Boolean functions and vectorial Boolean functions are the most important nonlinear components of stream ciphers. They should satisfy several criteria such as high nonlinearity, proper resiliency and so on to guarantee the security of the whole system...
Known and new results on free Boolean topological groups are collected. An account of the properties that these groups share with free or free Abelian topological groups and properties specific to free Boolean groups is given. Special emphasis is pla...
We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean netwo...
Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consi...
Various notions of large sets in groups, including the classical notions of thick, syndetic, and piecewise syndetic sets and the new notion of vast sets in groups, are studied with emphasis on the interplay between such sets in Boolean groups. Natura...
In several stream cipher designs, Boolean functions (BFs) play a crucial role as non-linear components, either serving as filtering functions or being used within the combining process. The overall strength of stream ciphers mainly depends on certain...
In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algeb...
Boolean chaotic systems solely composed of logic devices have been successfully applied in fields such as random number generation, reservoir computing, and radar detection because of their simple structure and amenability to integration. However, no...
In 2005, Guillot published a method for the construction of Boolean functions using linear codes through the Maiorana–McFarland construction of Boolean functions. In this work, we present a construction using Hermitian codes, starting from the...
The minimization of Boolean functions allows designers to make use of fewer components, thus reducing the cost of particular system. All procedures for reducing either two-level or multilevel Boolean networks into prime and irredundant form have O(2n...
We study some properties of arctic rank of Boolean matrices. We compare the arctic rank with Boolean rank and term rank of a given Boolean matrix. Furthermore, we obtain some characterizations of linear operators that preserve arctic rank on Boolean...
Boolean operations on geometric models are important in numerical simulation and serve as essential tools in the fields of computer-aided design and computer graphics. The accuracy of these operations is heavily influenced by finite precision arithme...
We revisit Probabilistic Boolean Networks as trainable function approximators. The key obstacle, non-differentiable structural choices (which predictors to read and which Boolean operators to apply), is addressed by casting the PBN’s structure...
This paper reviews recent approaches on how to accelerate Boolean Satisfiability (SAT) search by exploiting symmetries in the problem space. SAT search algorithms traverse an exponentially large search space looking for an assignment that satisfies a...
This paper introduces a novel quantum algorithm that is able to classify a hierarchy of classes of imbalanced Boolean functions. The fundamental characteristic of imbalanced Boolean functions is that the proportion of elements in their domain that ta...
Systems poised at a dynamical critical regime, between order and disorder, have been shown capable of exhibiting complex dynamics that balance robustness to external perturbations and rich repertoires of responses to inputs. This property has been ex...
The conversion between logic and algebraic expressions of Boolean control networks plays a worthy role in the analysis and design of digital circuits. In this paper, for a single Boolean function, a direct conversion between the minterm canonical for...
A bicluster consists of a subset of rows and columns of a given matrix, whose intersection defines the region (bicluster) of values of precisely defined condition. Through the decades, a variety of biclustering techniques have been successfully devel...
In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show th...
A probabilistic Boolean network (PBN) is well known as one of the mathematical models of gene regulatory networks. In a Boolean network, expression of a gene is approximated by a binary value, and its time evolution is expressed by Boolean functions....
In 2005, Philippe Guillot presented a new construction of Boolean functions using linear codes as an extension of the Maiorana–McFarland’s (MM) construction of bent functions. In this paper, we study a new family of Boolean functions with cryptograph...
We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and ca...
Biological systems are commonly constituted by a high number of interacting agents. This great dimensionality hinders biological modelling due to the high computational cost. Therefore, new modelling methods are needed to reduce computation time whil...
One of the fastest known general techniques for computing permanents is Ryser’s formula. On this note, we show that this formula over Sylvester Hadamard matrices of order
Existing algorithms for learning Boolean networks (BNs) have time complexities of at least O(N · n0:7(k+1)), where n is the number of variables, N is the number of samples and k is the number of inputs in Boolean functions. Some recent studies propos...
