita di Napoli (2004); An earlier version appeared in the Complexity Theory Column, SIGACT News 28(4), 2–15 (December 1997)Google Scholar
  2. 2.
    Allender, E., Jiao, J., Mahajan, M., Vinay, V.: Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science 209, 47–86 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: STOC, pp. 202–211 (2004)Google Scholar
  4. 4.
    Barrington, D.A.: Bounded-width polynomial size branching programs recognize exactly those languages in NC1. Journal of Computer and System Sciences 38, 150–164 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ben-Or, M., Cleve, R.: Computing algebraic formulas using a constant number of registers. SIAM Journal on Computing 21, 54–58 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Von Braunmuhl, B., Verbeek, R.: Input-driven languages are recognized in log n space. In: Ésik, Z. (ed.) FCT 1993. LNCS, vol. 710, pp. 40–51. Springer, Heidelberg (1993)Google Scholar
  7. 7.
    Brent, R.P.: The parallel evaluation of general arithmetic expressions. Journal of the ACM 21, 201–206 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Buss, S.: The Boolean formula value problem is in ALOGTIME. In: STOC, pp. 123–131 (1987)Google Scholar
  9. 9.
    Buss, S., Cook, S., Gupta, A., Ramachandran, V.: An optimal parallel algorithm for formula evaluation. SIAM Journal of Computation 21(4), 755–780 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC0 computation. Journal of Computer and System Sciences 57, 200–212 (1998); Preliminary version in Proceedings of the 11th IEEE Conference on Computational Complexity, pp. 12–21 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dymond, P.W.: Input-driven languages are in log n depth. Information processing letters 26, 247–250 (1988)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Limaye, N., Mahajan, M., Meyer, A.: On the complexity of membership and counting in height-deterministic pushdown automata. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) CSR 2008. LNCS, vol. 5010, pp. 240–251. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Limaye, N., Mahajan, M., Raghavendra Rao, B.V.: Arithmetizing classes around NC1 and L. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 477–488. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Limaye, N., Mahajan, M., Raghavendra Rao, B.V.: Arithmetizing classes around NC1 and L. Theory of Computing Systems 46(3), 499–522 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Andreas Krebs
    • 1
  • Nutan Limaye
    • 2
  • Meena Mahajan
    • 3
  1. 1.University of TübingenGermany
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia
  3. 3.The Institute of Mathematical SciencesChennaiIndia

Personalised recommendations