Rough set theory models uncertainty by approximating target concepts through lower and upper sets induced by indiscernibility, or more generally, by granulation relations in data tables. This perspective captures vagueness caused by limited observational resolution and supports set-theoretic reasoning about what can be determined with certainty and what remains only possible. This book is written as a map of models. Rather than developing a single algorithmic pipeline in depth, it provides a systematic survey of the main rough set paradigms and their extension routes. More specifically, representative variants are organized according to (i) the underlying granulation mechanism, such as equivalence-based, tolerance-based, covering-based, neighborhood-based, and probabilistic approximations, and (ii) the uncertainty semantics attached to data and relations, such as crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic settings. The book also explains how each choice changes the form of approximations and the interpretation of boundary regions. Throughout the book, small illustrative examples are used to clarify modeling intent and typical use cases in classification and decision support. Finally, an important clarification of scope should be noted. Since the main purpose of this book is to provide a map of models, the Abstract and Introduction should not lead readers to expect that feature reduction and rule induction are primary objectives. Although these topics are central in the rough set literature, they are treated here mainly as motivating applications and as entry points to the broader research landscape. The principal aim of the book is to survey and position rough set models and their extensions in a systematic and coherent manner.

Classical personality models (e.g., the Big Five, MBTI) reduce the richness of human tendencies to fixed, often binary, dimensions. Neutropsychic personality theory reconceptualises the self as a neutrosophic dynamic open system in which every trait is simultaneously present, absent, and indeterminate. By importing neutrosophic logic, where propositions possess independent degrees of truth (T), falsity (F) and indeterminacy (I), the framework yields a triadic trait model, a three layer memory architecture (conscious / aconscious / unconscious), and a dynamic evolution/neutrality/involution description of trait change.

Seismic hazard analysis plays a vital role in evaluating the potential earthquake risk in a given region. Northeast India is one of the most seismically active zones due to its tectonic positioning at the collision boundary of the Indian and Eurasian plates. This study aims to implement a comprehensive Seismic Hazard Assessment (SHA) framework using Fuzzy Multi-Criteria Decision Making (MCDM) techniques to improve the accuracy and reliability of Peak Ground Acceleration (PGA) estimates in Northeast India. The methodology integrates Trapezoidal Fuzzy Full Consistency Method (TrF-FUCOM) and Neutrosophic-TOPSIS under Single Valued Neutrosophic Set (SVNS) environment (Neutrosophic-TOPSIS), effectively addressing the limitations of traditional seismic hazard assessment methods, particularly in selecting and weighting Ground Motion Prediction Equations (GMPEs). An extensive earthquake catalogue covering the period from 1762 to 2024 has been analysed, and after declustering, fault zones have been delineated based on earthquake density along active faults. The analysis provides a detailed spatial distribution of Peak Ground Acceleration (PGA) across the region, with the highest PGA value reaching 1.43g using the Deterministic Seismic Hazard Assessment (DSHA) method. The findings of this study offer crucial insights for disaster preparedness, urban planning, and the design of earthquake-resistant infrastructure, helping to mitigate seismic risks and enhance the resilience of communities in Northeast India.

Real-world difficulties are defined as discrete or continuous, homogeneous or non-homogeneous, linear or nonlinear, etc. of model systems. The uncertainty results in imprecise system parameters. The values of the parameters may not always be considered real numbers. The real-world system must be emulated for stochastic, interval, fuzzy, intuitionistic, and other imprecise conditions to overcome these issues. Fuzzy sets are extended to intuitionistic fuzzy sets, while it is commonly known that fuzzy differential approaches have been employed in most texts. Intuitionistic fuzzy sets theory is applied in this study to develop a more realistic epidemic model. The population is divided into four categories based on this concept: susceptible (S), exposed (E), infected (I) and recovered (R). By treating each of the coefficients in the suggested model as a distinct type of triangular intuitionistic fuzzy number, we gave all of the epidemiological parameters intuitionistic fuzziness. The intuitionistic SEIR model has been examined using different weight assignments using the Utility Function Method (UFM) technique. The study explores non-negativity, boundedness, and physiologically realistic equilibrium points - all qualitative features of the SEIR model. The stability of the suggested model system has been examined using the intuitionistic fuzzy sets notion. Using MATLAB, all of the outputs and conclusions of the suggested model have been validated both visually and quantitatively.

