Interval time-varying delay is common in control process, e.g., automatic robot control system, and its stability analysis is of great significance to ensure the reliable control of industrial processes. In order to improve the conservation of the existing robust stability analysis method, this paper considers a class of linear systems with norm-bounded uncertainty and interval time-varying delay as the research object. Less conservative robust stability criterion is put forward based on augmented Lyapunov-Krasovskii functional method and reciprocally convex combination. Firstly, (...) the delay interval is partitioned into multiple equidistant subintervals, and a new Lyapunov-Krasovskii functional comprising quadruple-integral term is introduced for each subinterval. Secondly, a novel delay-dependent stability criterion in terms of linear matrix inequalities is given by less conservative Wirtinger-based integral inequality approach. Three numerical comparative examples are given to verify the superiority of the proposed approach in reducing the conservation of conclusion. For the first example about closed-loop control systems with interval time-varying delays, the proposed robust stability criterion could get MADB about 0.3 more than the best results in the previous literature; and, for two other uncertain systems with interval time-varying delays, the MADB results obtained by the proposed method are better than those in the previous literature by about 0.045 and 0.054, respectively. All the example results obtained in this paper clearly show that our approach is better than other existing methods. (shrink)

The paper describes and proves completeness theorems for a series of proof systems formalizing common sense reasoning about uncertain knowledge in the case where this consists of sets of linear constraints on a probability function.

The stabilization problem of a macroeconomic dynamical system is considered in this paper. The main features of this system are that the system uncertainties may be unknown functions of state and time but with known bounds. Furthermore, the control inputs are subject to constraints, which is a salient feature in an economic control problem. To ensure that the controls are within the specified boundaries, in our control design procedure, a creative diffeomorphism, which converts bounded controls into unbounded corresponding signals by (...) choosing an appropriate transformation function, is proposed. For the uncertain system, a deterministic robust control is designed to render the practical stability: uniform boundedness and uniform ultimate boundedness. The range of the input bounds is related to the uncertainties and can be designed according to the actual situation. Numerical simulations are performed to verify the effectiveness of the stabilization policy. (shrink)

The industrial product-service system is a kind of system engineering methodology, integration scheme, and business model to realize service value by adding intangible services in the whole life cycle. However, the design of the system involves many difficulties such as uncertain customer demands, strong subjectivity of the experience design, and long debugging times. Methods for solving upper problems are therefore essential. This paper presents a design model that integrates an improved affinity propagation clustering algorithm, quality function development, and axiomatic (...) design. The entire process of designing an iPSS can be split into three steps. First, uncertain customer demands is determined and standardized. Second, the functions of the product-service system are investigated. Finally, the structures of the system are determined. This paper examines the example of the control service of an iPSS for a water heater tank capping press. An improved AP clustering algorithm is used to determine standardized customer demands, the proposed QFD, and an AD integration model to initially establish a mapping between the customer demands domain and the function domain and clarify the design focus. Next, a QFD- and AD-integrated model is constructed to establish a mapping between the function domain and the structure domain and optimize the control scheme through the quality of its risk prediction. Finally the paper verifies that the upper process and methods can guide the design process effectively in production applications. (shrink)

The problem of determining an optimal benchmark to inefficient decision-making units is an important issue in the field of performance analysis. Previous methods for determining the projection points of inefficient DMUs have only focused on one objective and other features have been ignored. This paper attempts to determine the best projection point for each DMU when the inputs and outputs data are in stochastic form and presents an alternative definition for the best projection by considering three main aspects: technical efficient, (...) minimal cost, and maximal revenue as much as possible. Considering the important role of the electricity industry in the economic growth of each country, a practical example has been implemented on 16 regional electricity companies in Iran in 9 consecutive periods. The efficiency score along with the projection points of the three technical models ), cost, and stochastic revenue are compared with the projection point obtained from the model presented in this article, which simultaneously meets these three objectives, showing the improvement of companies’ performance. (shrink)

This paper addresses the adaptive asymptotic tracking control problem for nonlinear systems whose virtual control gains are unknown nonlinear functions of system states. Only in the first step, the Nussbaum gain technique is utilized to handle the uncertain virtual control gain. In the remaining steps, virtual control gains are dealt with by constructing novel control laws without the approximation of the uncertain nonlinear functions and external disturbances by neural networks or fuzzy logic. New adaptive laws are defined (...) to compensate for unknown virtual control gains, uncertain parameters, and external disturbances. Finally, an adaptive tracking controller is designed and applied to the control of a 3-order robot system, which guarantees the boundedness of all the signals in the closed-loop system and asymptotic stability of the tracking error. (shrink)

Earthquake is a kind of natural disaster, which will have a great impact on the building structure. In the vibration control field of building structures, the timeliness of system stability is extremely important. In traditional control methods, the timeliness is not paid enough attention for systems with uncertain seismic waves. For setting this problem, fuzzy adaptive compensation control for uncertain building structural systems by sliding-mode technology is proposed. It is combined with fuzzy adaptive control and sliding-mode (...) control to ensure that the system can be stable with satisfied timeliness. Also, saturation function is used to ensure the feasible physical implementation of the control system. Compared with the traditional LQR control, the simulation results showed that the proposed method can make the system reach a stable state with rapid convergence performance and has a feasible physical implementation. (shrink)

Standard adaptive control is the preferred approach for stabilization and synchronization of chaotic systems when the structure of such systems is a priori known but the parameters are unknown. However, in the presence of unmodeled dynamics and/or disturbance, this approach is not effective anymore due to the drift of the parameter estimations, which eventually causes the instability of the closed-loop system. In this paper, a robustifying term, which consists of a saturation function, is used to avoid this problem. (...) The robustifying term is added to the adaptive control law. Consequently, the learning law is also modified. The boundedness of the states and the parameter estimations is rigorously and thoroughly proven by means of Lyapunov like analysis based on Barbalat’s lemma. On these new conditions, the convergence to zero cannot be achieved due to the presence of unmodeled dynamics and/or disturbance. However, it is still possible to guarantee the asymptotic convergence to a bounded zone around zero. The width of this zone can be adjusted by the designer. The performance of this robust approach is verified by numerical simulations. Although, for simplicity, this strategy is only applied to the stabilization and synchronization of Zhang system, the procedure can easily be generalized to a broad class of chaotic systems. (shrink)

