Reducibility concepts are fundamental in complexity theory.
Usually, they are defined as follows: A problem P is reducible
to a problem S if P can be computed using a program or device
for S as a subroutine. However, in the case of such restricted
models as ... more >>>

We initiate a systematic study of linear sketching over $\mathbb F_2$. For a given Boolean function $f \colon \{0,1\}^n \to \{0,1\}$ a randomized $\mathbb F_2$-sketch is a distribution $\mathcal M$ over $d \times n$ matrices with elements over $\mathbb F_2$ such that $\mathcal Mx$ suffices for computing $f(x)$ with high ... more >>>

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function F$:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be F$_k:\{0,1\}^n \rightarrow \{0,2^{k-1}\}$ by $F_k(x) = 2^{k-F(x)}$ (mod $2^k$). We consider the fractional agreement (which we refer to as $\gamma_{d,k}(F)$) of $F_k$ with ... more >>>

Nisan and Szegedy showed that low degree Boolean functions are juntas. Kindler and Safra showed that low degree functions which are *almost* Boolean are close to juntas. Their result holds with respect to $\mu_p$ for every *constant* $p$. When $p$ is allowed to be very small, new phenomena emerge. ... more >>>

Given a Boolean function $f: \{-1,1\}^n\rightarrow \{-1,1\}$, define the Fourier distribution to be the distribution on subsets of $[n]$, where each $S\subseteq [n]$ is sampled with probability $\widehat{f}(S)^2$. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96] seeks to relate two fundamental measures associated with the Fourier distribution: does ... more >>>

Testing monotonicity of Boolean functions over the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic problem in property testing. When the range is real-valued, there are $\Theta(d\log n)$-query testers and this is tight. In contrast, the Boolean range qualitatively differs in two ways:
(1) Independence of $n$: There are testers ... more >>>

Nisan and Szegedy conjectured that block sensitivity is at most polynomial in sensitivity for any Boolean function. There is a huge gap between the best known upper bound on block sensitivity in terms of sensitivity - which is exponential, and the best known separating examples - which give only a ... more >>>

Recently, using spectral techniques, H. Huang proved that every subgraph of the hypercube of dimension n induced on more than half the vertices has maximum degree at least the square root of n. Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work we ... more >>>

We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL
Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a
new approach: looking at the first Fourier level of the function after a suitable random restriction and
applying the Log-Sobolev ... more >>>

Finding exact circuit size is a notorious optimization problem in practice. Whereas modern computers and algorithmic techniques allow to find a circuit of size seven in blink of an eye, it may take more than a week to search for a circuit of size thirteen. One of the reasons of ... more >>>

We describe an explicit and simple subset of the discrete hypercube which cannot be exactly covered by fewer than exponentially many hyperplanes. The proof exploits a connection to communication complexity, and relies heavily on Razborov's lower bound for disjointness.

Given a function $f:\mathbb F_2^n \to [-1,1]$, this work seeks to find a large affine subspace $\mathcal U$ such that $f$, when restricted to $\mathcal U$, has small nontrivial Fourier coefficients.

We show that for any function $f:\mathbb F_2^n \to [-1,1]$ with Fourier degree $d$, there exists an affine subspace ... more >>>

We describe a new construction of Boolean functions. A specific instance of our construction provides a 30-variable Boolean function having min-entropy/influence ratio to be 128/45 ? 2.8444 which is presently the highest known value of this ratio that is achieved by any Boolean function. Correspondingly, 128/45 is also presently the ... more >>>

The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory.
For example, symmetric functions, that is, functions that are invariant under the action of $S_n$ is an important class of functions in the study of Boolean functions.
A function $f:\{0,1\}^n \to \{0,1\}$ ... more >>>

For any Boolean functions $f$ and $g$, the question whether $\text{R}(f\circ g) = \tilde{\Theta}(\text{R}(f) \cdot \text{R}(g))$, is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether $\widetilde{\text{deg}}(f\circ g) = \tilde{\Theta}(\widetilde{\text{deg}}(f)\cdot\widetilde{\text{deg}}(g))$. These questions are two of the most important and ... more >>>

We study a natural complexity measure of Boolean functions known as the (exact) rational degree. For total functions $f$, it is conjectured that $\mathrm{rdeg}(f)$ is polynomially related to $\mathrm{deg}(f)$, where $\mathrm{deg}(f)$ is the Fourier degree. Towards this conjecture, we show that symmetric functions have rational degree at least $\mathrm{deg}(f)/2$ and ... more >>>