Mathematical systems theory

, Volume 29, Issue 3, pp 245–258

# On the correlation of symmetric functions

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

## Abstract

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.

## Keywords

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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© 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