Due to the excellent approximation ability, the neural networks based control method is used to achieve adaptive consensus of the fractional-order uncertain nonlinear multiagent systems with external disturbance. The unknown nonlinear term and the external disturbance term in the systems are compensated by using the radial basis function neural networks method, a corresponding fractional-order adaption law is designed to approach the ideal neural network weight matrix of the unknown nonlinear terms, and a control law is designed eventually. (...) According to the designed Lyapunov candidate function and the fractional theory, the systems stability is proved, and the adaptive consensus can be guaranteed by using the designed control law. Finally, two simulations are shown to illustrate the validity of the obtained results. (shrink)

This work focuses on a multiagent system with time-varying interval uncertainty in the system matrix, where multiple agents interact through an undirected topology graph and only the bounding matrices on the uncertainty in the system matrix are known. A reduced-order interval observer, which is named the reduced-order neighborhood interval observer, is designed to estimate the relative state of each agent and those of its neighbors. It is shown that the reduced-order IO can guarantee the consensus of the uncertain multiagent (...) system. Finally, simulation examples are proposed to verify the theoretical findings. (shrink)

EI objetivo de este artículo es presentar los principios de la programación lógica borrosa y de sus principales variantes, ilustrándolas a través de un conjunto de aproximaciones que, a nuestro entender, son representativas de los avances en esta área. También incluimos la descripción de otros sistemas de programación lógica que se sustentan en lógicas de la incertidumbre diferentes de la lógica borrosa. En esta presentación presuponemos que la mayoría de los lectores no son expertos en programación lógica; para seguirla sólo (...) se requiere un conocimiento básico de la lógica de predicados.The purpose of this paper is to present the principles of the fuzzy logic programming, exemplifying them by a couple of proposals that we think are representatives of the advances in this field. We include also the description of another systems of logic programming with uncertain information that are based on other logics of uncertainty which are different from fuzzy logic. This article only presuppose anelementary knowledge of the classical first-order logic. (shrink)

This paper proposes an integral sliding mode control method of a class of uncertain chaotic systems with saturation inputs. Firstly, fuzzy logic system is used to estimate the unknown nonlinear function. Then, a disturbance observer is constructed to estimate a compound disturbance, which contains the external disturbance, the error of saturation input and control output, and the fuzzy estimation error. Subsequently, a proposed integral sliding mode controller can ensure that all signals of the closed-loop system are ultimately bounded, (...) and based on the dynamic system of the integral sliding mode variable itself, the ultimate bound of the tracking error can be estimated. Simulation results show that the proposed ISMC method is more effective than the traditional ISMC method. (shrink)

This paper experimentally investigates the altruistic behavior of physicians and whether this behavior is affected by payment system and uncertainty in health outcome. Subjects in the experiment take on the role of physicians and decide on the provision of medical care for different types of patients, who are identical in all respects other than the degree to which a given level of medical treatment affects their health. We investigate physician altruism from the perspective of ethical principles, by categorizing physicians according (...) to how well their treatment decisions align with different principles for priority setting. The experiment shows that many physicians are altruistic toward their patients but also that the degree of altruism varies across patients with different medical needs. We find a strong effect of payment system that is overall unaffected by the introduction of risk and ambiguity in patients health outcomes. There is, however, substantial heterogeneity across individuals, in particular under the capitation payment system where physicians responses to the introduction of uncertainty in patient health are modulated by their own generic risk and ambiguity preferences. (shrink)

The fault ride-through capability and fault current issues are the main challenges in doubly fed induction generator- based wind turbines. Application of the bridge-type fault current limiter was recognized as a promising solution to cope with these challenges. This paper proposes a nonlinear sliding mode controller for the BFCL to enhance the FRT performance of the DFIG-based WT. This controller has robust performance in unpredicted voltage sag level and nonlinear features. Theoretical discussions, power circuit, and nonlinear control consideration of the (...) SMC-based BFCL are conducted, and then, its performance is verified through time-domain simulations in the PSCAD/EMTDC environment. To reduce the chattering phenomenon and decrease the reaching time, it used the exponential reaching law for designed SMC. Also, the SMC-based BFCL performance is compared with the conventional and PI controller-based BFCL for both symmetrical and asymmetrical short-circuit faults. Simulation results reveal that the SMC-based BFCL provides better performance compared with the conventional and PI controller-based BFCL to enhance the FRT. (shrink)

This paper proposes a novel robust fixed-time control for the robot manipulator system with uncertainties. Based on the uniform robust exact differentiator algorithm, a robust control term is constructed. Then, a robust fixed-time inverse dynamics control is proposed. For the proposed control method, the fixed-time stability of a closed-loop system with uncertainties is strictly proved. The newly proposed method exhibits the following two attractive features. First, the proposed control scheme extends the existing fixed-time IDC for the robot manipulator system to (...) the robust control scheme. Second, the proposed method is strictly nonsingular rather than the commonly used approximate approach. Simulation result demonstrates the effectiveness of the proposed control scheme. (shrink)

