Abstract
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.
Supported in part by the Indian DST and the German DAAD.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Allender, E., Beals, R., Ogihara, M.: The complexity of matrix rank and feasible systems of linear equations. Computational Complexity 8(2), 99–126 (1999)
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)
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)
Beigel, R., Reingold, N., Spielman, D.A.: PP is closed under intersection. Journal of Computer and System Sciences 50(2), 191–202 (1995)
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)
Fortnow, L., Reingold, N.: PP is closed under truth-table reductions. Inf. Comput. 124(1), 1–6 (1996)
Köbler, J., Schöning, U., Wagner, K.W.: The difference and truth-table hierarchies for NP. Theoretical Informatics and Applications 21(4), 419–435 (1987)
Lange, K.-J.: Unambiguity of circuits. Theor. Comput. Sci. 107(1), 77–94 (1993)
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)
Ogihara, M.: The PL hierarchy collapses. SIAM J. Comput. 27(5), 1430–1437 (1998)
Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, New York (1999)
von zur Gathen, J., Seroussi, G.: Boolean circuits versus arithmetic circuits. Information and Computation 91(1), 142–154 (1991)
Wilson, C.B.: Relativized circuit complexity. J. Comput. Syst. Sci. 31(2), 169–181 (1985)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Datta, S., Mahajan, M., Rao, B.V.R., Thomas, M., Vollmer, H. (2010). Counting Classes and the Fine Structure between NC 1 and L . In: Hliněný, P., Kučera, A. (eds) Mathematical Foundations of Computer Science 2010. MFCS 2010. Lecture Notes in Computer Science, vol 6281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15155-2_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-15155-2_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15154-5
Online ISBN: 978-3-642-15155-2
eBook Packages: Computer ScienceComputer Science (R0)