Andris Ambainis, Xiaoming Sun

In this note we give a new separation between sensitivity and block sensitivity of Boolean functions: $bs(f)=\frac{2}{3}s(f)^2-\frac{1}{3}s(f)$.

more >>>Meena Boppana

The Sensitivity Conjecture, posed in 1994, states that the fundamental measures known as the sensitivity and block sensitivity of a Boolean function $f$, $s(f)$ and $bs(f)$ respectively, are polynomially related. It is known that $bs(f)$ is polynomially related to important measures in computer science including the decision-tree depth, polynomial degree, ... more >>>

Andris Ambainis, Krisjanis Prusis

Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy, is whether sensitivity and block sensitivity are polynomially related. Motivated by the constructions of functions which achieve the largest known separations, we study the relation between 1-certificate complexity ... more >>>

Andris Ambainis, Jevgenijs Vihrovs

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of ... more >>>

Avishay Tal

The sensitivity of a Boolean function $f:\{0,1\}^n \to \{0,1\}$ is the maximal number of neighbors a point in the Boolean hypercube has with different $f$-value. Roughly speaking, the block sensitivity allows to flip a set of bits (called a block) rather than just one bit, in order to change the ... more >>>

Mitali Bafna, Satyanarayana V. Lokam, Sébastien Tavenas, Ameya Velingker

Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision ... more >>>

Shachar Lovett, Jiapeng Zhang

The sensitivity conjecture is one of the central open problems in boolean complexity. A recent work of Gopalan et al. [CCC 2016] conjectured a robust analog of the sensitivity conjecture, which relates the decay of the Fourier mass of a boolean function to moments of its sensitivity. We prove this ... more >>>

Pooya Hatami, Avishay Tal

A Boolean function is said to have maximal sensitivity $s$ if $s$ is the largest number of Hamming neighbors of a point which differ from it in function value. We construct a pseudorandom generator with seed-length $2^{O(\sqrt{s})} \cdot \log(n)$ that fools Boolean functions on $n$ variables with maximal sensitivity at ... more >>>

Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Pierre McKenzie, Shadab Romani

The Black-Box Hypothesis, introduced by Barak et al. (JACM, 2012), states that any property of boolean functions decided efficiently (e.g., in BPP) with inputs represented by circuits can also be decided efficiently in the black-box setting, where an algorithm is given an oracle access to the input function and an ... more >>>

Or Meir, Avi Wigderson

Consider a random sequence of $n$ bits that has entropy at least $n-k$, where $k\ll n$. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly distributed, that is, the coordinate “looks random”. In this work, we prove a stronger result that ... more >>>

Krishnamoorthy Dinesh, Jayalal Sarma

The well-known Sensitivity Conjecture regarding combinatorial complexity measures on Boolean functions states that for any Boolean function $f:\{0,1\}^n \to \{0,1\}$, block sensitivity of $f$ is polynomially related to sensitivity of $f$ (denoted by $\mathbf{sens}(f)$). From the complexity theory side, the XOR Log-Rank Conjecture states that for any Boolean function, $f:\{0,1\}^n ... more >>>

Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett

We propose a new framework for constructing pseudorandom generators for $n$-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in $[-1,1]^n$. Next, we use a fractional pseudorandom generator as steps of a random walk in $[-1,1]^n$ that ... more >>>

Krishnamoorthy Dinesh, Jayalal Sarma

In this paper, we study the Boolean function parameters sensitivity ($\mathbf{s}$), block sensitivity ($\mathbf{bs}$), and alternation ($\mathbf{alt}$) under specially designed affine transforms and show several applications. For a function $f:\mathbb{F}_2^n \to \{0,1\}$, and $A = Mx+b$ for $M \in \mathbb{F}_2^{n \times n}$ and $b \in \mathbb{F}_2^n$, the result of the ... more >>>

Sophie Laplante, Reza Naserasr, Anupa Sunny

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 >>>

Siddhesh Chaubal, Anna Gal

Nisan and Szegedy conjectured that block sensitivity is at most

polynomial in sensitivity for any Boolean function.

Until a recent breakthrough of Huang, the conjecture had been

wide open in the general case,

and was proved only for a few special classes

of Boolean functions.

Huang's result implies that block ...
more >>>