Counting Classes and the Fine Structure between NC1 and L

  • Samir Datta
  • Meena Mahajan
  • B. V. Raghavendra Rao
  • Michael Thomas
  • Heribert Vollmer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)


The class NC 1of problems solvable by bounded fan-in circuit families of logarithmic depth is known to be contained in logarithmic space L, but not much about the converse is known. In this paper we examine the structure of classes in between NC 1 and L based on counting functions or, equivalently, based on arithmetic circuits. The classes PNC 1 and C = NC 1, defined by a test for positivity and a test for zero, respectively, of arithmetic circuit families of logarithmic depth, sit in this complexity interval. We study the landscape of Boolean hierarchies, constant-depth oracle hierarchies, and logarithmic-depth oracle hierarchies over PNC 1 and C = NC 1. We provide complete problems, obtain the upper bound L for all these hierarchies, and prove partial hierarchy collapses—in particular, the constant-depth oracle hierarchy over PNC 1 collapses to its first level PNC 1, and the constant-depth oracle hierarchy over C = NC 1 collapses to its second level.


Language Class Arithmetic Circuit Polynomial Size Boolean Circuit Output Gate 
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  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., Beals, R., Ogihara, M.: The complexity of matrix rank and feasible systems of linear equations. Computational Complexity 8(2), 99–126 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Allender, E., Ogihara, M.: Relationships among PL, #L, and the determinant. RAIRO Theoretical Information and Applications 30, 1–21 (1996); Conference version in Proc. 9th IEEE Structure in Complexity Theory Conference, pp. 267–278 (1994)zbMATHMathSciNetGoogle 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.
    Beigel, R., Reingold, N., Spielman, D.A.: PP is closed under intersection. Journal of Computer and System Sciences 50(2), 191–202 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC 1 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)Google Scholar
  7. 7.
    Fortnow, L., Reingold, N.: PP is closed under truth-table reductions. Inf. Comput. 124(1), 1–6 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Köbler, J., Schöning, U., Wagner, K.W.: The difference and truth-table hierarchies for NP. Theoretical Informatics and Applications 21(4), 419–435 (1987)zbMATHGoogle Scholar
  9. 9.
    Lange, K.-J.: Unambiguity of circuits. Theor. Comput. Sci. 107(1), 77–94 (1993)zbMATHCrossRefGoogle Scholar
  10. 10.
    Mahajan, M., Raghavendra Rao, B.V.: Small-space analogues of Valiant’s classes. In: Gȩbala, M. (ed.) FCT 2009. LNCS, vol. 5699, pp. 250–261. Springer, Heidelberg (2009)Google Scholar
  11. 11.
    Ogihara, M.: The PL hierarchy collapses. SIAM J. Comput. 27(5), 1430–1437 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, New York (1999)Google Scholar
  13. 13.
    von zur Gathen, J., Seroussi, G.: Boolean circuits versus arithmetic circuits. Information and Computation 91(1), 142–154 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Wilson, C.B.: Relativized circuit complexity. J. Comput. Syst. Sci. 31(2), 169–181 (1985)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Samir Datta
    • 1
  • Meena Mahajan
    • 2
  • B. V. Raghavendra Rao
    • 3
  • Michael Thomas
    • 4
  • Heribert Vollmer
    • 4
  1. 1.Chennai Mathematical InstituteIndia
  2. 2.The Institute of Mathematical SciencesChennaiIndia
  3. 3.Universität des SaarlandesSaarbrückenGermany
  4. 4.Leibniz UniversitätHannoverGermany

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