טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentTal Avishay
SubjectOn The Minimal Fourier Degree of Symmetric Boolean
Functions
DepartmentDepartment of Computer Science
Supervisor Professor Amir Shpilka
Full Thesis textFull thesis text - English Version


Abstract

In this thesis we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f:{0,1}k {0,1} there exists a non-empty subset S of {1,...,n} such that |S| = O(Γ(k) √k), and the S-fourier coefficient of f is non-zero, where Γ(m) ≤ m0.525 is the largest gap between consecutive prime numbers in {1,?,m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [MOS04]. Namely, we show that the running time of their algorithm is at most nO(Γ(k) √k) poly(n, 2k, log(1)), where n is the number of variables, k is the size of the junta (i.e. number of relevant variables) and δ is the error probability. In particular, for k ≥ log(n)1/(1-0.525) ≈ log(n)2.1 our analysis matches the lower bound 2k (up to polynomial factors).


Our bound on the degree greatly improves the previous result of Kolountzakis et al. [KLMMV09] who proved that |S| = O(k / log(k)).