Granular rough sets approximate a target concept using granules rather than individual objects. Three-way decision based on granular rough sets follows this idea by constructing granular positive, boundary, and negative regions. However, when deriving three-way decision rules from these regions, existing studies typically rely on qualitative relationships between a test object and the granules within a region, resulting in a binary yes/no mechanism. Although this strict condition ensures high precision, its inflexibility tends to favor the boundary region under uncertainty, which may lead to low coverage or recall. To address this issue, this paper quantifies the relationship between objects and granules and accordingly constructs ensemble three-way decision rules. Specifically, we introduce a quantitative satisfaction degree and an object-granule similarity measure to characterize these relationships. The resulting quantities are then aggregated across all granules within each region to generate ensemble three-way decision rules, enabling the identification of more reasonable positive instances.

Optimizing the feature selection process by combining labeled data with unlabeled data is a core topic of semi-supervised feature selection (SFS). However, a handful of labeled data and an abundance of unlabeled data are bound to affect consistency among data, knowledge, and models. The central challenge of SFS still lies in robustness and uncertainty in data modeling and knowledge discovery. Accordingly, SFS needs to be concerned with three problems: 1) how to represent the knowledge structure of semi-supervised data, 2) how to enhance the robustness of knowledge modeling, and 3) how to effectively reason and measure uncertainty in modeling processing. To accomplish these, this study proposes a consistency-enhanced multi-granularity knowledge modeling (Ce-MGK) framework for semi-supervised feature selection. Specifically, an affinity-separation-aware soft learning mechanism with consistent object selection is established, which can effectively enhance the performance of subsequent multi-granularity knowledge acquisition based on fuzzy rough approximation modeling. Furthermore, multi-granularity structures and inter-level interactions are discussed for multi-level uncertainty analysis, i.e., zentropy uncertainty measure. Unlike existing measures, zentropy fully utilizes inter-level relationships among multi-granularity knowledge structures, which can effectively exploit hidden information in semi-supervised data. Finally, two inner and outer significance measures are designed for feature evaluation, and the optimal feature subset is determined based on consistent information gain. Extensive experiments on multiple benchmark datasets demonstrate the superior robustness and effectiveness of the Ce-MGK framework compared with state-of-the-art methods.

Multigranularity knowledge modeling is an influential study for information processing and knowledge discovery in artificial intelligence (AI). A central research focus is the multigranularity representation and learning of knowledge structures. Among them, fuzzy rough sets (FRSs) have emerged as a representative method for characterizing uncertain knowledge. However, the existing FRS studies still exhibit two limitations: low robustness in knowledge acquisition and incomplete characterization of uncertainty. Hence, this article proposes a zentropy-enhanced multigranularity knowledge modeling framework for robust feature selection (ZeMG-FS). Specifically, we design a fast and adaptive multigranularity information granulation mechanism based on generalized granular-ball generation to effectively capture data distributions embedded in complex data. Then, the fuzzy rough approximation method is incorporated into the representation of multigranularity knowledge. Furthermore, we analyze the fundamental relationships and structures of the multigranularity knowledge model to introduce a novel multilevel zentropy. Unlike existing entropy measures, the primary consideration of the proposed zentropy is to match and enhance the performance of the proposed model. Finally, we design two feature evaluation criteria grounded in the model and apply them to feature selection. Extensive experiments demonstrate that our proposed methods achieve superior robustness and effectiveness compared with state-of-the-art approaches.

Multi-granularity computing for knowledge discovery has emerged as a remarkable paradigm in data mining and machine learning. As a representative method, granular-ball computing has attracted considerable attention due to its efficiency and adaptability in handling complex data distributions. However, most existing granularity-based approaches focus on intra-granular mutual information while neglecting the heterogeneity and overlapping phenomena across granularities. This limitation often leads to imprecise knowledge space construction and inaccurate uncertainty estimation in feature evaluation. To overcome this problem, this study proposes a novel and high-efficiency multi-granularity knowledge fusion framework for feature selection, incorporating an enhanced granular-ball generation mechanism and a newly designed granular-ball entropy (GB-E) uncertainty measure. Specifically, we first develop an enhanced granular-ball generation mechanism to construct multi-granularity knowledge space by incorporating class distribution information, thus achieving more accurate and flexible data partitioning. Subsequently, by jointly analyzing the separation and aggregation among granular balls, a novel granular-ball entropy is proposed to quantify uncertainty in the multi-granularity knowledge space. Compared with existing uncertainty measure methods, it provides a dual-perspective uncertainty characterization and effectively improves the accuracy of granularity information fusion. Furthermore, two feature significance measures based on the proposed GB-E measure are introduced for feature evaluation, and then a corresponding feature selection method is developed. Extensive experiments on multiple public datasets demonstrate the proposed method’s superior classification performance compared with several state-of-the-art approaches.

