Advertisement

An Exact Characterization of Symmetric Functions in qAC0[2]

  • Chi-Jen Lu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1449)

Abstract

qAC0[2] is the class of languages computable by circuits of constant depth and quasi-polynomial \( (2^{\log ^{o(1)_{ n} } } ) \) size with unbounded fan-in AND, OR, and PARITY gates. Symmetric functions are those functions that are invariant under permutations of the input variables. Thus a symmetric function fn: 0, 1n #x2192; 0, 1 can also be seen as a function fn: 0, 1, ..., n} → 0, 1. We give the following characterization of symmetric functions in qAC0[2], according to how fn(x) changes as x grows from 0 to n. A symmetric function f = (fn) is in qAC0[2] if and only if fn has period 2t(n) = logO(1)n except within both ends of length logO(1)n.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Ajtai, Σ11-formula on finite structures, Annals of Pure and Applied Logic, 24, pages 1–48, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Aspnes, R. Beigel, M. Furst, and S. Rudich, The expressive power of voting polynomials, in Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pages 402–409, 1991.Google Scholar
  3. 3.
    D. A. Mix Barrington and H. Straubing, Complex polynomials and circuit lower bounds for modular counting, In Proceedings of the 1st Latin Amercan Symposium on Theoretical Informatics, pages 24–31, 1992.Google Scholar
  4. 4.
    B. Brustmann and I. Wegener, The complexity of symmetric functions in boundeddepth circuits, Information Processing Letters, 25, pages 217–219, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    C. Damm and K. Lenz, Symmetric functions in AC 0[2], manuscript.Google Scholar
  6. 6.
    R. Fagin, M. M. Klawe, N. J. Pippenger, and L. Stockmeyer, Bounded depth, polynomial size circuits for symmetric functions, Theoretical Computer Science, 36, pages 239–250, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Furst, J. Saxe, and M. Sipser, Parity, circuits, and the polynomial time hierarchy, Mathematical System Theory, 17, pages 13–27, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    F. Green, An oracle separating ⊕P from PP PH, In Proceedings of Fifth Annual Conference on Structure in Complexity Theory, pages 295–298, 1990.Google Scholar
  9. 9.
    J. Håstad, Computational limitations of small-depth circuits, MIT Press, 1986.Google Scholar
  10. 10.
    A. A. Razborov, Lower bounds for the size of bounded depth with basis ∧, ⊕, Mathematical Notes of the Academy of Sciences of the USSR, 41, pages 598–607, 1987.CrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Smolensky, Algebraic methods in the theory of lower bounds for boolean circuit complexity, In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pages 77–82, 1987.Google Scholar
  12. 12.
    A. C.-C. Yao, Separating the polynomial-time hierachy by oracles, In Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, pages 1–10, 1985.Google Scholar
  13. 13.
    Z.-L. Zhang, D. A. Mix Barrington, and J. Tarui, Computing symmetric functions with AND/OR circuits and a single MAJORITY gate, In Proceedings of the 10th Annual Symposium on Theoretical Aspects of Computing, pages 535–544, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Chi-Jen Lu
    • 1
  1. 1.Computer Science DepartmentUniversity of Massachusetts at AmherstUSA

Personalised recommendations