This book represents the fourth volume in the series Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of the HyperUncertain Set, building on the foundational advancements introduced in previous volumes. The series aims to explore the ongoing evolution of uncertain combinatorics through innovative methodologies such as graphization, hyperization, and uncertainization. These approaches integrate and extend core concepts from fuzzy, neutrosophic, soft, and rough (...) set theories, providing robust frameworks to model and analyze the inherent complexity of real-world uncertainties. At the heart of this series lies combinatorics and set theory—cornerstones of mathematics that address the study of counting, arrangements, and the relationships between collections under defined rules. Traditionally, combinatorics has excelled in solving problems involving uncertainty, while advancements in set theory have expanded its scope to include powerful constructs like fuzzy and neutrosophic sets. These advanced sets bring new dimensions to uncertainty modeling by capturing not just binary truth but also indeterminacy and falsity. In this fourth volume, the integration of set theory with graph theory takes center stage, culminating in "graphized" structures such as hypergraphs and superhypergraphs. These structures, paired with innovations like Neutrosophic Oversets, Undersets, Offsets, and the Nonstandard Real Set, extend the boundaries of mathematical abstraction. This fusion of combinatorics, graph theory, and uncertain set theory creates a rich foundation for addressing the multidimensional and hierarchical uncertainties prevalent in both theoretical and applied domains. The book is structured into thirteen chapters, each contributing unique perspectives and advancements in the realm of HyperUncertain Sets and their related frameworks. -/- The first chapter (Advancing Traditional Set Theory with Hyperfuzzy, Hyperneutrosophic, and Hyperplithogenic Sets) explores the evolution of classical set theory to better address the complexity and ambiguity of real-world phenomena. By introducing hierarchical structures like hyperstructures and superhyperstructures—created through iterative applications of power sets—it lays the groundwork for more abstract and adaptable mathematical tools. The focus is on extending three foundational frameworks: Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets into their hyperforms: Hyperfuzzy Sets, Hyperneutrosophic Sets, and Hyperplithogenic Sets. These advanced concepts are applied across diverse fields such as statistics, clustering, evolutionary theory, topology, decision-making, probability, and language theory. The goal is to provide a robust platform for future research in this expanding area of study. The second chapter (Applications and Mathematical Properties of Hyperneutrosophic and SuperHyperneutrosophic Sets) extends the work on Hyperfuzzy, Hyperneutrosophic, and Hyperplithogenic Sets by delving into their advanced applications and mathematical foundations. Building on prior research, it specifically examines Hyperneutrosophic and SuperHyperneutrosophic Sets, exploring their integration into: Neutrosophic Logic, Cognitive Maps,Graph Neural Networks, Classifiers, and Triplet Groups. The chapter also investigates their mathematical properties and applicability in addressing uncertainties and complexities inherent in various domains. These insights aim to inspire innovative uses of hypergeneralized sets in modern theoretical and applied research. The third chapter (New Extensions of Hyperneutrosophic Sets – Bipolar, Pythagorean, Double-Valued, and Interval-Valued Sets) studies advanced variations of Neutrosophic Sets, a mathematical framework defined by three membership functions: truth (T), indeterminacy (I), and falsity (F). By leveraging the concepts of Hyperneutrosophic and SuperHyperneutrosophic Sets, the study extends: Bipolar Neutrosophic Sets, Interval-Valued Neutrosophic Sets, Pythagorean Neutrosophic Sets, and Double-Valued Neutrosophic Sets. These extensions address increasingly complex scenarios, and a brief analysis is provided to explore their potential applications and mathematical underpinnings. Building on prior research, the fourth chapter (Hyperneutrosophic Extensions of Complex, Single-Valued Triangular, Fermatean, and Linguistic Sets) expands on Neutrosophic Set theory by incorporating recent advancements in Hyperneutrosophic and SuperHyperneutrosophic Sets. The study focuses on extending: Complex Neutrosophic Sets, Single-Valued Triangular Neutrosophic Sets, Fermatean Neutrosophic Sets, and Linguistic Neutrosophic Sets. The analysis highlights the mathematical structures of these hyperextensions and explores their connections with existing set-theoretic concepts, offering new insights into managing uncertainty in multidimensional challenges. The fifth chapter (Advanced Extensions of Hyperneutrosophic Sets – Dynamic, Quadripartitioned, Pentapartitioned, Heptapartitioned, and m-Polar) delves deeper into the evolution of Neutrosophic Sets by exploring advanced frameworks designed for even more intricate applications. New extensions include: Dynamic Neutrosophic Sets, Quadripartitioned Neutrosophic Sets, Pentapartitioned Neutrosophic Sets, Heptapartitioned Neutrosophic Sets, and m-Polar Neutrosophic Sets. These developments build upon foundational research and aim to provide robust tools for addressing multidimensional and highly nuanced problems. The sixth chapter (Advanced Extensions of Hyperneutrosophic Sets – Cubic, Trapezoidal, q-Rung Orthopair, Overset, Underset, and Offset) builds upon the Neutrosophic framework, which employs truth (T), indeterminacy (I), and falsity (F) to address uncertainty. Leveraging advancements in Hyperneutrosophic and SuperHyperneutrosophic Sets, the study extends: Cubic Neutrosophic Sets, Trapezoidal Neutrosophic Sets, q-Rung Orthopair Neutrosophic Sets, Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets. The chapter provides a brief analysis of these new set types, exploring their properties and potential applications in solving multidimensional problems. The seventh chapter (Specialized Classes of Hyperneutrosophic Sets – Support, Paraconsistent, and Faillibilist Sets) delves into unique classes of Neutrosophic Sets extended through Hyperneutrosophic and SuperHyperneutrosophic frameworks to tackle advanced theoretical challenges. The study introduces and extends: Support Neutrosophic Sets, Neutrosophic Intuitionistic Sets, Neutrosophic Paraconsistent Sets, Neutrosophic Faillibilist Sets, Neutrosophic Paradoxist and Pseudo-Paradoxist Sets, Neutrosophic Tautological and Nihilist Sets, Neutrosophic Dialetheist Sets, and Neutrosophic Trivialist Sets. These extensions address highly nuanced aspects of uncertainty, further advancing the theoretical foundation of Neutrosophic mathematics. The eight chapter (MultiNeutrosophic Sets and Refined Neutrosophic Sets) focuses on two advanced Neutrosophic frameworks: MultiNeutrosophic Sets, and Refined Neutrosophic Sets. Using Hyperneutrosophic and nn-SuperHyperneutrosophic Sets, these extensions are analyzed in detail, highlighting their adaptability to multidimensional and complex scenarios. Examples and mathematical properties are provided to showcase their practical relevance and theoretical depth. The ninth chapter (Advanced Hyperneutrosophic Set Types – Type-m, Nonstationary, Subset-Valued, and Complex Refined) explores extensions of the Neutrosophic framework, focusing on: Type-m Neutrosophic Sets, Nonstationary Neutrosophic Sets, Subset-Valued Neutrosophic Sets, and Complex Refined Neutrosophic Sets. These extensions utilize the Hyperneutrosophic and SuperHyperneutrosophic frameworks to address advanced challenges in uncertainty management, expanding their mathematical scope and practical applications. The tenth chapter (Hyperfuzzy Hypersoft Sets and Hyperneutrosophic Hypersoft Sets) integrates the principles of Fuzzy, Neutrosophic, and Soft Sets with hyperstructures to introduce: Hyperfuzzy Hypersoft Sets, and Hyperneutrosophic Hypersoft Sets. These frameworks are designed to manage complex uncertainty through hierarchical structures based on power sets, with detailed analysis of their properties and theoretical potential. The eleventh chapter (A Review of SuperFuzzy, SuperNeutrosophic, and SuperPlithogenic Sets) revisits and extends the study of advanced set concepts such as: SuperFuzzy Sets, Super-Intuitionistic Fuzzy Sets,Super-Neutrosophic Sets, and SuperPlithogenic Sets, including their specialized variants like quadripartitioned, pentapartitioned, and heptapartitioned forms. The work serves as a consolidation of existing studies while highlighting potential directions for future research in hierarchical uncertainty modeling. Focusing on decision-making under uncertainty, the tweve chapter (Advanced SuperHypersoft and TreeSoft Sets) introduces six novel concepts: SuperHypersoft Rough Sets,SuperHypersoft Expert Sets, Bipolar SuperHypersoft Sets, TreeSoft Rough Sets, TreeSoft Expert Sets, and Bipolar TreeSoft Sets. Definitions, properties, and potential applications of these frameworks are explored to enhance the flexibility of soft set-based models. The final chapter (Hierarchical Uncertainty in Fuzzy, Neutrosophic, and Plithogenic Sets) provides a comprehensive survey of hierarchical uncertainty frameworks, with a focus on Plithogenic Sets and their advanced extensions: Hyperplithogenic Sets, SuperHyperplithogenic Sets. It examines relationships with other major concepts such as Intuitionistic Fuzzy Sets, Vague Sets, Picture Fuzzy Sets, Hesitant Fuzzy Sets, and multi-partitioned Neutrosophic Sets, consolidating their theoretical interconnections for modeling complex systems. This volume not only reflects the dynamic interplay between theoretical rigor and practical application but also serves as a beacon for future research in uncertainty modeling, offering advanced tools to tackle the intricacies of modern challenges. (shrink)