A fundamental notion of three-way decision is a triadic structure, namely, a triad of three elements (things) equipped with a structure. In this paper, I give a triadic explanation and formulation of three-way decision and examine the structures, patterns, and models of triangular thinking. A triangular representation of a triad uses the three corners (vertexes) of a triangle to represent the three elements of the triad and the three sides (edges) to represent the pairwise relations of elements. There are two important families of triangular thinking. One family focuses on the space of all points inside the triangle for discussion. The main idea is to represent and process any point in terms of the three elements or the three relations. The other family focuses on the interactions of the three elements and the three relations. According to the presence, absence, and the direction of an edge in the triangle, there are sixteen patterns of interaction for triangular thinking. With respect to the elements and the relations of elements, there are three basic modes of interaction, namely, element-element (i.e., vertex-vertex), relation-relation (i.e., edge-edge), and element-relation (i.e., vertex-edge) interactions. By combining the two notions, I discuss models of triangular thinking and review the uses of these models across various disciplines and fields. Finally, I discuss dynamic triangular thinking through the change and revision of triangles. It is evident that triangular thinking offers new insights into three-way decision and may lead to new theories, methods, and tools.

Decision problems in the real world often face challenges arising from incomplete information and the resulting decision risk. To address these issues, this paper develops a novel grey three-way decision model by leveraging grey system theory, which manages incomplete information, with three-way decision theory, which minimizes the decision risk. First, within the context of grey sets, the study defines the fundamental concepts of grey kernel, grey support, and grey boundary by considering both the characteristic function value and the degree of greyness. These concepts establish the theoretical foundation for constructing a unified framework that progresses from qualitative to quantitative modelling. Subsequently, a qualitative grey three-way decision model is proposed. This model employs the newly introduced concepts to achieve a highly reliable partition of the most definite components of the information; however, its partitioning rules remain overly rigid. To overcome this limitation, a more refined quantitative grey three-way decision model is further developed. Its core innovation lies in in troducing a grey transformation function and the concept of grey three-way approximation, which convert grey information into a clear three-valued representation. Moreover, by incorporating adjustable threshold parameters and the principle of minimum cost, the proposed model enables a more flexible and adaptive decision-making process for grey information. Theoretical analysis shows that the qualitative model is a special case of the quan titative model, ensuring internal consistency within the theoretical system. Finally, a case study on enterprise green performance evaluation in an eco-industrial park is conducted. Multi-perspective analyzes demonstrate the effectiveness, robustness, and advantages of the proposed model in complex and uncertain decision environments.

A disjoint trisection consists of pairwise disjoint positive, boundary, and negative region, which is a fundamental notion in three-way classification. The boundary region may have different semantics, depending on how a trisection is formed and interpreted under different types of uncertainty. In this paper, we investigate three representative models, namely, interval sets, rough sets, and fuzzy sets, with a focus on how boundary regions are interpreted in each model. For interval sets, the boundary region consists of objects whose membership states are unknown due to incomplete information or knowledge. For rough sets, the boundary region consists of objects whose membership states cannot be decided based on other description using a description logic language. For fuzzy sets, the boundary region consists of objects having partial membership. A clear understanding of the different semantics of the boundary region under different types of uncertainty ensures meaningful uses and avoids misuses of various theories of uncertainty. By following the same discussion, one can study other types of uncertainty.

One of the claims regarding the power of granular computing is the computational efficiency and practical applicability in solving complex problems. The majority of discussions supporting this claim are typically made based on intuitive arguments or through examples by equating the complexity and the granularity of granules and granular structures. In 2019, Matthew Yao (Knowledge-Based Systems 163 (2019) 885-897) argued that the granularity and complexity are two related but different concepts. For quantifying the complexity, he proposed a class of interaction-based measures. Unfortunately, this direction of research has not received its due attention. To further promote theoretical studies on the concept of the complexity in granular computing, in this paper, we conduct an in-depth and systematic analysis of complexity measures in a progressive partitioning model of granular computing. By taking interactions among components as the underlying notion for explaining the complexity of problem-solving using granular computing, we investigate a class of interaction-based complexity measures. As a step towards systematic research in pursuit of a deeper understanding of the fundamental notion of complexity in granular computing, we move beyond intuitive arguments and aim at a sound theoretical foundation.

