Division Is In Uniform TC0

  • William Hesse
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2076)

Abstract

Integer division has been known since 1986 [41312] to be in slightly non-uniform TC0, i.e., computable by polynomial-size, constant depth threshold circuits. This has been perhaps the outstanding natural problem known to be in a standard circuit complexity class, but not known to be in its uniform version. We show that indeed division is in uniform TC0. A key step of our proof is the discovery of a first-order formula expressing exponentiation modulo any number of polynomial size.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Ajtai and M. Ben-Or. A theorem on probabilistic constant depth computations. In ACM Symposium on Theory of Computing (STOC’ 84), pages 471–474, 1984. ACM Press.Google Scholar
  2. 2.
    E. Allender, D. A. Mix Barrington, and W. Hesse. Uniform circuits for division: Consequences and problems. To appear in Proceedings of the 16th Annual IEEE Conference on Computational Complexity (CCC-2001), 2001. IEEE Computer Society.Google Scholar
  3. 3.
    D. A. M. Barrington, N. Immerman, and H. Straubing. On uniformity within NC1. Journal of Computer and System Sciences, 41:274–306, 1990.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994–1003, 1986.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Chiu, G. Davida, and B. Litow. NC1 division. online at http://www.cs.jcu.edu.au/~bruce/papers/crr00_3.ps.gz.
  6. 6.
    G. I. Davida and B. Litow. Fast Parallel Arithmetic via Modular Representation. SIAM Journal of Computing, 20(4):756–765, 1991.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R. Fagin, M. M. Klawe, N. J. Pippenger, and L. Stockmeyer. Bounded-depth, polynomial-size circuits for symmetric functions. Theoretical Computer Science, 36(2-3):239–250, 1985.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. In 22nd Annual Symposium on Foundations of Computer Science, 260–270, 1981. IEEE.Google Scholar
  9. 9.
    J. Hastad. Almost optimal lower bounds for small depth circuits. In Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, 6–20, 1986.Google Scholar
  10. 10.
    N. Immerman. Descriptive Complexity. Springer-Verlag, New York, 1999.MATHGoogle Scholar
  11. 11.
    N. Immerman and S. Landau. The complexity of iterated multiplication. Information and Computation, 116(1):103–116, 1995.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. H. Reif. On threshold circuits and polynomial computation. In Proceedings, Structure in Complexity Theory, Second Annual Conference, pages 118–123, IEEE Computer Society Press.Google Scholar
  13. 13.
    J. H. Reif and S. R. Tate. On threshold circuits and polynomial computation. SIAM Journal on Computing, 21(5):896–908, 1992.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • William Hesse
    • 1
  1. 1.Department of Computer ScienceUniversity of MassachusettsMA

Personalised recommendations