In this paper, the stabilization problem of nonholonomic chained-form systems is addressed with uncertain constants. In this paper, the active disturbance rejection control is designed to solve this problem. The proposed control strategy combines extended state observer and adaptive sliding mode controller. The control of nonholonomic chained-form systems with dynamic nonlinear uncertain terms and uncertain constants is first discussed in this paper. In comparison with existing methods, the proposed method in this paper has better performance. (...) It is proved that, with the application of the proposed control strategy, semiglobal finite-time stabilization of the systems is achieved. An example is given to illustrate the effectiveness of the proposed method. (shrink)

The Belgian party system is aften considered to be or to have been very stable. This article investigates the possibility of the Belgian parties and party systems to 'go Italian ', i.e. to be confronted with a radicial change resulting from a fundamental lack of legitimacy. This problem of legitimation can be expected from the fact that Belgium is a very consociational democracy, in which the parties play a very important role, but tend to become very entangled with the (...) state. The split of the national parties and the federal reform of the state have made the decision-making structures even more complex than before, and have therefore not at all reduced the 'partitocratic' nature of the system. For these reasons a future Italian-style collapse of the parties and the party systems is certainly not to be excluded. (shrink)

Reasoning about mental states and processes is important in various subareas of the legal domain. A trial lawyer might need to reason and the beliefs, reasoning and other mental states and processes of members of a jury; a police officer might need to reason about the conjectured beliefs and reasoning of perpetrators; a judge may need to consider a defendant's mental states and processes for the purposes of sentencing and so on. Further, the mental states in question may themselves be (...) about the mental states and processes of other people. Therefore, if AI systems are to assist with reasoning tasks in law, they may need to be able to reason about mental states and processes. Such reasoning is riddled with uncertainty, and this is true in particular in the legal domain. The article discusses how various different types of uncertainty arise, and shows how they greatly complicate the task of reasoning about mental states and processes. The article concentrates on the special case of states of belief and processes of reasoning, and sketches an implemented, prototype computer program (ATT-Meta) that copes with the various types of uncertainty in reasoning about beliefs and reasoning. In particular, the article outlines the system's facilities for handling conflict between different lines of argument, especially when these lie within the reasoning of different people. The system's approach is illustrated by application to a real-life mugging example. [NB: The date on the archived preprint avail via this Phil* page has an incorrect date, 2011, on it. The correct date is 2001.]. (shrink)

