Search Results (34)

The growing volume of electronic health records (EHRs) has made cloud-assisted storage increasingly important, but outsourced records require both encrypted storage and controlled retrieval. Attribute-based keyword searchable encryption (ABKS) combines search with fine-grained authorization; nevertheless, many existing constructions provide only restricted Boolean queries, particularly when negation is involved. This paper presents a ciphertext-policy ABKS search layer for EHR sharing. The proposed construction operates over a fixed application dictionary and assumes a non-colluding honest-but-curious cloud server. Each record is represented by one signed literal per dictionary term, where a positive literal denotes presence and a negated literal denotes absence. Before query evaluation, Boolean formulas involving AND, OR, and NOT are converted into negation normal form using De Morgan’s laws, so that NOT appears only at leaf nodes. The resulting query is then evaluated as a monotone threshold tree. An authorized user can generate a valid query token only when the user’s multi-valued attribute vector matches the policy selected by the data owner. Keyed pseudorandom tags hide literal and policy values from the cloud server while enabling efficient component lookup. We define a selective chosen-keyword security model, state the resulting leakage, and give a game-based proof under the decisional bilinear Diffie–Hellman and pseudorandom-function assumptions. Theoretical and implementation-based comparisons show that the pairing-dominant costs grow linearly with the attribute and dictionary sizes. The construction therefore improves Boolean expressiveness while retaining practical search-layer overhead under the stated assumptions. Full article

Biclustering is a two-dimensional data analysis paradigm that aims to identify subsets of rows and columns in a data matrix whose intersection forms a submatrix satisfying predefined properties. In the case of binary data, a bicluster is typically defined as a submatrix containing exclusively ones or exclusively zeros. Numerous approaches to binary biclustering have been proposed over the last several decades. Among them, a distinctive line of research relies on Boolean reasoning, transforming the biclustering task into the problem of finding prime implicants of a data-dependent Boolean function. From a theoretical perspective, it has been shown that every prime implicant of the data-dependent Boolean function corresponds to an inclusion–maximal bicluster in the original dataset, and vice versa. Here, inclusion maximality means that no additional row or column can be added without violating the bicluster property. For discrete datasets, the corresponding Boolean function can be represented in Conjunctive Normal Form (CNF), where clauses contain up to three variables, reducing the general biclustering problem to an instance of the 3-SAT problem. Furthermore, the computational complexity of discrete-data biclustering can be reduced to the analysis of binary matrices. In this case, the associated Boolean functions consist of clauses containing at most two literals, yielding monotone 2-CNF formulas. This paper focuses on improving the efficiency of computations required to derive a disjunctive normal form (DNF) representation composed exclusively of prime implicants. Since the extraction of prime implicants constitutes the computational core of the Boolean biclustering framework, accelerating this process directly enhances the scalability and practical applicability of biclustering methods based on Boolean reasoning. Full article

In this paper, we consider the groups of linear operators that preserve idempotent matrices over the binary Boolean algebra. We prove that this set forms an automorphism group under composition and determine its exact algebraic structure as a direct product $S_n\times C_2$ for $n\ge 2$. Furthermore, we distinguish between elementary transposition matrices and involutory permutation matrices. We establish a recurrence relation, a closed combinatorial formula and asymptotic estimates for the number of involutory permutation matrices, showing that these numbers coincide with the involution numbers in the symmetric group. Structural properties such as center, solvability and generating sets of this group are fully characterized. Full article

Statistical classification is traditionally studied using probability, decision theory, information theory, and optimisation. We show that classification problems also possess a natural combinatorial and topological structure. Given a binary classification problem, we introduce the notion of a decision-sufficient feature set, namely a collection of features that preserves the optimal Bayes decision rule. Rather than focusing on a single optimal feature subset, we study the entire family of decision-sufficient feature sets. We prove that this family forms an order filter in the Boolean lattice of feature subsets and that its complement forms an abstract simplicial complex. Consequently, every classification problem canonically determines a topological object, called the decision complex, the faces of whichh correspond to feature subsets that fail to preserve optimal decisions. This viewpoint leads naturally to three invariants: the decision core, the decision dimension, and the decision complex. General structural results are established, and a complete characterisation is obtained for Gaussian-discriminant models. In that setting, the decision core coincides with the support of the discriminant vector, yielding explicit formulae for the decision core and decision dimension. These results establish a new connection between statistical decision theory and combinatorial topology. Full article

