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Reports tagged with Boolean functions:
TR96-010 | 9th February 1996
Christoph Meinel, Anna Slobodova

#### An Adequate Reducibility Concept for Problems Defined in Terms of Ordered Binary Decision Diagrams

Revisions: 1

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 >>>

TR16-174 | 7th November 2016
Elchanan Mossel, Sampath Sampath Kannan, Grigory Yaroslavtsev

#### Linear Sketching over $\mathbb F_2$

Revisions: 5 , Comments: 2

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 >>>

TR17-013 | 23rd January 2017
Abhishek Bhrushundi, Prahladh Harsha, Srikanth Srinivasan

#### On polynomial approximations over $\mathbb{Z}/2^k\mathbb{Z}$

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 >>>

TR17-180 | 26th November 2017
Irit Dinur, Yuval Filmus, Prahladh Harsha

#### Low degree almost Boolean functions are sparse juntas

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 >>>

TR18-167 | 25th September 2018
Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucky, Nitin Saurabh, Ronald de Wolf

#### Improved bounds on Fourier entropy and Min-entropy

Revisions: 1

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 >>>

TR18-187 | 4th November 2018

#### Domain Reduction for Monotonicity Testing: A $o(d)$ Tester for Boolean Functions on Hypergrids

Revisions: 4

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 >>>

TR18-205 | 3rd December 2018
Siddhesh Chaubal, Anna Gal

#### New Constructions with Quadratic Separation between Sensitivity and Block Sensitivity

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 >>>

TR20-002 | 6th January 2020
Sophie Laplante, Reza Naserasr, Anupa Sunny

#### Sensitivity lower bounds from linear dependencies

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 >>>

TR20-009 | 6th February 2020
Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, Muli Safra

#### Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality

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 >>>

TR21-044 | 14th February 2021
Alexander Kulikov, Nikita Slezkin

#### SAT-based Circuit Local Improvement

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 >>>

TR21-148 | 3rd November 2021
Benjamin Diamond, Amir Yehudayoff

#### Explicit Exponential Lower Bounds for Exact Hyperplane Covers

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.

more >>>

TR22-109 | 27th July 2022
Siddharth Iyer, Michael Whitmeyer

#### Searching for Regularity in Bounded Functions

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 >>>

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