The second volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond” presents a deep exploration of the progress in uncertain combinatorics through innovative methodologies like graphization, hyperization, and uncertainization. This volume integrates foundational concepts from fuzzy, neutrosophic, soft, and rough set theory, among others, to further advance the field. Combinatorics and set theory, two central pillars of mathematics, focus on counting, arrangement, and the study of collections under defined rules. Combinatorics excels (...) in handling uncertainty, while set theory has evolved with concepts such as fuzzy and neutrosophic sets, which enable the modeling of complex real-world uncertainties by addressing truth, indeterminacy, and falsehood. These advancements, when combined with graph theory, give rise to novel forms of uncertain sets in "graphized" structures, including hypergraphs and superhypergraphs. Innovations such as Neutrosophic Oversets, Undersets, and Offsets, as well as the Nonstandard Real Set, build upon traditional graph concepts, pushing both theoretical and practical boundaries. The synthesis of combinatorics, set theory, and graph theory in this volume provides a robust framework for addressing the complexities and uncertainties inherent in both mathematical and real-world systems, paving the way for future research and application. In the first chapter, “A Review of the Hierarchy of Plithogenic, Neutrosophic, and Fuzzy Graphs: Survey and Applications”, the authors investigate the interrelationships among various graph classes, including Plithogenic graphs, and explore other related structures. Graph theory, a fundamental branch of mathematics, focuses on networks of nodes and edges, studying their paths, structures, and properties. A Fuzzy Graph extends this concept by assigning a membership degree between 0 and 1 to each edge and vertex, representing the level of uncertainty. The Turiyam Neutrosophic Graph is introduced as an extension of both Neutrosophic and Fuzzy Graphs, while Plithogenic graphs offer a potent method for managing uncertainty. The second chapter, “Review of Some Superhypergraph Classes: Directed, Bidirected, Soft, and Rough”, examines advanced graph structures such as directed superhypergraphs, bidirected hypergraphs, soft superhypergraphs, and rough superhypergraphs. Classical graph classes include undirected graphs, where edges lack orientation, and directed graphs, where edges have specific directions. Recent innovations, including bidirected graphs, have sparked ongoing research and significant advancements in the field. Soft Sets and their extension to Soft Graphs provide a flexible framework for managing uncertainty, while Rough Sets and Rough Graphs address uncertainty by using lower and upper approximations to handle imprecise data. Hypergraphs generalize traditional graphs by allowing edges, or hyperedges, to connect more than two vertices. Superhypergraphs further extend this by allowing both vertices and edges to represent subsets, facilitating the modeling of hierarchical and group-based relationships. The third chapter, “Survey of Intersection Graphs, Fuzzy Graphs, and Neutrosophic Graphs”, explores the intersection graph models within the realms of Fuzzy Graphs, Intuitionistic Fuzzy Graphs, Neutrosophic Graphs, Turiyam Neutrosophic Graphs, and Plithogenic Graphs. The chapter highlights their mathematical properties and interrelationships, reflecting the growing number of graph classes being developed in these areas. Intersection graphs, such as Unit Square Graphs, Circle Graphs, and Ray Intersection Graphs, are crucial for understanding complex graph structures in uncertain environments. The fourth chapter, “Fundamental Computational Problems and Algorithms for SuperHyperGraphs”, addresses optimization problems within the SuperHypergraph framework, such as the SuperHypergraph Partition Problem, Reachability, and Minimum Spanning SuperHypertree. The chapter also adapts classical problems like the Traveling Salesman Problem and the Chinese Postman Problem to the SuperHypergraph context, exploring how hypergraphs, which allow hyperedges to connect more than two vertices, can be used to solve complex hierarchical and relational problems. The fifth chapter, “A Short Note on the Basic Graph Construction Algorithm for Plithogenic Graphs”, delves into algorithms designed for Plithogenic Graphs and Intuitionistic Plithogenic Graphs, analyzing their complexity and validity. Plithogenic Graphs model multi-valued attributes by incorporating membership and contradiction functions, offering a nuanced representation of complex relationships. The sixth chapter, “Short Note of Bunch Graph in Fuzzy, Neutrosophic, and Plithogenic Graphs”, generalizes traditional graph theory by representing nodes as groups (bunches) rather than individual entities. This approach enables the modeling of both competition and collaboration within a network. The chapter explores various uncertain models of bunch graphs, including Fuzzy Graphs, Neutrosophic Graphs, Turiyam Neutrosophic Graphs, and Plithogenic Graphs. In the seventh chapter, “A Reconsideration of Advanced Concepts in Neutrosophic Graphs: Smart, Zero Divisor, Layered, Weak, Semi, and Chemical Graphs”, the authors extend several fuzzy graph classes to Neutrosophic graphs and analyze their properties. Neutrosophic Graphs, a generalization of fuzzy graphs, incorporate degrees of truth, indeterminacy, and falsity to model uncertainty more effectively. The eighth chapter, “Short Note of Even-Hole-Graph for Uncertain Graph”, focuses on Even-Hole-Free and Meyniel Graphs analyzed within the frameworks of Fuzzy, Neutrosophic, Turiyam Neutrosophic, and Plithogenic Graphs. The study investigates the structure of these graphs, with an emphasis on their implications for uncertainty modeling. The ninth chapter, “Survey of Planar and Outerplanar Graphs in Fuzzy and Neutrosophic Graphs”, explores planar and outerplanar graphs, as well as apex graphs, within the contexts of fuzzy, neutrosophic, Turiyam Neutrosophic, and plithogenic graphs. The chapter examines how these types of graphs are used to model uncertain parameters and relationships in mathematical and real-world systems. The tenth chapter, “General Plithogenic Soft Rough Graphs and Some Related Graph Classes”, introduces and explores new concepts such as Turiyam Neutrosophic Soft Graphs and General Plithogenic Soft Graphs. The chapter also examines models of uncertain graphs, including Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Graphs, all designed to handle uncertainty in diverse contexts. The eleventh chapter, “Survey of Trees, Forests, and Paths in Fuzzy and Neutrosophic Graphs”, provides a comprehensive study of Trees, Forests, and Paths within the framework of Fuzzy and Neutrosophic Graphs. This chapter focuses on classifying and analyzing graph structures like trees and paths in uncertain environments, contributing to the ongoing development of graph theory in the context of uncertainty. (shrink)

