Related papers: The More the Merrier: On Evolving Five-valued Spec…

A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The…

We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linear equations, we focus…

We develop a method for training neural networks on Boolean data in which the values at all nodes are strictly $\pm 1$, and the resulting models are typically equivalent to networks whose nonzero weights are also $\pm 1$. The method…

We consider the problem of computing the Boolean convolution (with wraparound) of $n$~vectors of dimension $m$, or, equivalently, the problem of computing the sumset $A_1+A_2+\ldots+A_n$ for $A_1,\ldots,A_n \subseteq \mathbb{Z}_m$. Boolean…

We consider polynomial transforms (polyspectra) of Berry's model -- the Euclidean Random Wave model -- and of Random Hyperspherical Harmonics. We determine the asymptotic behavior of variance for polyspectra of any order in the…

We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends…

In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these…

In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and…

Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision, and many others. The general solver has a complexity of $O(n^3 \log^2 n . E +n^4 {\log}^{O(1)}…

In this paper, we consider decision trees that use both queries based on one attribute each and queries based on hypotheses about values of all attributes. Such decision trees are similar to ones studied in exact learning, where not only…

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof method yields an improved bound of $\widetilde{O}(\sqrt{s})$ assuming a conjecture of…

The subject of this textbook is the analysis of Boolean functions. Roughly speaking, this refers to studying Boolean functions $f : \{0,1\}^n \to \{0,1\}$ via their Fourier expansion and other analytic means. Boolean functions are perhaps…

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…

We consider the Stochastic Boolean Function Evaluation (SBFE) problem where the task is to efficiently evaluate a known Boolean function $f$ on an unknown bit string $x$ of length $n$. We determine $f(x)$ by sequentially testing the…

Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this…

This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a H\"older continuous function with H\"older exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a…

We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main…

Boolean functions are important primitives in different domains of cryptology, complexity and coding theory. In this paper, we connect the tools from cryptology and complexity theory in the domain of Boolean functions with low polynomial…

Finding the optimal parameter setting (i.e. the optimal population size, the optimal mutation probability, the optimal evolutionary model etc) for an Evolutionary Algorithm (EA) is a difficult task. Instead of evolving only the parameters…

Estimation of distribution algorithms are evolutionary algorithms that use extracted information from the population instead of traditional genetic operators to generate new solutions. This information is represented as a probabilistic…