This paper develops the foundations of Quantum Granular Computing (QGC), extending classical granular computing including fuzzy, rough, and shadowed granules to the quantum regime. Quantum granules are modeled as effects on a finite dimensional Hilbert space, so granular memberships are given by Born probabilities. This operator theoretic viewpoint provides a common language for sharp (projective) and soft (nonprojective) granules and embeds granulation directly into the standard formalism of quantum information theory. We establish foundational results for effect based quantum granules, including normalization and monotonicity properties, the emergence of Boolean islands from commuting families, granular refinement under Luders updates, and the evolution of granules under quantum channels via the adjoint channel in the Heisenberg picture. We connect QGC with quantum detection and estimation theory by interpreting the effect operators realizing Helstrom minimum error measurement for binary state discrimination as Helstrom type decision granules, i.e., soft quantum counterparts of Bayes optimal decision regions. Building on these results, we introduce Quantum Granular Decision Systems (QGDS) with three reference architectures that specify how quantum granules can be defined, learned, and integrated with classical components while remaining compatible with near term quantum hardware. Case studies on qubit granulation, two qubit parity effects, and Helstrom style soft decisions illustrate how QGC reproduces fuzzy like graded memberships and smooth decision boundaries while exploiting noncommutativity, contextuality, and entanglement. The framework thus provides a unified and mathematically grounded basis for operator valued granules in quantum information processing, granular reasoning, and intelligent systems.

Many modern systems, such as financial, transportation, and telecommunications systems, are time-sensitive in the sense that they demand low-latency predictions for real-time decision-making. Such systems often have to contend with continuous unbounded data streams as well as concept drift, which are challenging requirements that traditional regression techniques are unable to cater to. There exists a need to create novel data stream regression methods that can handle these scenarios. We present a database-inspired datastream regression model that (a) uses inspiration from R*-trees to create granules from incoming datastreams such that relevant information is retained, (b) iteratively forgets granules whose information is deemed to be outdated, thus maintaining a list of only recent, relevant granules, and (c) uses the recent data and granules to provide low-latency predictions. The R*-tree-inspired approach also makes the algorithm amenable to integration with database systems. Our experiments demonstrate that the ability of this method to discard data produces a significant order-of-magnitude improvement in latency and training time when evaluated against the most accurate state-of-the-art algorithms, while the R*-tree-inspired granulation technique provides competitively accurate predictions

Stellar light curves contain valuable information about oscillations and granulation, offering insights into stars' internal structures and evolutionary states. Traditional asteroseismic techniques, primarily focused on power spectral analysis, often overlook the crucial phase information in these light curves. Addressing this gap, recent machine learning applications, particularly those using Convolutional Neural Networks (CNNs), have made strides in inferring stellar properties from light curves. However, CNNs are limited by their localized feature extraction capabilities. In response, we introduce $\textit{Astroconformer}$, a Transformer-based deep learning framework, specifically designed to capture long-range dependencies in stellar light curves. Our empirical analysis centers on estimating surface gravity ($\log g$), using a dataset derived from single-quarter Kepler light curves with $\log g$ values ranging from 0.2 to 4.4. $\textit{Astroconformer}$ demonstrates superior performance, achieving a root-mean-square-error (RMSE) of 0.017 dex at $\log g\approx3$ in data-rich regimes and up to 0.1 dex in sparser areas. This performance surpasses both K-nearest neighbor models and advanced CNNs. Ablation studies highlight the influence of receptive field size on model effectiveness, with larger fields correlating to improved results. $\textit{Astroconformer}$ also excels in extracting $\nu_{\max}$ with high precision. It achieves less than 2% relative median absolute error for 90-day red giant light curves. Notably, the error remains under 3% for 30-day light curves, whose oscillations are undetectable by a conventional pipeline in 30% cases. Furthermore, the attention mechanisms in $\textit{Astroconformer}$ align closely with the characteristics of stellar oscillations and granulation observed in light curves.

