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Monotone Boolean functions (MBFs) are Boolean functions $f: {0,1}^n \rightarrow {0,1}$ satisfying the monotonicity condition $x \leq y \Rightarrow f(x) \leq f(y)$ for any $x,y \in {0,1}^n$. The number of MBFs in n variables is known as the…
In many classification tasks there is a requirement of monotonicity. Concretely, if all else remains constant, increasing (resp. decreasing) the value of one or more features must not decrease (resp. increase) the value of the prediction.…
Semi-bent Boolean functions are interesting from a cryptographic standpoint, since they possess several desirable properties such as having a low and flat Walsh spectrum, which is useful to resist linear cryptanalysis. In this paper, we…
A fuzzy Boolean function is a map $f:\cube^n\to [0,1]$, where $n\in\mathbb N$. We introduce and compare three ways of saying that such a function has bounded complexity. The first is a sampling property: the value $f(x)$ can be recovered,…
Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is…
In this paper an algorithm is designed which generates in-equivalent Boolean functions of any number of variables from the four Boolean functions of single variable. The grammar for such set of Boolean function is provided. The Turing…
Polynomial threshold gates are basic processing units of an artificial neural network. When the input vectors are binary vectors, these gates correspond to Boolean functions and can be analyzed via their polynomial representations. In…
We associate to each Boolean function a polynomial whose evaluations represents the distances from all possible Boolean affine functions. Both determining the coefficients of this polynomial from the truth table of the Boolean function 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)}…
A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we…
This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean n-tuples is introduced. Stochastic boolean vectors and…
We prove that the fully asynchronous dynamics of a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ without negative loop can be simulated, in a very specific way, by a monotone Boolean network with $2n$ components. We then use this result to…
It is consistent that there is a partial order (P,<) of size aleph_1 such that every monotone (unary) function from P to P is first order definable in (P,<). The partial order is constructed in an extension obtained by finite support…
A novel approach to Boolean matrix factorization (BMF) is presented. Instead of solving the BMF problem directly, this approach solves a nonnegative optimization problem with the constraint over an auxiliary matrix whose Boolean structure…
Bernstein-Vazirani algorithm (the one-query algorithm) can identify a completely specified linear Boolean function using a single query to the oracle with certainty. The first aim of the paper is to show that if the provided Boolean…
The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$.…
Boolean networks have been used in a variety of settings, as models for general complex systems as well as models of specific systems in diverse fields, such as biology, engineering, and computer science. Traditionally, their properties as…
Monotone Boolean functions are a structurally important class of Boolean functions, but their restricted form imposes strong limitations on achievable nonlinearity. In this paper, we investigate whether evolutionary computation can evolve…
We address the problem of finding optimal strategies for computing Boolean symmetric functions. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we…
We consider the problem of studying the simulation capabilities of the dynamics of arbitrary networks of finite states machines. In these models, each node of the network takes two states 0 (passive) and 1 (active). The states of the nodes…