Numerous facts have confirmed that rough set (RS) is a valuable tool for data analysis. However, RS theory still faces the following two challenges. Firstly, most RS models focus on a single performance indicator and lack excellent overall performance. In addition, existing RS models are suitable for data with simple labels, but cannot handle dual mixed data with complex structures. Three-way decision is a very practical methodology widely used to tackle various learning tasks. Inspired by the three-way decision, we develop a novel RS model to analyze dual mixed data. Specifically, we first split a dual mixed dataset into multiple classic mixed data subsets. Then, based on each mixed data subset, we develop an RS sub-model. Finally, we fuse all sub-models to build a novel RS model. Through comparative experiments, the developed model can not only effectively work with dual mixed data, but also has significant advantages in computational efficiency, knowledge representation ability, knowledge classification accuracy, and data dimensionality compression.

High-dimensional and weakly supervised (HiDWS) data present significant challenges for traditional machine learning and pattern recognition. Although semi-supervised feature selection has shown effectiveness in improving the quality of HiDWS data, existing methods remain sensitive and lack robustness due to the unreliability of unlabeled data learning and the uncertainty in modeling processes. Hence, this study focuses on a multi-granularity zentropy modeling (Ze-MGM) framework with model-agnostic for highly-accuracy and robust semi-supervised feature selection. Unlike existing methods, Ze-MGM does not rely on specific settings such as rough or fuzzy set assumptions and can effectively capture the granularity of information under high-dimensional and weakly supervised data scenarios. Specifically, we first introduce a strategic soft label ($S2-$S2-Label) learning method that integrates object proximity and classification certainty to reduce uncertainty between features and labels. This method also enables the selection of compatible instances, thereby mitigating the negative impact of incompatible objects on label learning. Subsequently, a multi-granularity knowledge space and zentropy uncertainty measure are constructed by analyzing the hierarchical relationships among labels, decisions, and specific classes, which enables accurate multi-granularity knowledge representation and multi-granularity uncertainty characterization in HiDWS data modeling processing. Finally, two multi-granularity significance measures based on multi-granularity uncertainty are defined for feature evaluation and selection via a semi-supervised paradigm. Extensive experiments on multiple benchmark datasets demonstrate that the proposed Ze-MGM method achieves superior generalization performance and robustness compared to state-of-the-art methods.

This study presents an ensemble decision-making approach to evaluate Blockchain and Internet of Things (BCT-IoT) integrations for enhancing supply chain (SCs) performance. While existing evaluations focus on individual functional aspects, they often overlook interdependencies and fail to provide reliable ranking and grading under uncertainty. To overcome these limitations, a novel multidimensional Ensemble Generalized Three-Way-Decision (EG3WD) approach is developed. The first dimension represents the bi-level hierarchical Half-Quadratic Programming (HQP) structure, while the second dimension involves the weighting process. Weights are determined using a reformulated Fuzzy-Weighted-Zero-Inconsistency-Based-Interrelationship-Process (FWZICbIP) technique within a Mixed-Normal-Hesitant-Fuzzy-Sets (MNHFS) environment referred to as FWZICbIP-MNHFS and are integrated with an HQP-flat based ensemble technique. This approach enables robust ranking and grading of BCT-IoT alternatives within SCs performance evaluations. Key contributions include: (1) integrating expert evaluations across the supply chain, blockchain, and IoT domains; (2) enhancing expert and criterion weighting under uncertainty; (3) introducing a dynamic fuzzy-opinion matrix to link BCT-IoT alternatives with functional aspects using a utility-based dynamic localization technique; and (4) applying a probabilistic multidimensional EG3WD approach with adaptive Bayesian thresholds for robust triadic classification. The approach was applied to 40 BCT-IoT alternatives. Results show that alternative A11 achieved the highest probability (0.9728) and was consistently classified as Positive, while A31 had the lowest (0.0460), falling in the NEG region. Compared to existing approaches, multidimensional EG3WD demonstrated superior performance in managing uncertainty, consistency, and decision reliability.