The objective of this work is to investigate the $\mathbb{T}=(s,t)$-deformed free convolution ${⊞}_{\mathbb{T}}$ for $s>0$ and $t\in \mathbb{R}$ and to clarify its structural and asymptotic properties within the framework of Cauchy–Stieltjes kernel (CSK) families. The methodology is based on the analysis of the associated variance functions (VFs), which provide an effective analytic tool for describing deformation mechanisms, invariance properties, and convolution structures. In particular, we derive an explicit formula for the VF of convolution powers and exploit this representation to develop approximation procedures for distributions in CSK families generated by the $(s,t)$-deformed free Gaussian and free Poisson laws. We also establish several limit theorems describing the asymptotic behavior of the deformation. These findings highlight intrinsic symmetry and scaling properties and reveal connections with free additive, Boolean additive, and free multiplicative convolutions, thereby placing the $(s,t)$-deformation within a unified probabilistic framework governed by transformation, invariance, and structural regularity. Full article

We construct and analyze a family of support-defined Brauer-type configurations canonically associated with the Boolean geometry underlying the Grassmann algebra. The construction is governed by an x-support map on monomial labels, which identifies the vertex set with the Boolean lattice $\mathcal{P}\left(\right[n\left]\right)$. This identification yields a Boolean support quiver isomorphic to the directed Hasse diagram of $\mathcal{P}\left(\right[n\left]\right)$, equivalently, to an oriented hypercube. We then equip the family with a canonical cyclic ordering at each vertex and obtain a genuine connected reduced Brauer configuration in the standard sense, together with its associated Brauer configuration algebra and its standard Brauer quiver. A ghost-variable mechanism is introduced to obtain a connected realization without altering any support-controlled invariants. We prove that polygon membership, valencies, multiplicities, Boolean stratification, and the support quiver are invariant under support-preserving ghost relabelings. We also give an explicit description of the standard Brauer quiver and show that it is different from the Boolean support quiver. On the algebraic side, we derive closed formulas for the center dimension, the algebra dimension, and the normalization constant of the induced weighted distribution. On the probabilistic side, we distinguish the vertex entropy from the layer entropy, establish an exact decomposition of the former by Hamming layers, and show that the layer distribution is asymptotically concentrated on the middle layers, while extremal vertices and any fixed maximal path contribute a negligible fraction of the total weight. As a consequence, the layer entropy satisfies a logarithmic asymptotic law. We also investigate geometric consequences of the Boolean model transported through the support identification. Coordinate projections produce a rigidity phenomenon for antipodal pairs, providing a combinatorial analogue of Greenberger–Horne–Zeilinger (GHZ)-type fragility, whereas the first Boolean layer exhibits a persistence property analogous to W-type robustness. Together, these results exhibit a concrete bridge between Grassmann combinatorics, Brauer configuration theory, hypercube geometry, and entropy asymptotics. Full article

This manuscript proves specific theorems for transforming Boolean expressions of logical formulas when moving from one basis to another, simplifying the solution of complex equations, especially for cryptographic applications. The paper develops methods for solving specific nonlinear systems of Boolean equations used in cryptographic S-boxes using transformations to simpler forms, such as disjunctive normal forms (DNFs) and Zhegalkin polynomials. The main contributions include a mathematical basis for transforming formulas, a complexity-reducing grouping method, and the RLSY program for practical implementation. A rigorous theory, cryptographic relevance, and a detailed description of the algorithm are proposed. The grouping method reduces the system complexity by a factor of $2^{11}$, as shown in a test example, improving computational efficiency. A solution to a special class of systems of nonlinear Boolean equations of the second degree, which are a logical model of algebraic cryptanalysis, is also proposed. Test examples of logical formula transformations are given. Full article

