Related papers: Spectral Norm, Economical Sieve, and Linear Invari…
Let $f$ and $g$ be Boolean functions over a finite Abelian group $\mathcal{G}$, where $g$ is fully known, and we have {\em query access} to $f$, that is, given any $x \in \mathcal{G}$ we can get the value $f(x)$. We study the tolerant…
In this short note, we initiate the study of the Linear Isomorphism Testing Problem in the setting of communication complexity, a natural linear algebraic generalization of the classical Equality problem. Given Boolean functions $f, g :…
The standard model of Boolean function property testing is not well suited for testing $\textit{sparse}$ functions which have few satisfying assignments, since every such function is close (in the usual Hamming distance metric) to the…
The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [CDH+24][CPPS25a][CPPS25b] In this paper we consider the problem of relative-error testing an unknown and arbitrary…
Several recent works [DHLNSY25, CPPS25a, CPPS25b] have studied a model of property testing of Boolean functions under a \emph{relative-error} criterion. In this model, the distance from a target function $f: \{0,1\}^n \to \{0,1\}$ that is…
We give a non-adaptive algorithm that makes $2^{\tilde{O}(\sqrt{k\log(1/\varepsilon_2 - \varepsilon_1)})}$ queries to a Boolean function $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$ and distinguishes between $f$ being $\varepsilon_1$-close to some…
We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra~(SICOMP 2018) for Boolean functions to the case of real-valued functions $f \colon \{0,1\}^d\to\mathbb{R}$. Our main tool in the proof of the generalized…
We present an adaptive tester for the unateness property of Boolean functions. Given a function $f:\{0,1\}^n \to \{0,1\}$ the tester makes $O(n \log(n)/\epsilon)$ adaptive queries to the function. The tester always accepts a unate function,…
We give the first agnostic, efficient, proper learning algorithm for monotone Boolean functions. Given $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$ uniformly random examples of an unknown function $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$, our…
A Boolean function $f:\{0,1\}^n\to \{0,1\}$ is $k$-linear if it returns the sum (over the binary field $F_2$) of $k$ coordinates of the input. In this paper, we study property testing of the classes $k$-Linear, the class of all $k$-linear…
Let a Boolean function be available as a black-box (oracle) and one likes to devise an algorithm to test whether it has certain property or it is $\epsilon$-far from having that property. The efficiency of the algorithm is judged by the…
The framework of distribution testing is currently ubiquitous in the field of property testing. In this model, the input is a probability distribution accessible via independently drawn samples from an oracle. The testing task is to…
Let $f: T\to \{ 0,1 \}$ be a Boolean function on the Boolean half-slice, $T$, \ie elements of $\{0,1\}^n$ with Hamming weight $n/2$. We show that if $f(x)+f(y)=f(x+y)$ holds with probability $\frac{1+\delta}{2}$ over a uniform pair $(x,y)$…
A natural measure of smoothness of a Boolean function is its sensitivity (the largest number of Hamming neighbors of a point which differ from it in function value). The structure of smooth or equivalently low-sensitivity functions is still…
A Boolean function is symmetric if it is invariant under all permutations of its arguments; it is quasi-symmetric if it is symmetric with respect to the arguments on which it actually depends. We present a test that accepts every…
The relative-error property testing model was introduced in [CDHLNSY24] to facilitate the study of property testing for "sparse" Boolean-valued functions, i.e. ones for which only a small fraction of all input assignments satisfy the…
We consider the problem of testing whether an unknown Boolean function $f$ is monotone versus $\epsilon$-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied…
We prove a $k^{-\Omega(\log(\varepsilon_2 - \varepsilon_1))}$ lower bound for adaptively testing whether a Boolean function is $\varepsilon_1$-close to or $\varepsilon_2$-far from $k$-juntas. Our results provide the first superpolynomial…
The goal in the area of functions property testing is to determine whether a given black-box Boolean function has a particular given property or is $\varepsilon$-far from having that property. We investigate here several types of properties…
We study the relative-error property testing model for Boolean functions that was recently introduced in the work of Chen et al. (SODA 2025). In relative-error testing, the testing algorithm gets uniform random satisfying assignments as…