Uncertain information processing in the big data era faces dual challenges of data fuzziness and knowledge uncertainty. Classical rough set theory provides mathematical rigor through equivalence relations, though these relations cannot adequately represent fuzzy boundaries. Fuzzy set theory describes graded membership but lacks systematic granularity analysis mechanisms. The combination into fuzzy rough sets faces limitations from conventional covering models. The introduction of fuzzy /3-covering rough sets (/3-FCRS) in 2016 added a confidence-level parameter /3 that enables flexible granularity regulation, leading to increased development of model variants. However, existing literature lacks surveys that systematically integrate theoretical development, model classification, measurement systems, and application trends. This paper presents a systematic analysis of the decade-long development with five main contributions: (1) a hierarchical theoretical framework tracing evolution from classical covering to fuzzy /3-covering approximation spaces, examining core operators, axiomatic foundations, and reduction principles; (2) a classification of eleven model variants organized into neighborhood-driven, logical-operator-enhanced, and structurally-expanded categories, with examination of mathematical foundations, development motivations, and applicability; (3) a three-level measurement system consisting of Choquet-integral-based fuzzy measures, noise-tolerant discrimination indexes, and variable-precision distinguishability indicators; (4) analysis of applications in feature selection, multi-attribute decision-making, outlier detection, and three-way decisions, examining how practical requirements have influenced theoretical developments; and (5) bibliometric analysis of core publications identifying research communities, temporal patterns, and interdisciplinary trends, while assessing theoretical limitations, computational challenges, and interpretability issues. This survey concludes that fuzzy /3-covering rough set theory provides a parametric framework linking fuzzy sets, rough sets, and multigranular analysis. Further development of unified axiomatic frameworks, efficient algorithms, and enhanced interpretability is needed to support broader practical adoption.

By integrating granular computing with rough set theory, granular rough sets enhance the semantics and effectiveness of decision-making through granule-based representations. Existing research has not thoroughly explored the issues of inducing three-way decision rules with granular rough sets, partly due to the challenge of meaningfully describing granules. To address these gaps, this paper proposes a unified framework for threeway decision models based on granular rough sets. Additionally, we introduce a generalized formulation for granule descriptions. It extends traditional representations to include all possible descriptions within a given domain. Through the lens of the proposed framework and granular descriptions, we formulate a three-way decision model in generalized granular rough sets and further demonstrate its instantiation potential across three specific types of granular spaces: quotient spaces, neighborhood-induced granular spaces, and maximal-clique-induced granular spaces. The effectiveness of the proposed models is illustrated through examples using set-valued information tables and experiments on real-world datasets. The results show that the proposed models have good performance and practical applicability.

The existence of noise in data may decrease the performance of the classification model. An intuitive assumption suggests that samples with a higher local density have a greater probability of being correctly classified. Moreover, there are differences in the importance of features. It should further increase the influence of important features on the classification task. Motivated by these issues, we propose a sample-driven adaptive feature selection method. It aims to the impact of noise and obtain the adaptive importance of features. In this paper, the Weighted Fuzzy Rough Sets model (WFRS) is proposed by utilizing more reliable samples to address noisy data. Firstly, the approximation margin is derived by applying the WFRS model within the original feature space from the sample's perspective. It serves as a basis for deriving the feature weights. On this basis, the weighted feature space is obtained. This space compresses the features with the lower weight and stretches the features with higher weight. Then, we obtain a specific fuzzy relation to maximize the within-class samples and minimize the between-class samples in the weighted feature space. Further, an evaluation function of features is constructed to model the uncertainty of fuzzy positive and non-negative regions. A Sample-driven adaptive Feature selection algorithm based on the Weighted fuzzy rough sets model (SFW) is designed to capture the varying importance of features by considering the local density of samples. The weighted features are subsequently selected to facilitate the downstream classification tasks. Experimental results show the WFRS model's robustness, the effectiveness of SFW, and its superiority compared to alternative methods.

