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Florentin Smarandache

Publications and source records attributed to Florentin Smarandache.

At least 19 recordsLinked to original sources
arXiv2026

Representing Higher-Order Networks: A Survey of Graph-Based Frameworks

Many real-world phenomena can be naturally represented using graphs and networks. Classical graph models, however, are generally restricted to pairwise interactions and may therefore be insufficient for capturing the richer structural relationships encountered in practical systems. Higher-order graph formalisms extend the classical framework by incorporating multiway, hierarchical, temporal, multilayer, recursive, and tensor-based interactions, thereby enabling more expressive representations of complex systems. This book provides a comprehensive overview of mathematical concepts and frameworks for modeling higher-order networks. It surveys foundational notions, extended frameworks, and newly introduced formalisms, with particular emphasis on their underlying structural principles, interrelationships, and respective modeling roles. Its primary objective is to offer a unified perspective that enables readers to compare diverse higher-order network models and to identify suitable mathematical tools for both theoretical investigation and practical application. Now in its fourth edition, this volume presents an expanded, revised, and corrected treatment of foundational concepts, established extensions, and recently introduced structures, while maintaining a unifying focus on the relationships among these models. It is intended to serve as a reference for researchers and students studying and comparing higher-order network models in combinatorics, topology, applied mathematics, and related fields.

cs.SI↗
arXiv2026

Institutional Prestige as Geographic Bias in Large Language Models: Evidence from Three Factorial Experiments with Bootstrap Confidence Intervals

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% bootstrap CI: +0.175 to +0.422), while name-origin effects are negligible and non-significant (95% CI crosses zero). Study 2 (2x2 Prestige x Country design) breaks the prestige-geography confound: the prestige effect (+0.185; 95% CI: +0.093 to +0.275) exceeds the country-of-origin effect (+0.126; 95% CI: +0.037 to +0.218) by 1.5x. Study 3 (2x2 Journal x Institution design) reveals that journal prestige (Nature vs. a peripheral open-access journal) dominates institutional prestige by 5.7x: journal effect +1.937 (95% CI: +1.811 to +2.062) vs. institution effect +0.341 (95% CI: +0.184 to +0.504). A "rescue effect" is confirmed: publishing in Nature compensates for low institutional prestige more strongly for candidates from the University of Guayaquil (+2.127) than from MIT (+1.745). Results are quantified using the Neutrosophic Bias Index NBI ; the I component reveals elevated evaluation inconsistency for low-prestige profiles, an epistemic disadvantage not captured by mean-only metrics. Code and data: https://github.com/mleyvaz/geo-bias-llm

cs.CL↗
arXiv2026

The α -Discounting (α-DMCDM) as an extension of AHP, TOPSIS, VIKOR, PROMETHEE, and Weighted Sum

The Analytic Hierarchy Process (AHP) and other classic Multi-Criteria Decision Making (MCDM) techniques excel when decision makers can provide consistent pair wise judgments. Real world problems, however, often involve inconsistent, n-wise, or non-linear preference structures that render traditional methods inadequate. The α-Discounting MCDM (α-D MCDM) extends AHP by embedding a global discounting parameter α that transforms an inconsistent system of preference equations into a solvable algebraic system.

math.GM↗
arXiv2026

Breaking the Chains of Probability: Neutrosophic Logic as a New Framework for Epistemic Uncertainty in Large Language Models

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% of evaluations, hyper-truth emerged spontaneously, predominantly under ethical contradiction and logical paradox. We demonstrate that this approach preserves truth values in fuzzy contexts and offers a robust method for identifying and quantifying internal model conflict. We conclude that the integration of neutrosophic evaluation layers is a critical step toward more transparent, reliable, and ethically aware AI systems.

cs.AI↗
arXiv2026

Introduction of Over/Under/Off Masses

In this paper, for the first time, we extend the Over/Under/Off Set/Logic/Probability used in uncertain theories (such as: fuzzy, neutrosophic and extensions) to the Over/Under/Off Mass that could be used in Information Fusion. The approach is exemplified in three scenarios: (1) wildfire evacuation and resource allocation with satellite, IoT, and social media; (2) coverage gaps where indeterminacy must be managed; and (3) security monitoring where contradictory or erroneous reports are discounted.

math.GM↗
arXiv2026

Fuzzy, Neutrosophic, and Uncertain Graph Theory: Properties and Applications

This book presents a comprehensive and systematic survey of graph theory under uncertainty, with particular emphasis on the unifying role of the uncertain graph framework. It reviews fundamental concepts, structural properties, graph classes, and graph parameters within fuzzy, neutrosophic, and related models, while also introducing a wide range of extensions such as uncertain digraphs, hypergraphs, superhypergraphs, and dynamic graphs. In addition to theoretical developments, the book explores practical applications, including uncertain molecular graphs, decision-making systems, graph neural networks, knowledge graphs, and cognitive maps. By organizing diverse uncertainty-aware graph models within a common perspective, this work provides a coherent framework for understanding their relationships, capabilities, and applications in complex systems.

cs.AI↗
arXiv2026

Handbook of Rough Set Extensions and Uncertainty Models

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.