This book is the sixth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Building upon the foundational contributions of previous volumes, this edition focuses on the exploration and development of Various New Uncertain Concepts, further enriching the study of uncertainty and complexity through innovative theoretical advancements and practical applications. The volume is meticulously organized into 15 chapters, each presenting unique perspectives and contributions to the (...) field. From theoretical explorations to real-world applications, these chapters provide a cohesive and comprehensive overview of the state of the art in uncertain combinatorics, emphasizing the versatility and power of the newly introduced concepts and methodologies. The first chapter (SuperHypertree-depth – Structural Analysis in SuperHyperGraphs) explores the concept of SuperHypertree-depth, an extension of the classical graph parameter Tree-depth and its hypergraph counterpart Hypertree-depth. By introducing hierarchical nesting within SuperHyperGraphs, where both vertices and edges can represent recursive subsets, this study investigates the mathematical properties and structural implications of these extended parameters. The findings highlight the relationships between SuperHypertree-depth and its traditional graph-theoretic equivalents, providing a deeper understanding of their applicability to hierarchical and complex systems. The second chapter (Obstructions for Hypertree-width and SuperHypertree-width) examines the role of ultrafilters as obstructions in determining Hypertree-width and extends the concept to SuperHypertree-width. Building on hypergraph theory, which abstracts traditional graph frameworks into more complex domains, the study investigates how recursive structures within SuperHyperGraphs redefine the computational and structural properties of these parameters. Ultrafilters, with their broad mathematical significance, serve as critical tools for understanding the limitations and potentials of these advanced graph metrics. The third chapter (SuperHypertree-Length and SuperHypertree-Breadth in SuperHyperGraphs) investigates the extension of the graph-theoretic parameters Tree-length and Tree-breadth to the realms of hypergraphs and SuperHyperGraphs. By leveraging the hierarchical nesting of SuperHyperGraphs, the study explores how these parameters adapt to increasingly complex and multi-level structures. Comparative analyses between these extended parameters and their classical counterparts reveal new insights into their relevance and utility in advanced graph and hypergraph theory. Plithogenic Sets, which generalize Fuzzy and Neutrosophic Sets, are extended in the fourth chapter (Extended HyperPlithogenic Sets and Generalized Plithogenic Graphs) to Extended Plithogenic Sets, HyperPlithogenic Sets, and SuperHyperPlithogenic Sets. This study further investigates their application to graph theory through the concepts of Extended Plithogenic Graphs and Generalized Extended Plithogenic Graphs. The chapter provides a concise exploration of these frameworks, offering insights into their potential for addressing uncertainty and complexity in graph structures. Soft Sets provide an effective framework for decision-making by mapping parameters to subsets of a universal set, addressing uncertainty and vagueness. The fifth chapter (Double-Framed Superhypersoft Set and Double-Framed Treesoft Set) introduces the Double-Framed SuperHypersoft Set and the Double-Framed Treesoft Set as extensions of traditional and advanced soft set frameworks, such as Hypersoft and SuperHypersoft Sets. The chapter explores their relationships with existing concepts, offering new tools to handle complex decision-making scenarios with enhanced structural flexibility. The sixth paper (HyperPlithogenic Cubic Set and SuperHyperPlithogenic Cubic Set) introduces the concepts of the HyperPlithogenic Cubic Set and SuperHyperPlithogenic Cubic Set, which extend the Plithogenic Cubic Set by integrating both interval-valued and single-valued fuzzy memberships. These sets leverage multi-attribute aggregation techniques inherent to plithogenic structures, allowing for nuanced representations of uncertainty. Additionally, related constructs such as the HyperPlithogenic Fuzzy Cubic Set, HyperPlithogenic Intuitionistic Fuzzy Cubic Set, and HyperPlithogenic Neutrosophic Cubic Set are explored, further enriching the theoretical and practical applications of this framework. The seventh chapter (L-Neutrosophic Sets and Nonstationary Neutrosophic Sets) extends the foundational concepts of fuzzy sets by integrating Neutrosophic and Plithogenic frameworks. By introducing L-Neutrosophic Sets and Nonstationary Neutrosophic Sets, the study enhances the representation of uncertainty through independent membership components: truth, indeterminacy, and falsity. These advanced constructs also incorporate multi-dimensional and contradictory attributes, providing a robust means of modeling complex decision-making and uncertain data. Plithogenic and Rough Sets, known for generalizing uncertainty modeling and classification, are extended in the eight chapter (Forest HyperPlithogenic and Forest HyperRough Sets) to Forest HyperPlithogenic Sets, Forest SuperHyperPlithogenic Sets, Forest HyperRough Sets, and Forest SuperHyperRough Sets. These frameworks incorporate hierarchical and recursive structures to advance existing set-theoretic paradigms. The chapter explores their applications in multi-level data analysis and uncertainty classification, demonstrating their adaptability to complex systems. Building on Fuzzy, Neutrosophic, and Plithogenic Sets, the tenth chapter (Symbolic HyperPlithogenic Sets) introduces Symbolic HyperPlithogenic Sets and Symbolic n-SuperHyperPlithogenic Sets. These sets incorporate symbolic components and algebraic coefficients, enabling flexible operations within a defined prevalence order. By extending symbolic representation into hyperplithogenic and superhyperplithogenic domains, the chapter opens new pathways for addressing uncertainty and hierarchical complexity in mathematical modeling. Soft Sets, designed to manage uncertainty and imprecision, have evolved through various extensions like Hypersoft Sets and SuperHypersoft Sets. The eleventh chapter (N-SuperHypersoft and Bijective SuperHypersoft Sets) introduces N-SuperHypersoft Sets, N-Treesoft Sets, Bijective SuperHypersoft Sets, and Bijective Treesoft Sets. These new constructs enhance decision-making frameworks by incorporating advanced hierarchical and bijective relationships, building on existing theories and expanding their applications. Plithogenic Sets, known for integrating multi-valued attributes and contradictions, and Rough Sets, which partition data into definable approximations, are combined in the twelfth chapter (Plithogenic Rough Sets) to form Plithogenic Rough Sets. This fusion provides a powerful framework for addressing uncertainty in dynamic and complex decision-making scenarios, offering a novel approach to uncertainty modeling. Expanding on Neutrosophic Sets, which represent truth, indeterminacy, and falsehood, this chapter introduces Plithogenic Duplets and Plithogenic Triplets. These constructs leverage the Plithogenic framework to incorporate attributes, values, and contradiction measures. The thirteenth chapter (Plithogenic Duplets and Triplets) examines their relationships with Neutrosophic Duplets and Triplets, offering new tools for multi-dimensional data representation and decision-making. Building on foundational concepts like Rough Sets and Vague Sets, the fourteenth chapter (SuperRough and SuperVague Sets) introduces SuperRough Sets and SuperVague Sets. These generalized frameworks extend uncertainty modeling by incorporating hierarchical structures. The study also demonstrates that SuperRough Sets can evolve into SuperHyperRough Sets, providing further generalizations for advanced data classification and analysis. The fifteenth chapter (Neutrosophic TreeSoft Expert and ForestSoft Sets) revisits the Neutrosophic TreeSoft Set, which combines the hierarchical structure of TreeSoft Sets with the Neutrosophic framework for uncertainty representation. Additionally, it introduces the Neutrosophic TreeSoft Expert Set, incorporating expert knowledge into the model. The chapter also explores the ForestSoft Set and its extension, the Neutrosophic ForestSoft Set, to provide multi-level, tree-structured approaches for complex data representation and analysis. (shrink)