Dempster-Shafer Theory (DST) and Dezert-Smarandache Theory (DSmT) are prominent frameworks within evidence theory for managing uncertainty and fusing information. While both utilize basic belief assignments (BBAs) on sets of hypotheses, they diverge critically in their foundational mathematical assumptions and conflict handling. This article provides a comprehensive comparison, detailing the philosophical and practical shift from DST to its generalized superset, DSmT. DST is built upon the power set 2Θ of a mutually exclusive frame of discernment, forcing conflict to be resolved via global normalization (Dempster's Rule), which can lead to counter-intuitive results in high-conflict scenarios (e.g., Zadeh's paradox). DSmT, conversely, operates on the hyper-power set DΘ of an exhaustive but non-exclusive frame, allowing for the representation of vague, paradoxical, or overlapping concepts (e.g., A∩B). Crucially, DSmT employs the Proportional Conflict Redistribution (PCR) family of rules, which resolves disagreement by locally redistributing conflicting mass back only to the propositions that generated it. This mechanism ensures stability and interpretability, even under extreme conflict. We analyze the trade-offs in expressiveness and computational cost, illustrating that the choice between the two theories hinges on two core factors: the expected level of conflict and the degree of conceptual overlap in the problem domain. Ultimately, DSmT is demonstrated to fully encompass DST, functioning as a robust, flexible alternative for modern multi-sensor fusion, intelligence analysis, and decision-making systems characterized by ambiguity and high disagreement.

Neutrosophication converts a crisp value into a triple (T, I, F) of truth, indeterminacy and falsity degrees. A common way of arguing that a method is "truly neutrosophic" is to observe that T + I + F ≠ 1 or that F ≠ 1-T. Neither observation shows that the third coordinate carries information not already contained in the other two. We audit five neutrosophication transformations (a proposed K-Means + sigmoid method, and the parabolic, threshold distance, kernel-density and triangular-fuzzy methods) on six clinical attributes of the Cleveland heart-disease data (n = 299), separating normalization, complementarity, data adaptivity, empirical non-redundancy, stability and downstream utility as distinct, separately tested properties. An equation-level audit shows that all five transformations are functions of a single scalar and that four of them derive I analytically from T. Cross-validated regressions of I on (T, F) give out-of-sample R 2 of 1.00, 0.99, 1.00 and 0.98 for the four complementary methods and 1.00 for K-Means at full numerical precision, falling to 0.48 when T and F are recorded to two decimals: the K-Means coordinate is non-redundant only through the numerically negligible tails of saturated sigmoids (T, F < 0.01 for 63 % of observations). A nearest-neighbour test confirms the pattern. In a leakage-safe, same-classifier ablation, adding I to (T, F) changed the ROC-AUC of a logistic regression by-0.009 (95 %-0.035 to +0.017) for K-Means and by at most 0.008 for the other methods, and no neutrosophic representation exceeded the raw six attributes by more than 0.006 AUC with any of three classifiers. The K-Means I correlated weakly with external uncertainty signals (AUC 0.56 against misclassification), and two ambiguity-based alternatives were not better. K-Means results were stable across seeds with k-means++ but not with the original random initialisation, and sensitive to the sigmoid slope, to the choice of K and to scaling. The main lesson is methodological: not every three-component representation contains three independent dimensions of information, and neutrosophication methods should not be evaluated by T + I + F alone. Because a univariate transformation cannot add information about its input, the constructive consequence is that future indeterminacy coordinates must be grounded in separately auditable evidence (reliability, additional sources, resampling, time, modalities, agents); the paper closes with a prior-art-audited taxonomy of such evidence-grounded designs, ten design principles and a validation checklist.

