#### Document Type

Dissertation

#### Degree

Doctor of Philosophy (PhD)

#### Major/Program

Computer Science

#### First Advisor's Name

Ning Xie

#### First Advisor's Committee Title

Committee Chair

#### Second Advisor's Name

Dong Chen

#### Second Advisor's Committee Title

Committee Member

#### Third Advisor's Name

Mohammad Hadi Amini

#### Third Advisor's Committee Title

Committee Member

#### Fourth Advisor's Name

S. S. Iyengar

#### Fourth Advisor's Committee Title

Committee Member

#### Fifth Advisor's Name

Xiaosheng Li

#### Fifth Advisor's Committee Title

Committee Member

#### Keywords

function analysis, fourier analysis, additive combinatorics

#### Date of Defense

7-1-2021

#### Abstract

Boolean function is one of the most fundamental computation models in theoretical computer science. The two most common representations of Boolean functions are Fourier transform and real polynomial form. Applying analytic tools under these representations to the study Boolean functions has led to fruitful research in many areas such as complexity theory, learning theory, inapproximability, pseudorandomness, metric embedding, property testing, threshold phenomena, social choice, etc. In this thesis, we focus on \emph{sparse representations} of Boolean function in both Fourier transform and polynomial form, and obtain the following new results. A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the Fourier support of $f$, then $f$ must be the indicator function of some affine subspace of dimension $n-k$. Here we slightly generalize their result, and show that Boolean functions whose Fourier coefficients take values in the set $\{-2/2^k, -1/2^k, 0, 1/2^k, 2/2^k\}$ are indicator functions of two disjoint affine subspaces of dimension $n-k$ or four disjoint affine subspaces of dimension $n-k-1$. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of $\mathbb{F}_2^{n}$ when the doubling constant of the subset is small. For polynomial representation of Boolean function, we study the distribution of the number of non-zero coefficients of \emph{random} Boolean functions. For a random Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$, i.e., a function whose value at each point on the Boolean cube is chosen independently and uniformly at random from $\{0,1\}$, in real polynomial representation, we give several bounds and concentration results about the distribution of the sparsity of $f$.

#### Identifier

FIDC010235

#### Previously Published In

Part of the work was presented as *A Generalization of a Theorem of Rothschild and van Lint* by The 16th International Computer Science Symposium in Russia CSR(2021).

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

#### Recommended Citation

Xu, Yekun, "A Study of Sparse Representation of Boolean Functions" (2021). *FIU Electronic Theses and Dissertations*. 4712.

https://digitalcommons.fiu.edu/etd/4712

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