The present paper is part of the research on the description of rings with a given property of the lattice of left (right) annihilators. The anti-isomorphism of lattices of left and right annihilators in any ring gives some kind of symmetry: the latt...
Probabilistic Boolean Networks can capture the dynamics of complex biological systems as well as other non-biological systems, such as manufacturing systems and smart grids. In this proof-of-concept manuscript, we propose a Probabilistic Boolean Netw...
Applying the correct set of Boolean operations is a fundamental task in constructive solid geometry (CSG), which is a staple in automated manufacturing systems. Although textual buttons and icons are the most common interfaces to apply such operation...
In this paper, we present four product operations to construct cryptographic boolean functions from smaller ones with predictable Walsh spectrum. A lot of cryptographic properties of boolean functions can be presented by their Walsh spectrum. In our...
This article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols,...
We consider finite B-fuzzy topologies, that is, collections of mappings from a finite set to the Boolean algebra B, satisfying analogous conditions of classical topologies. We establish a one-to-one correspondence between the class of B-fuzzy topolog...
24 November 2023
We propose and implement an interpretable machine learning classification model for Explainable AI (XAI) based on expressive Boolean formulas. Potential applications include credit scoring and diagnosis of medical conditions. The Boolean formula defi...
The solution of nonlinear Boolean equations in a binary field plays a crucial part in cryptanalysis and computational mathematics. To speed up the process of solving Boolean equations is an urgent task that needs to be addressed. In this paper, we pr...
Control of gene regulatory networks is one of the fundamental topics in systems biology. In the last decade, control theory of Boolean networks (BNs), which is well known as a model of gene regulatory networks, has been widely studied. In this review...
Resolution of fuzzy relational inequalities (FRIs) plays a significant role in decision-making, image compression and fuzzy control. This paper studies the resolution of a kind of FRIs with Boolean semi-tensor product composition. First, by resorting...
Boolean network (BN) is a simple model widely used to study complex dynamic behaviour of biological systems. Nonetheless, it might be difficult to gather enough data to precisely capture the behavior of a biological system into a set of Boolean funct...
Search engines use significant hardware and energy resources to process billions of user queries per day, where Boolean query processing for document retrieval is an essential ingredient. Considering the huge number of users and large scale of the ne...
In this paper, we present a library with sequential and parallel functions for computing some of the most important cryptographic characteristics of Boolean and vectorial Boolean functions. The library implements algorithms to calculate the nonlinear...
Let X be a nonempty set and
This paper proposes a methodology for the design of fuzzy inference systems based on Boolean relations. The approach using Boolean sets presents limited performance due to the abrupt transitions that occur during its functioning, therefore, fuzzy set...
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open proble...
This paper studies the robust stabilization of impulsive Boolean control networks (IBCNs) with function perturbation. A Boolean control network (BCN) with a state-dependent impulsive sequence is converted to an equivalent BCN by the semi-tensor produ...
In this paper, we address an NPN Boolean matching algorithm. The proposed structural difference signature (SDS) of a Boolean function significantly reduces the search space in the Boolean matching process. The paper analyses the size of the search sp...
We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures...
Stream ciphers usually rely on highly secure Boolean functions to ensure safe communication within unsafe channels. However, discovering secure Boolean functions is a non-trivial optimization problem that has been addressed by many optimization techn...
Knowledge space theory (KST) is a mathematical framework for the assessment of knowledge and learning in education. An important task of KST is to achieve all of the atoms. With the development of KST, considering its relationship with formal concept...
This article focuses on the relationships between fuzzy logic and classical logic properties using fuzzy t-norms, t-conorms, and fuzzy implications. It aims to contribute to fuzzy set theory by extending the Boolean laws in classical logic to fuzzy l...
Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltje...
Biological pathways are thought to be robust against a variety of internal and external perturbations. Fail-safe mechanisms allow for compensation of perturbations to maintain the characteristic function of a pathway. Pathways can undergo changes dur...
There are several goals of the two-dimensional data analysis: one may be interested in searching for groups of similar objects (clustering), another one may be focused on searching for some dependencies between a specified one and other variables (cl...
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