Virtual reality (VR) is increasingly employed to investigate human-environment interactions; however, current evaluation methods remain fragmented. Physical–functional metrics, biofeedback signals, and self-report data are frequently analyzed in isolation, leaving the symbolic–narrative dimension of architecture largely implicit. This Hypothesis and Theory article introduces the Neutrosophic Technarrative Architecture (NTA): a robust methodological framework and computational pipeline for the symbolic–emotional evaluation of virtual architectural spaces. The proposed framework integrates four core components: (1) the Neutrosophic Technarrative Model (NTM), which encodes architectural spaces through symbolic nodes such as justice, identity, hope, resistance, and eco-symbiosis; (2) immersive VR scenarios acting as controlled symbolic environments; (3) psychophysiological measures (EEG, GSR, HRV) as indicators of affective engagement; and (4) an AI-ready computational model that links symbolic design variables, biofeedback features, and user evaluations via neutrosophic inference. Rather than introducing new sensing technologies, the novelty of NTA lies in the symbolic–computational integration of existing VR and affective computing methods. We define the conceptual structure of NTA, formulate testable hypotheses, and provide a reproducible methodological blueprint—supported by open-source Python implementations—for multimodal analysis. We demonstrate that NTA (a) offers a systematic pipeline for connecting symbolic design intentions with measurable VR experiences, (b) employs neutrosophic representation to quantify ambiguity and indeterminacy in user responses, and (c) enables predictive modelling that transcends traditional self-report-only approaches. This work establishes a foundational research agenda at the intersection of VR user experience, neuroarchitecture, and ethically oriented design.

In recent years, there has been a growing interest in neutrosophic probability distributions as effective tools for modeling data that involve uncertainty, ambiguity, or vagueness-limitations that classical probability models often fail to address. In addition, the simulation of interval data has been misapplied in neutrosophic analysis by assuming a uniform distribution over the interval. In this study, a neutrosophic extension of the Weibull distribution is used to generate neutrosophic data. From this data, the indeterminacy component, referred to as "indeterminacy factor," is extracted and estimated. To understand the behavior of this indeterminacy factor, several continuous probability distributions are fitted to its values. This paper makes three main contributions: (1) it presents a novel Neutrosophic Weibull distribution that can capture non-uniform indeterminacy patterns; (2) it offers a comparative analysis of several candidate distributions to assess the probabilistic framework of indeterminacy; and (3) it supports the suggested model using simulations and real-life data sets, proving its outstanding goodness-of-fit and practical importance. These findings emphasize the need for more suitable probabilistic models when dealing with neutrosophic data. Finally, the proposed neutrosophic Weibull distribution is applied to two real-world datasets containing uncertain observations. In both cases, the Weibull model shows the best fit. The corresponding indeterminacy values are then modeled using different probability distributions, and the results reflect similar patterns to those observed in the simulated neutrosophic data. Based on the analysis, it is concluded that the existing simulation-originally developed for interval analysis under a uniform distribution assumption-is not suitable for neutrosophic analysis.

The fundamental focus of operations research is the existence of a problem requiring decision-making. The need for operations research methods increases as the complexity of the problem increases. One important method in operations research is linear programming, which relies on translating the actual situation under study into a linear mathematical model consisting of an objective function and constraints. This method uses data collected from the situation by experts. As we know, this data is suitable for operating conditions similar to those in which it was collected. In other words, this data is subject to change depending on the surrounding conditions. In light of this uncertainty, it was necessary to devise scientific methods suitable for all circumstances. In classical studies, researchers in the field of operations research introduced sensitivity analysis and parametric programming. This method is an expansion of sensitivity analysis because parametric programming studies the effect of simultaneous changes in the data when the coefficients change as a function of a single parameter. It also examines the effect of continuous changes in the coefficients of the objective function and the right-hand side of the constraints on the optimal solution. It provides us with a set of optimal acceptable solutions to the problem under study. In this research, we present a new approach to the product mix problem that aims to reformulate the mathematical model of this problem using the parametric programming method, which is an extension of sensitivity analysis, where the effect of simultaneous changes in the data is studied when the coefficients change as a function of a single parameter, Double treatment in product mixture problem data hyperparametric function and superhyperparametric function and the concept of a Hyperfunction, which associates each of the acceptable values provided by the model study using parametric programming to a subset of outputs. This generalizes classical parametric programming to represent multi-valued results. We will also reformulate it using parametric programming and SuperHyperFunction, through which sets (or groups of sets) are associated with values of higher-order power sets, which enables us to capture complex hierarchical or layered uncertainties. This enables us to obtain solutions that fit all the conditions that the operating environment of the system under study may experience.

