Positive region plays a fundamental role in rough set-based attribute reduction. We study positive regions of decision systems and of binary relations in rough set theory within the framework of reverse mathematics and computability theory. First, we propose the notion of infinite decision systems and prove that the existence of positive regions of decision systems is equivalent to arithmetic comprehension over the weak base theory RCA0. We also show that the complexity of positive regions of computable decision systems lies exactly in π02 of the arithmetic hierarchy. Next, we study positive regions of equivalence relations and binary relations. We show that the existence of each of the two positive regions is equivalent to arithmetic comprehension over RCA0; however, the exact complexity of positive regions of computable equivalence relations lies in π01 and the exact complexity of positive regions of computable binary relations lies in ∑02 of the arithmetic hierarchy.

Maximal consistent blocks-based optimistic and pessimistic probabilistic rough fuzzy sets and their applications in three-way multiple attribute decision-making

Optimizations of approximation operators in covering rough set theory

Fusing fuzzy rough sets and mean shift for anomaly detection

An OWA-based multi-granulation fuzzy rough set model using Choquet integrals and its applications

A two-stage feature selection approach with fuzzy covering-based rough sets based on discernibility matrix

Parameterized fuzzy β-covering relations-based fuzzy rough set models and their applications to three-way decision and attribute reduction

A novel mixed-attribute data anomaly detection method based on granular-ball multi-kernel fuzzy rough sets

Rapid attribute and scale selection with adaptive three-way sampling and neighborhood rough set

A multi-granularity decision tree algorithm based on variable precision rough sets and Zentropy

Combined variable precision fuzzy rough set and its application in medical diagnosis

The neighborhood rough set based on division-mining-fusion strategy

Weighted multi-granularity fuzzy probabilistic rough set based on semi-overlapping function and its application in three-way decision

Acquisition of representative data sets by filtering out redundant objects and attributes with fuzzy preference-based rough sets and dominance principles

Dynamic decision-making paradigm for multi-modal information in a human–computer interaction perspective: Fusing composite rough set and incremental learning

Various set-theoretic frameworks have been widely recognized for their effectiveness in handling uncertainty, including Fuzzy Sets, Neutrosophic Sets, Plithogenic Sets, Rough Sets, and Soft Sets. These foundational models have been further extended through the use of hyperstructures—based on the powerset construction—and superhyperstructures—based on the n-th-order powerset, obtained by iteratively applying the powerset operation [1, 2]. These extended constructs are collectively referred to as HyperUncertain Sets and SuperHyperUncertain Sets. Research on SuperHyperUncertain Sets is still in its early stages, and investigations into their properties, extended forms, and potential applications are expected to become increasingly sig-nificant in the future. In this paper, we propose two new, more general frameworks: the (m, n)-SuperHyperUncertain Set and the (h, k)-ary (m, n)-SuperHyperUncertain Set. These new structures represent a concrete and refined reconsideration of the foundational concepts introduced in [1, 2]. It is anticipated that the concepts developed in this work can be effectively applied tothe modeling of more hierarchical forms of uncertainty, as well as to scenarios requiring complex membership functions. Since this paper conducts only theoretical analysis, we also hope that quantitative analysis using computational methods will be carried out in the future.

In order to solve problems and provide practical solutions, researchers attempt to accurately describe societal difficulties and obstacles. An efficient method for handling complicated real-world data is rough set theory. The method finds confirmed and likely data from subsets using rough approximation operators. In order to increase accuracy while following Pawlak’s conventional approximation axioms, earlier research has created rough approximation models based on neighborhood systems. Based on cardinality rough neighborhoods and grills, we present new rough set notions in this study. These models are a suitable approach for a number of scenarios, including computational analysis, comparisons on medical datasets, real-world data analysis challenges, and classification accuracy metrics.We thoroughly examine the fundamental components of these ideas and clarify how they relate to one another as well as to earlier paradigms. Next, we describe boundary regions and assess the accuracy of the data using a topological technique. Additionally, we look at how well our models handle heart failure disease in certain individuals and come to the conclusion that the suggested rough set concepts improve upon the characteristics of the earlier approach spaces.Finally, we identify the shortcomings of the current concepts and show their advantages in terms of extending the verified information gleaned from subsets of data while preserving the key elements of Pawlak’s original paradigms that were destroyed by the models that went before them.

Graph theory offers a powerful language for encoding relationships via vertices and edges [1]. Hypergraphs extend this by introducing hyperedges that may join any number of vertices at once [2], and superhypergraphs go further still by iteratively applying the powerset construction to capture multi-level, hierarchical connections [3]. Meanwhile, a variety of frameworks for managing uncertainty—fuzzy sets [4], soft sets [5], intuitionistic fuzzy sets [6], rough sets[7], neutrosophic sets[8], and plithogenic sets[9]—have been proposed to handle imprecision and indeterminacy. In directed settings, one encounters directed graphs[10], directed hypergraphs[11], and directed superhypergraphs[12], as well as soft directed graphs[13]. In this work, we unify these ideas by introducing the Directed Soft SuperHyperGraph, marrying the notions of directionality, hierarchical hyperstructure, and soft-set parameterization. We anticipate that this new model will be especially well suited to representing complex, multi-layered, directed networks—such as those arising in urban infrastructure, transportation planning, and information flow—where both hierarchy and uncertainty must be managed simultaneously.

A variety of grill-based topologies are developed and contrasted with earlier topologies. The results demonstrate that the present ones exceed their predecessors. This study distinguishes itself by highlighting the advantages of certain topologies and identifying both the minimum and maximum values. These structural topologies are later utilized to conduct a more thorough investigation of extended rough sets. Compared to earlier models, the suggested approximate models reduce vagueness and uncertainty, which makes them especially important when applied to rough sets (rhss). Furthermore, the suggested models differ from their predecessors in that they exhibit all of Pawlak’s properties, including the capability of contrasting various approximations (Aprs), and have the quality of monotonicity across all relations. In addition, the importance of new discoveries was highlighted by demonstrating their use for human health. Besides examining its limitations, the benefits of the chosen technique were assessed. The paper ends with a summary of the main ideas of the proposed methodology and recommendations for future research paths.

Cavitation phenomenon in piston pumps not only causes vibration and noise but also leads to component damage. Conventional diagnostic methods suffer from low accuracy, while deep learning approaches lack interpretability. To address these limitations, this paper proposes an intelligent fault diagnosis method based on the rough set Attribute Weighted Convolutional Neural Network (RSAW-CNN). First, based on cavitation mechanisms and the mathematical model, the computational fluid dynamics model of the piston pump is established to simulate the failure condition. Subsequently, employing rough set theory, an original fault decision table is constructed, discretized, and subjected to attribute reduction. A weight matrix is generated according to the importance of each data channel in the classification decision and embedded into the input layer of the Convolutional Neural Network (CNN) to enhance the influence of key features. Decision rules are also extracted to provide interpretable decision support for fault diagnosis. Experimental results demonstrate that the proposed RSAW-CNN method achieves an average diagnostic accuracy of over 99.2%. Compared to the backpropagation neural network, residual neural network, CNN, and the CNN with squeeze-and-excitation networks, its average accuracy has improved by 15.87%, 10.83%, 7.48%, and 5.40%. The proposed method not only exhibits high diagnostic accuracy but also offers strong interpretability and reliability.