Guided by the principles of three-way decision theory, emphasizing thinking, problem-solving, and computing in threes, we examine several triadic structures that provide insights for eXplainable Artificial Intelligence (XAI). In this paper, we propose a framework to explain XAI at three levels. At the top level, we conceptualize XAI through the Social-Machine-Human triad, addressing three foundational concerns regarding, the social considerations of what AI should or should not do, the machine capabilities of AI systems of what machines can do, and the human-centric concerns of what human can easily understand and trust. At the middle level, we highlight the machine aspect of XAI by adopting the input-process-output (IPO) triad to describe the functioning of an AI system. At the bottom level, we introduce three specialized triads to further explain the IPO triad. The volume-veracity-validity triad refines the input component, the theory-algorithm-implementation triad clarifies the process within computation, and the symbol-meaningvalue triad describes the output. This trilevel framework aims to provide a holistic, structured foundation of explaining XAI as a field of research.

Classifying a pair of agents, based on their ratings on an issue, into one of alliance, conflict, or neutral relations is a basic issue in Pawlak conflict analysis. In this paper, we first review three models that are based on an auxiliary function, a distance function, and a pair of alliance and conflict measures, respectively. We show that the differences among the three models lie in their use of different partial orderings of the nine pairs of ratings given by two agents. Based on such an observation, in this paper, we give a decision-theoretic analysis by introducing a loss function. The loss function specifies the costs of assigning a pair of ratings to alliance, conflict, and neutrality. For any pair of ratings, we take an action with the minimum cost. The decision-theoretic analysis enables us to unify existing models. In other words, by using different loss functions, we can derive the existing models.

Over the past two decades, Web Intelligence (WI) has emerged as a key field driving the evolution of AI in the connected world, addressing the demands of a future intelligent society. This paper provides a comprehensive review of WI’s contributions since its inception in 2000, spanning three distinct phases: Wisdom World Wide Web (WI 1.0, 2000–2009), Wisdom Web of Things (WI 2.0, 2010–2017), and Wisdom Web of Everything (WI 3.0, since 2018). For each phase, we examine key advancements, challenges, and future directions from the perspectives of both intelligent machines and human experts, highlighting significant societal impacts. To advance WI research, we propose a large language model-based learning framework for topic analysis and trend prediction. Moving beyond single-perspective approaches, we emphasize the Connected Intelligence Ecosystem defined by the HIGH5 scheme comprising one goal, two twins, three fundamentals, four functions, and five services that are realized through WI 3.0. This vision serves as a bridge from localized models to a global reference framework for addressing sustainability challenges in future societies. To illustrate the real-world implications of WI 3.0, we present case studies focusing on brain-inspired research, particularly in the intersection of brain intelligence, brain health, and brainternet-fostering interdisciplinary collaboration across diverse research communities.

We propose a trilevel framework for studying rough sets and granular rough sets by applying the principles of three-way decision as thinking in threes. The framework builds and interprets any model of rough sets at three levels: the binary relations level concerning the relationships between objects, the granular space level concerning granules of objects, namely, sets of objects called granular objects, and the approximation level concerning the approximations of sets of objects by granular objects. We identify and characterize eight classes of rough set models, including Pawlak, covering-based, and granular rough sets. By reviewing the existing studies within the framework, we find that there is a lack of investigations on three classes. To fill in these gaps, we investigate two types of granular spaces induced by any binary relations: neighborhood-induced granular spaces and maximal-clique-induced granular spaces. We examine the properties of the two types of granular space and the properties of rough set approximations in the corresponding two classes of models. We also consider a third class of models of granular rough sets based on granular spaces without referencing a binary relation.

The three-way approximation of fuzzy sets represents membership values using a three-valued set $\lbrace \mathbf{1}, \mathbf{m}, \mathbf{0}\rbrace$, where 1 indicates total belongingness, 0 total nonbelongingness, and m an intermediate state. This approach elevates values of membership function above a threshold $\alpha$ to 1, reduces those below $\beta$ to 0, and assigns the remaining ones to an intermediate value m. A key challenge lies in determining the thresholds $\alpha$ and $\beta$ and selecting the value of m, as existing models often lack analytical solutions and fail to fully explore the relationship between m and membership structures. This study introduces an asymmetric three-way approximation model for fuzzy sets, removing the constraint $\alpha + \beta = 1$. Analytical formulas are derived for the thresholds $ \alpha $ and $ \beta $ by minimizing information loss, and the relationship between m and membership structures is thoroughly examined. An adaptive optimizer is proposed to learn the approximate optimal value of m by minimizing the information loss. The experimental results show that information loss decreases initially before increasing as m grows. Besides, our model achieves the best classification across most datasets.