Two-Fold Algebra (TFA) was recently developed to bridge classical algebraic operations with fuzzy and fuzzy-extension (especially neutrosophic) components, allowing for the simultaneous modeling of objects and their associated uncertainty descriptors. However, as real-world systems increasingly demand the integration of multiple, independent qualification dimensions—such as risk, sustainability, and reliability, the binary nature of TFA becomes a limiting factor. This paper introduces two generalized frameworks: the Horizontal and respectively Vertical Generalization n-Fold Algebra (n-FA), and from 2-valued to m-values operations, m ≥ 2. We formally define the n-FA structure as a coupling of a classical backbone (1) with (n-1) independent or interdependent component sub-laws. We provide rigorous systematic construction, explore various specializations (including fuzzy and intuitionistic-fuzzy cases), and derive the essential algebraic properties—such as closure, associativity, and monotonicity—required for coherent multi-component operations. Finally, we demonstrate the versatility of n-FA through numerical examples in supply-chain risk and multi-criteria decision-making, establishing it as a robust mathematical language for complex, high-dimensional uncertainty modeling.

Soft set theory provides a direct framework for parameterized decision modeling by assigning to each attribute (parameter) a subset of a given universe, thereby representing uncertainty in a structured way [1, 2]. Over the past decades, the theory has expanded into numerous variants-including hypersoft sets, superhypersoft sets, TreeSoft sets, bipolar soft sets, and dynamic soft sets-and has been connected to diverse areas such as topology and matroid theory. In this book, we present a survey-style overview of soft sets and their major extensions, highlighting core definitions, representative constructions, and key directions of current development.

Classical dynamical systems describe the evolution of states through deterministic or stochastic laws. However, many real-world systems involve uncertainty, contradiction, incompleteness, and indeterminacy that cannot be adequately represented within traditional frameworks. This paper introduces Neutrosophic-Plithogenic Dynamical Systems (NPDS), extending classical dynamics through neutrosophic states and plithogenic contradiction structures. A system state is represented by the neutrosophic triple , where T, I, and F denote truth, indeterminacy, and falsehood components. Plithogenic attributes and contradiction degrees are then incorporated into the dynamics. New concepts are introduced, including neutrosophic attractors, neutrosophic repellers, neutrosophic neutral sets, neutrosophic partial attractors, neutrosophic partial repellers, neutrosophic Lyapunov exponents, and neutrosophic chaos measures. The resulting framework provides a natural mathematical representation for systems whose behavior is simultaneously attractive, neutral, and repulsive. Applications to scientific theory evolution, social systems, economics, artificial intelligence, and complex decision processes are discussed. The concepts of Partial Attractors, Partial Neutrals, Partial Repellers and respectively of Refined Attractors, Refined Neutrals, Refined Neutral Sets, were introduced by F.Smarandache in 2026, and they are richer and closer to reality than the ordinary classical versions of Attractors, Repellers.

Large Language Models (LLMs) are predominantly governed by probabilistic frameworks in which the sum of outcome probabilities is constrained to unity. This architectural limitation, often imposed by Softmax layers, leads to a collapse of uncertainty that makes it difficult to differentiate between epistemic uncertainty, paradox, and vagueness. We present an empirical investigation of the application of Neutrosophic Logic, a framework that treats Truth (T), Indeterminacy (I), and Falsity (F) as three independent dimensions, to model epistemic states in LLMs. We conducted experiments on a family of four OpenAI GPT models across five linguistic phenomena: logical paradoxes, epistemic ignorance, vagueness, ethical contradictions, and future contingencies, under three prompting strategies: neutrosophic, probabilistic, and entropy-derived. Our findings reveal that the neutrosophic approach, by allowing T+I+F > 1, a state we term hyper-truth, provides a richer representation of a model's internal state. In 35

We investigate whether large language models (LLMs) systematically discriminate in candidate evaluations based on applicant name ethnicity and/or institutional prestige and geographic location. Three factorial experiments are reported (4,320 API calls, four LLMs, five professional domains). Study 1 (3x4 design) finds a statistically robust institution-tier gradient of +0.297 points on a 10-point scale (95

Graph theory provides a fundamental framework for modeling relationships using vertices and edges. Hypergraphs extend this framework by permitting hyperedges that connect any number of vertices simultaneously, and Super-hypergraphs further generalize hypergraphs through iterated powerset constructions to represent hierarchical connections. Concurrently, a variety ofuncertainty-modeling paradigms—such as fuzzy sets, soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy offsets, hyper-intuitionistic fuzzy sets, and intuitionistic fuzzy multidirected sets—have enriched classical graph concepts. In this paper, we introduce the Intuitionistic Fuzzy Soft n-Super-hypergraph, a unified n-th order structure that seamlessly integrates intuitionistic fuzzy Super-hypergraphs with soft Superhypergraphs. We present rigorous definitions, explore key structural properties, and highlight potential applications in decision-making under uncertainty, demonstrating how this framework effectively captures both complex hierarchical organization and nuanced uncertainty in real-world networks.