The third volume of “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond” presents an in-depth exploration of the cutting-edge developments in uncertain combinatorics and set theory. This comprehensive collection highlights innovative methodologies such as graphization, hyperization, and uncertainization, which enhance combinatorics by incorporating foundational concepts from fuzzy, neutrosophic, soft, and rough set theories. These advancements open new mathematical horizons, offering novel approaches to managing uncertainty within complex systems. Combinatorics, a discipline focused (...) on counting, arrangement, and structure, often faces challenges when uncertainty is present. Set theory, which underpins combinatorial problems, has evolved to tackle these challenges. The introduction of fuzzy and neutrosophic sets has expanded the toolkit for modeling uncertainty by incorporating elements of truth, indeterminacy, and falsehood into decision-making processes. These innovations seamlessly intersect with graph theory, providing new ways to represent uncertain structures through "graphized" forms such as hypergraphs and superhypergraphs. This volume also introduces advanced concepts like Neutrosophic Oversets, Undersets, and Offsets, which push the boundaries of classical graph theory and offer deeper insights into the mathematical and practical challenges posed by real-world systems. By blending combinatorics, set theory, and graph theory, the authors have created a robust framework for addressing uncertainty in both mathematical systems and their real-world applications. This foundation sets the stage for future breakthroughs in combinatorics, set theory, and related fields. Each chapter in this volume contributes both theoretical foundations and practical applications, demonstrating the power of integrating graph theory, set theory, and uncertainty models. The new ideas, algorithms, and mathematical tools presented here will drive the future of combinatorial research and its applications in uncertain environments. In the first chapter, “Introduction to Upside-Down Logic: Its Deep Relation to Neutrosophic Logic and Applications”, the authors present Upside-Down Logic, a novel logical framework that systematically transforms truths into falsehoods and vice versa, based on contextual shifts. Introduced by F. Smarandache, this paper provides a mathematical definition of Upside-Down Logic, including applications related to the Japanese language. The chapter also introduces Contextual Upside-Down Logic, an extension that adjusts logical connectives alongside flipped truth values, as well as Indeterm-Upside-Down Logic and Certain Upside-Down Logic to address indeterminacy. A simple algorithm is also proposed to demonstrate the computational aspects of this logic. In the second chapter, “Local-Neutrosophic Logic and Local-Neutrosophic Sets: Incorporating Locality with Applications”, the authors introduce Local-Neutrosophic Logic and Local-Neutrosophic Sets, which integrate the concept of locality into Neutrosophic Logic. By defining locality as the influence of immediate surroundings on an object or system, this chapter explores how it affects indeterminacy in real-world problems. The paper also examines potential applications and provides mathematical definitions for these new concepts. The third chapter, “A Review of Fuzzy and Neutrosophic Offsets: Connections to Some Set Concepts and Normalization Function”, extends the concept of offsets in uncertain set-theoretic frameworks, such as Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets. This chapter introduces several advanced types of offsets, including Nonstationary Fuzzy Offset, Multi-valued Plithogenic Offset, and Subset-valued Neutrosophic Offset, offering deeper insights into handling uncertainty in mathematical models. In the fourth chapter, “Review of Plithogenic Directed, Mixed, Bidirected, and Pangene OffGraph”, the authors build upon Plithogenic Graphs to propose extensions such as Plithogenic Directed OffGraph, Plithogenic BiDirected OffGraph, and Plithogenic Mixed OffGraph. These new concepts, including the Plithogenic Pangene OffGraph, are explored in detail, with a focus on their mathematical properties and potential applications in uncertain graph theory. The fifth chapter, “Short Note on Neutrosophic Closure Matroids”, explores the extension of matroid concepts into Neutrosophic and Turiyam Neutrosophic set theories, introducing Neutrosophic closure matroids. This concept integrates uncertainty, indeterminacy, and liberal states into matroid theory, enhancing its applicability in optimization and combinatorial problems. In the sixth chapter, “Some Graph Parameters for Superhypertree-width and Neutrosophic Tree-width”, the authors discuss graph parameters such as Superhypertree-width and Neutrosophic tree-width. These parameters play a crucial role in the study of graph characteristics, particularly in algorithms and real-world applications. The chapter explores the generalization of hypergraphs to SuperHyperGraphs and examines how these concepts extend tree-width parameters within the context of Neutrosophic logic. (shrink)