In this research, new concepts of existential granules that determine themselves are invented, and are characterized from algebraic, topological, and mereological perspectives. Existential granules are those that determine themselves initially, and interact with their environment subsequently. Examples of the concept, such as those of granular balls, though inadequately defined, algorithmically established, and insufficiently theorized in earlier works by others, are already used in applications of rough sets and soft computing. It is shown that they fit into multiple theoretical frameworks (axiomatic, adaptive, and others) of granular computing. The characterization is intended for algorithm development, application to classification problems and possible mathematical foundations of generalizations of the approach. Additionally, many open problems are posed and directions provided.

Theories of rough mereology have originated from diverse semantic considerations from contexts relating to study of databases, to human reasoning. These ideas of origin, especially in the latter context, are intensely complex. In this research, concepts of rough contact relations are introduced and rough mereologies are situated in relation to general spatial mereology by the present author. These considerations are restricted to her rough mereologies that seek to avoid contamination.

The concepts of rough and definite objects are relatively more determinate than those of granules and granulation in general rough set theory (RST) [1]. Representation of rough objects can however depend on the dialectical relation between granulation and definiteness. In this research, we make this exact in the context of RST over proto-transitive approximation spaces. This approach can be directly extended to many other types of RST. These are used for formulating an extended concept of knowledge interpretation (KI)(relative the situation for classical RST) and the problem of knowledge representation (KR) is solved. These will be of direct interest in granular KR in RST as developed by the present author [2] and of rough objects in general. In [3], these have already been used for five different semantics by the present author. This is an extended version of [4] with key examples and more results.

To capture the uncertainty of information or knowledge in information systems, various information granulations, also known as knowledge granulations, have been proposed. Recently, several axiomatic definitions of information granulation have been introduced. In this paper, we try to improve these axiomatic definitions and give a universal construction of information granulation by relating information granulations with a class of functions of multiple variables. We show that the improved axiomatic definition has some concrete information granulations in the literature as instances.

This study describes application of some approximate reasoning methods to analysis of hydrocyclone performance. In this manner, using a combining of Self Organizing Map (SOM), Neuro-Fuzzy Inference System (NFIS)-SONFIS- and Rough Set Theory (RST)-SORST-crisp and fuzzy granules are obtained. Balancing of crisp granules and non-crisp granules can be implemented in close-open iteration. Using different criteria and based on granulation level balance point (interval) or a pseudo-balance point is estimated. Validation of the proposed methods, on the data set of the hydrocyclone is rendered.

In this study, we reproduce two new hybrid intelligent systems, involve three prominent intelligent computing and approximate reasoning methods: Self Organizing feature Map (SOM), Neruo-Fuzzy Inference System and Rough Set Theory (RST),called SONFIS and SORST. We show how our algorithms can be construed as a linkage of government-society interactions, where government catches various states of behaviors: "solid (absolute) or flexible". So, transition of society, by changing of connectivity parameters (noise) from order to disorder is inferred.

ABSTRACT: This paper describes application of information granulation theory, on the design of rock engineering flowcharts. Firstly, an overall flowchart, based on information granulation theory has been highlighted. Information granulation theory, in crisp (non-fuzzy) or fuzzy format, can take into account engineering experiences (especially in fuzzy shape-incomplete information or superfluous), or engineering judgments, in each step of designing procedure, while the suitable instruments modeling are employed. In this manner and to extension of soft modeling instruments, using three combinations of Self Organizing Map (SOM), Neuro-Fuzzy Inference System (NFIS), and Rough Set Theory (RST) crisp and fuzzy granules, from monitored data sets are obtained. The main underlined core of our algorithms are balancing of crisp(rough or non-fuzzy) granules and sub fuzzy granules, within non fuzzy information (initial granulation) upon the "open-close iterations". Using different criteria on balancing best granules (information pockets), are obtained. Validations of our proposed methods, on the data set of in-situ permeability in rock masses in Shivashan dam, Iran have been highlighted.