The mass migration of human populations to urban areas has resulted in unprecedented challenges for city services. To address and find solutions for these emerging issues, decision-makers must embrace the smart city and Society 5.0 paradigms, which comprehensively tackle various dimensions of the problem and ensure adaptability to evolving citizen needs. Central to the success of these paradigms is technology, particularly artificial intelligence (AI). AI's transformative capabilities enable the expansion of services, automation of tasks, efficient operationalization and processing vast amounts of data to address urban challenges, aligning with several sustainable development goals (SDGs) such as sustainable cities and communities (SDG 11). Municipalities require strategic plans that empower them to adapt to the AI, Society 5.0 and smart city paradigms, involving multiple stakeholders, including businesses. This study presents a multi-criteria analysis system designed to support decision-making in this complex context, considering the subjective nature and inherent complexity of the decision problem. The system development involved input from key decision-makers with relevant expertise, utilizing methodologies such as cognitive mapping and the decision-making trial and evaluation laboratory technique applied in a neutrosophic environment to analyze cause-and-effect relationships between factors affecting adaptation initiatives. Based on a constructivist, process-oriented approach, the developed analysis system can assist decision-makers in navigating uncertainty during evaluations of technology integration. This holistic and comprehensive system promotes informed decision-making within the AI, Society 5.0 and smart city contexts, contributing to the achievement of relevant SDGs.

Hypergraphs generalize this framework by allowing hyperedges that connect more than two vertices [1]. Superhypergraphs further enrich the model through iterated powerset constructions, capturing hierarchical and self-referential structures among hyperedges [2]. An (m, n)-SuperHyperGraph is a mathematical structure in which each vertex corresponds to an (m, n n)-superhyperfunction defined on a base set, while the hyperedges group such functions together to represent higher-order relationships and contextual connections. Systematic research on SuperHyperGraphs is still relatively limited compared with the extensive literature on graphs and hypergraphs. To help bridge this gap, this book presents a survey of fundamental and advanced concepts related to SuperHy-perGraphs. Our aim is twofold: (i) to increase the visibility and accessibility of SuperHyperGraph theory and thereby stimulate further research, and (ii) to deepen the mathematical understanding of their structures among researchers and practitioners who work with graph- and hypergraph-based models.

Real-world phenomena often exhibit vagueness, partial truth, and incomplete information. To model such uncertainty in a mathematically rigorous way, many generalized set-theoretic frameworks have been introduced, including Fuzzy Sets [1], Intuitionistic Fuzzy Sets [2], Neutrosophic Sets [3,4], Vague Sets [5], Hesitant Fuzzy Sets [6], Picture Fuzzy Sets [7], Quadripartitioned Neutrosophic Sets [8], Penta-Partitioned Neutrosophic Sets [9], Plithogenic Sets [10], HyperFuzzy Sets [11], and HyperNeutrosophic Sets [12]. Within these frameworks, a wide range of notions has been proposed and studied, particularly in the settings of fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic set theories. This extensive literature underscores both the significance of these theories and the breadth of their application areas. As a result, many ideas, constructions, and structural patterns recur across these four major families of uncertainty-oriented models. In this book, we provide a comprehensive, large-scale survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets. Our goal is to give readers a systematic overview of existing developments and, through a unified exposition, to stimulate new insights, further conceptual extensions, and additional applications across a wide range of disciplines.

This paper proposes an extension of the Neutrosophic Theory of Evolution into the field of evolutionary biogeography. Using the freshwater crab Potamon fluviatile as a case study, we analyze how evolutionary, involutionary, and indeterminate processes interact across geological, ecological, and spatial scales. The species originated from marine ancestors in the ancient Tethys Ocean and later colonized freshwater ecosystems across the Mediterranean basin. A long-isolated population inhabiting subterranean canals beneath the ruins of Rome exhibits unusual body size and longevity, suggesting a form of island-like gigantism. We argue that such evolutionary phenomena can be interpreted within a neutrosophic framework where adaptation involves simultaneous processes of evolution, involution, and indeterminacy. This study represents the first formal applications of neutrosophic theory to evolutionary biogeography.