As a contribution to automated set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms $x=y\setminus z$, $x\ne y\setminus z$, and $z=\left\{x\right\}$, where $x,y,z$ stand for variables ranging over the von Neumann universe of sets, into quantifier-free Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness, and inclusion. Moreover, the result of each translation is a conjunction of literals of the forms $x=y\setminus z$ and $x\ne y\setminus z$ and of implications whose antecedents are isolated literals and whose consequents are either inclusions (strict or non-strict) between variables, or equalities between variables. Besides reflecting a simple and natural semantics, which ensures satisfiability preservation, the proposed translation has quadratic algorithmic time complexity and bridges two languages, both of which are known to have an $\mathsf{NP}$-complete satisfiability problem. Full article

This paper is a theoretical contribution that introduces a new reasoning framework for SAT solving based on resolvable networks (RNs). RNs provide a graph-based representation of propositional satisfiability in which clauses are interpreted as directed reaches between disjoint subsets of Boolean variables (nodes). Building on this framework, we introduce a novel RN-based SAT solver, called RN-Solver, which replaces local assignment-driven branching by global reasoning over token distributions. Token distributions, interpreted as truth assignments, are generated by monotone CNF–DNF dualization applied to white (all-positive) clauses. New white clauses are derived via resolution along private-pivot chains, and the solver’s progression is governed by a taxonomy of token distributions (black-blocked, terminal, active, resolved, and non-resolved). The main results establish the soundness and completeness of the RN-Solver. Experimentally, the solver performs very well on pigeonhole formulas, where the separation between white and black clauses enables effective global reasoning. In contrast, its current implementation performs poorly on random 3-SAT instances, highlighting both practical limitations and significant opportunities for optimization and theoretical refinement. The presented RN-Solver implementation is a proof-of-concept which validates the underlying theory rather than a state-of-the-art competitive solver. One promising direction is the generalization of strongly connected components from directed graphs to resolvable networks. Finally, the token-based perspective naturally suggests a connection to token-superposition Petri net models. Full article

The concept of r-free convolution (which is represented by $\begin{array}{|c|}\hline r\\ \hline\end{array}$) was introduced for $0\le r\le 1$. It is equal to the Boolean additive convolution ⊎ if $r=0$ and reduced to the free additive convolution ⊞ when $r=1$. This paper presents certain features of the r-free convolution in relation to the CSK families and their associated variance functions. We provide the variance function formula under $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution power. We then estimate members of the r-free Gaussian and r-free Poisson CSK families using the variance function and the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution, respectively. Additionally, a novel limit theorem for the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution is provided utilizing the variance functions and the free multiplicative convolution. Full article

We propose SAT-GATv2, a graph neural network (GNN)-based model designed to solve the Boolean satisfiability problem (SAT) through graph-based deep learning techniques. SAT-GATv2 transforms SAT formulas into graph structures, leveraging message-passing neural networks (MPNNs) to propagate local information and dynamic attention mechanisms (GATv2) to accurately capture inter-node dependencies and enhance node feature representations. Unlike traditional heuristic-driven SAT solvers, SAT-GATv2 adopts a data-driven approach, learning structural patterns directly from graph representations and providing a complementary framework to existing methods. Experimental results demonstrate that SAT-GATv2 achieves an accuracy improvement of 1.75–5.51% over NeuroSAT on challenging random 3-SAT(n) instances, highlighting its effectiveness in handling difficult problem distributions, and outperforms other GNN-based models on SR(n) datasets, showcasing its scalability and adaptability. Ablation studies validate the critical roles of MPNNs and GATv2 in improving prediction accuracy and scalability. While SAT-GATv2 does not yet surpass CDCL-based solvers in overall performance, it addresses their limitations in scalability and adaptability to complex instances, offering an efficient graph-based alternative for tackling larger and more complex SAT problems. This study establishes a foundation for integrating deep learning with combinatorial optimization, emphasizing its potential for applications in artificial intelligence and operations research. Full article

