Mathematical systems theory

, Volume 29, Issue 3, pp 245–258 | Cite as

On the correlation of symmetric functions

  • Jin -Yi Cai
  • F. Green
  • T. Thierauf


Thecorrelation between two Boolean functions ofn inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper we compute, in closed form, the correlation between any twosymmetric Boolean functions. As a consequence of our main result, we get that every symmetric Boolean function having an odd period has anexponentially small correlation (inn) with the parity function. This improves a result of Smolensky [12] restricted to symmetric Boolean functions: the correlation between parity and any circuit consisting of a Mod q gate over AND gates of small fan-in, whereq is odd and the function computed by the sum of the AND gates is symmetric, is bounded by 2−Ω(n).

In addition, we find that for a large class of symmetric functions the correlation with parity isidentically zero for infinitely manyn. We characterize exactly those symmetric Boolean functions having this property.


Boolean Function Symmetric Function Parity Function Random Oracle Symmetric Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    M. Ajtai, Σ11-formulae on finite structures,Annals of Pure and Applied Logic 24 (1983), 1–48.CrossRefMathSciNetzbMATHGoogle Scholar
  2. [2]
    L. Babai, A random oracle separates PSPACE from the polynomial-time hierarchy,Information Processing Letters 26 (1987), 51–53.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [3]
    D. M. Barrington, R. Beigel, and S. Rudich, Representing Boolean functions as polynomials modulo composite numbers,Proceedings of the 24th ACM Symposium on Theory of Computing, 1992, pp. 455–461.Google Scholar
  4. [4]
    J.-Y. Cai, With probability one, a random oracle separates PSPACE from the polynomial-time hierarchyJournal of Computer and System Science 38 (1989), 68–85.CrossRefzbMATHGoogle Scholar
  5. [5]
    M. Furst, J. B. Saxe, and M. Sipser, Parity, circuits, and the polynomial-time hierarchy,Mathematical Systems Theory 17 (1984), 13–27.CrossRefMathSciNetzbMATHGoogle Scholar
  6. [6]
    F. Green, An oracle separating ⊕P from PPPH,Information Processing Letters 37 (1991), 149–153.CrossRefMathSciNetzbMATHGoogle Scholar
  7. [7]
    J. Håstad,Computational Limitations of Small-Depth Circuits, MIT Press, Cambridge, MA, 1987.Google Scholar
  8. [8]
    K. Ireland and M. Rosen,A Classical Introduction to Modern Number Theory, 2nd edn., Springer-Verlag, New York, 1990.zbMATHGoogle Scholar
  9. [9]
    N. Katz, Sommes Exponentielles,Astérisque 79 (1980).Google Scholar
  10. [10]
    A. A. Razborov, Lower bounds on the size of bounded depth networks over a complete basis with logical addition,Matematicheskie Zametki 41 (1987), 598–607. English translation:Mathematical Notes of the Academy of Sciences of the USSR 41 (1987), 333–338.MathSciNetGoogle Scholar
  11. [11]
    W. M. Schmidt,Equations over Finite Fields: An Elementary Approach, Lecture Notes in Mathematics, vol. 536, Springer-Verlag, New York, 1976.Google Scholar
  12. [12]
    R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity,Proceedings of the 19thAnnual ACM Symposium on Theory of Computing, 1987, pp. 77–82.Google Scholar
  13. [13]
    C. Smorynski,Logical Number Theory, Vol. I, Springer-Verlag, New York, 1991.Google Scholar
  14. [14]
    A. C. Yao, Separating the polynomial-time hierarchy by oracles,Proceedings of the 26thAnnual IEEE Symposium on Foundations of Computer Science, 1985, pp. 1–10.Google Scholar
  15. [15]
    S. Zabek, Sur la périodicité modulom des suites de nombres (k n),Annals Universitatis Mariae Curie-Sklodowska Sectio A 10 (1956), 37–47.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1996

Authors and Affiliations

  • Jin -Yi Cai
    • 1
  • F. Green
    • 2
  • T. Thierauf
    • 3
  1. 1.Department of Computer ScienceState University of New York at BuffaloBuffaloUSA
  2. 2.Department of Mathematics & Computer ScienceClark UniversityWorcesterUSA
  3. 3.Abteilung Theoretische InformatikUniversität UlmOberer EselsbergGermany

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