This book is the fifth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of Various SuperHyperConcepts, building on the foundational advancements introduced in previous volumes. The series aims to explore the ongoing evolution of uncertain combinatorics through innovative methodologies such as graphization, hyperization, and uncertainization. These approaches integrate and extend core concepts from fuzzy, neutrosophic, soft, and rough (...) set theories, providing robust frameworks to model and analyze the inherent complexity of real-world uncertainties. At the heart of this series lies combinatorics and set theory—cornerstones of mathematics that address the study of counting, arrangements, and the relationships between collections under defined rules. Traditionally, combinatorics has excelled in solving problems involving uncertainty, while advancements in set theory have expanded its scope to include powerful constructs like fuzzy and neutrosophic sets. These advanced sets bring new dimensions to uncertainty modeling by capturing not just binary truth but also indeterminacy and falsity. In this fifth volume, the exploration of Various SuperHyperConcepts provides an innovative lens to address uncertainty, complexity, and hierarchical relationships. It synthesizes key methodologies introduced in earlier volumes, such as hyperization and neutrosophic extensions, while advancing new theories and applications. From pioneering hyperstructures to applications in advanced decision-making, language modeling, and neural networks, this book represents a significant leap forward in uncertain combinatorics and its practical implications across disciplines. -/- The book is structured into 17 chapters, each contributing unique perspectives and advancements in the realm of Various SuperHyperConcepts and their related frameworks: Chapter 1 introduces the concept of Body-Mind-Soul-Spirit Fluidity within psychology and phenomenology, while examining established social science frameworks like PDCA and DMAIC. It extends these frameworks using Neutrosophic Sets, a flexible extension of Fuzzy Sets, to improve their adaptability for mathematical and programming applications. The chapter emphasizes the potential of Neutrosophic theory to address multi-dimensional challenges in social sciences. Chapter 2 delves into the theoretical foundation of Hyperfunctions and their generalizations, such as Hyperrandomness and Hyperdecision-Making. It explores higher-order frameworks like Weak Hyperstructures, Hypergraphs, and Cognitive Hypermaps, aiming to establish their versatility in addressing multi-layered problems and setting a foundation for further studies. Chapter 3 extends traditional decision-making methodologies into HyperDecision-Making and n-SuperHyperDecision-Making. By building on approaches like MCDM and TOPSIS, this chapter develops frameworks capable of addressing complex decision-making scenarios, emphasizing their applicability in dynamic, multi-objective contexts. Chapter 4 explores integrating uncertainty frameworks, including Fuzzy, Neutrosophic, and Plithogenic Sets, into Large Language Models (LLMs). It proposes innovative models like Large Uncertain Language Models and Natural Uncertain Language Processing, integrating hierarchical and generalized structures to advance the handling of uncertainty in linguistic representation and processing. Chapter 5 introduces the Natural n-Superhyper Plithogenic Language by synthesizing natural language, plithogenic frameworks, and superhyperstructures. This innovative construct seeks to address challenges in advanced linguistic and structural modeling, blending attributes of uncertainty, complexity, and hierarchical abstraction. Chapter 6 defines mathematical extensions such as NeutroHyperstructures and AntiHyperstructures using the Neutrosophic Triplet framework. It formalizes structures like neutro-superhyperstructures, advancing classical frameworks into higher-dimensional realms. Chapter 7 explores the extension of Binary Code, Gray Code, and Floorplans through hyperstructures and superhyperstructures. It highlights their iterative and hierarchical applications, demonstrating their adaptability for complex data encoding and geometric arrangement challenges. Chapter 8 investigates the Neutrosophic TwoFold SuperhyperAlgebra, combining classical algebraic operations with neutrosophic components. This chapter expands upon existing algebraic structures like Hyperalgebra and AntiAlgebra, exploring hybrid frameworks for advanced mathematical modeling. Chapter 9 introduces Hyper Z-Numbers and SuperHyper Z-Numbers by extending the traditional Z-Number framework with hyperstructures. These extensions aim to represent uncertain information in more complex and multidimensional contexts. Chapter 10 revisits category theory through the lens of hypercategories and superhypercategories. By incorporating hierarchical and iterative abstractions, this chapter extends the foundational principles of category theory to more complex and layered structures. Chapter 11 formalizes the concept of n-SuperHyperBranch-width and its theoretical properties. By extending hypergraphs into superhypergraphs, the chapter explores recursive structures and their potential for representing intricate hierarchical relationships. Chapter 12 examines superhyperstructures of partitions, integrals, and spaces, proposing a framework for advancing mathematical abstraction. It highlights the potential applications of these generalizations in addressing hierarchical and multi-layered problems. Chapter 13 revisits Rough, HyperRough, and SuperHyperRough Sets, introducing new concepts like Tree-HyperRough Sets. The chapter connects these frameworks to advanced approaches for modeling uncertainty and complex relationships. Chapter 14 explores Plithogenic SuperHyperStructures and their applications in decision-making, control, and neuro systems. By integrating these advanced frameworks, the chapter proposes innovative directions for extending existing systems to handle multi-attribute and contradictory properties. Chapter 15 focuses on superhypergraphs, expanding hypergraph concepts to model complex structural types like arboreal and molecular superhypergraphs. It introduces Generalized n-th Powersets as a unifying framework for broader mathematical applications, while also touching on hyperlanguage processing. Chapter 16 defines NeutroHypergeometry and AntiHypergeometry as extensions of classical geometric structures. Using the Geometric Neutrosophic Triplet, the chapter demonstrates the flexibility of these frameworks in representing multi-dimensional and uncertain relationships. Chapter 17 establishes the theoretical groundwork for SuperHyperGraph Neural Networks and Plithogenic Graph Neural Networks. By integrating advanced graph structures, this chapter opens pathways for applying neural networks to more intricate and uncertain data representations. (shrink)