In the study of algebraic structures related to logical systems, ideals and filters have different meanings and are algebraic notions related to logical provable formulas. Unlike the classical Boolean lattice theory, ideals and filters are not dual notions in residuated lattices. An interesting subclass of residuated lattices is the class of triangle algebras, which is an equational representation of interval-valued residuated lattices that provides an algebraic framework for using closed intervals as truth values in fuzzy logic. The main aim of this article is to introduce and study the concept of ideals in triangle algebras and investigate the connection between ideals and filters. We first point out that the construction procedure for the filter generated by a subset of a triangle algebra established by another study is incorrect, and we proceed to give an alternative characterization. Full article

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 defines a rule with tunable complexity (or interpretability) according to which input data are classified. Such a formula can include any operator that can be applied to one or more Boolean variables, thus providing higher expressivity compared to more rigid rule- and tree-based approaches. The classifier is trained using native local optimization techniques, efficiently searching the space of feasible formulas. Shallow rules can be determined by fast Integer Linear Programming (ILP) or Quadratic Unconstrained Binary Optimization (QUBO) solvers, potentially powered by special-purpose hardware or quantum devices. We combine the expressivity and efficiency of the native local optimizer with the fast operation of these devices by executing non-local moves that optimize over the subtrees of the full Boolean formula. We provide extensive numerical benchmarking results featuring several baselines on well-known public datasets. Based on the results, we find that the native local rule classifier is generally competitive with the other classifiers. The addition of non-local moves achieves similar results with fewer iterations. Therefore, using specialized or quantum hardware could lead to a significant speedup through the rapid proposal of non-local moves. Full article

Self-adhesive resin cements (SARCs) are used because of their mechanical properties, ease of cementation protocols, and lack of requirements for acid conditioning or adhesive systems. SARCs are generally dual-cured, photoactivated, and self-cured, with a slight increase in acidic pH, allowing self-adhesiveness and increasing resistance to hydrolysis. This systematic review assessed the adhesive strength of SARC systems luted to different substrates and computer-aided design and manufacturing (CAD/CAM) ceramic blocks. The PubMed/MedLine and Science Direct databases were searched using the Boolean formula [((dental or tooth) AND (self-adhesive) AND (luting or cement) AND CAD-CAM) NOT (endodontics or implants)]. Of the 199 articles obtained, 31 were selected for the quality assessment. Lava Ultimate (resin matrix filled with nanoceramic) and Vita Enamic (polymer-infiltrated ceramic) blocks were the most tested. Rely X Unicem 2 was the most tested resin cement, followed by Rely X Unicem > Ultimate > U200, and μTBS was the test most used. The meta-analysis confirmed the substrate-dependent adhesive strength of SARCs, with significant differences between them and between SARCs and conventional resin-based adhesive cement (α < 0.05). SARCs are promising. However, one must be aware of the differences in the adhesive strengths. An appropriate combination of materials must be considered to improve the durability and stability of restorations. Full article

The purpose of the current study is to investigate a special case of art gallery problem, namely a sculpture garden problem. In this problem, for a given polygon P, the ultimate goal is to place the minimum number of guards (landmarks) to define the interior polygon P by applying a monotone Boolean formula composed of the guards. Using this problem, it can replace the operation-based method with time-consuming, pixel-based algorithms. So, the processing time of some problems in the fields of machine vision, image processing and gamification can be strongly reduced. The problem has also many applications in mobile device localization in the Internet of Things (IoT). An open problem in this regard is the proof of Eppstein’s conjecture, which has remained an open problem since 2007. According to his conjecture, in the worst case, $n-2$ vertex guards are required to describe any n-gon. In this paper, a lower bound is introduced for the special case of this problem (natural vertex guard), which shows that if a polygon can be defined with natural vertex guards, then $n-2$ is a lower bound. Full article