cs.AI↗
arXiv2026

Survey of Various Fuzzy and Uncertain Decision-Making Methods

Decision-making in real applications is often affected by vagueness, incomplete information, heterogeneous data, and conflicting expert opinions. This survey reviews uncertainty-aware multi-criteria decision-making (MCDM) and organizes the field into a concise, task-oriented taxonomy. We summarize problem-level settings (discrete, group/consensus, dynamic, multi-stage, multi-level, multiagent, and multi-scenario), weight elicitation (subjective and objective schemes under fuzzy/linguistic inputs), and inter-criteria structure and causality modelling. For solution procedures, we contrast compensatory scoring methods, distance-to-reference and compromise approaches, and non-compensatory outranking frameworks for ranking or sorting. We also outline rule/evidence-based and sequential decision models that produce interpretable rules or policies. The survey highlights typical inputs, core computational steps, and primary outputs, and provides guidance on choosing methods according to robustness, interpretability, and data availability. It concludes with open directions on explainable uncertainty integration, stability, and scalability in large-scale and dynamic decision environments.

cs.AI↗
arXiv2026

A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets

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.

cs.AI↗
arXiv2026

Theoretical Foundations of Superhypergraph and Plithogenic Graph Neural Networks

Hypergraphs generalize classical graphs by allowing a single edge to connect multiple vertices, providing a natural language for modeling higher-order interactions. Superhypergraphs extend this paradigm further by accommodating nested, set-valued entities and relations, enabling the representation of hierarchical, multi-level structures beyond the expressive reach of ordinary graphs or hypergraphs. In parallel, neural networks-especially Graph Neural Networks (GNNs)-have become a standard tool for learning from relational data, and recent years have seen rapid progress on Hypergraph Neural Networks (HGNNs) and their theoretical properties. To model uncertainty and multi-aspect attributes in complex networks, several graded and multi-valued graph frameworks have been developed, including fuzzy graphs and neutrosophic graphs. The plithogenic graph framework unifies and refines these approaches by incorporating multi-valued attributes together with membership and contradiction mechanisms, offering a flexible representation for heterogeneous and partially inconsistent information. This book develops the theoretical foundations of SuperHyperGraph Neural Networks (SHGNNs) and Plithogenic Graph Neural Networks, with the goal of extending message-passing principles to these advanced higher-order structures. We provide rigorous definitions, establish fundamental structural properties, and prove well-definedness results for key constructions, with particular emphasis on strengthened formulations of Soft Graph Neural Networks and Rough Graph Neural Networks.

cs.AI↗
arXiv2026

A Dynamic Survey of Soft Set Theory and Its Extensions

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.

cs.AI↗
arXiv2026

Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond

Combinatorics studies how discrete objects can be counted, arranged, and combined under specified rules. Motivated by uncertainty in real-world data and decisions, modern set-theoretic formalisms such as fuzzy sets, neutrosophic sets, rough sets, soft sets, and plithogenic sets have been developed. In particular, neutrosophic sets model uncertainty by assigning to each element degrees of truth, indeterminacy, and falsity. In parallel, these uncertainty frameworks are increasingly investigated in graphized and hyperized forms, where generalized graph models encompass classical graphs, hypergraphs, and higher-order "superhyper" structures; related hyper- and superhyper-concepts also arise beyond graph theory. This book (Edition 2.0) surveys and consolidates recent developments at the intersection of combinatorics, uncertain sets, uncertain graphs, and hyper/superhyper frameworks, while introducing several new graph and set concepts. As representative contributions, we extend graph-theoretic notions via Neutrosophic Oversets, Neutrosophic Undersets, Neutrosophic Offsets, and the Nonstandard Real Set. The second edition adds newly introduced concepts, corrects typographical issues, and re-examines mathematical consistency, aiming to serve as a compact reference and a source of inspiration for further research.

cs.AI↗
arXiv2023

Pura Vida Neutrosophic Algebra

We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic structures included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors

math.GM↗
arXiv2022

Improved Definition of NonStandard Neutrosophic Logic and Introduction to Neutrosophic Hyperreals

O the third version of this response-paper to Imamura criticism, we recall that NonStandard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century old neutrosophic operators (1995) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the NonStandard Analysis hyperreals, monads and binads to tiny intervals with the desired accuracy, otherwise they would be inapplicable. We point out several errors and false statements by Imamura with respect to the inf/sup of nonstandard subsets, also Imamura 'rigorous definition of neutrosophic logic' is wrong and the same for his definition of nonstandard unit interval, and we prove that there is not a total order on the set of hyperreals (because of the newly introduced Neutrosophic Hyperreals that are indeterminate), whence the transfer principle is questionable.

cs.AI↗
arXiv2019

Neutrosophic Set is a Generalization of Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set, q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, etc

In this paper we prove that Neutrosophic Set (NS) is an extension of Intuitionistic Fuzzy Set (IFS) no matter if the sum of single-valued neutrosophic components is < 1, or > 1, or = 1. For the case when the sum of components is 1 (as in IFS), after applying the neutrosophic aggregation operators one gets a different result from that of applying the intuitionistic fuzzy operators, since the intuitionistic fuzzy operators ignore the indeterminacy, while the neutrosophic aggregation operators take into consideration the indeterminacy at the same level as truth-membership and falsehood-nonmembership are taken. NS is also more flexible and effective because it handles, besides independent components, also partially independent and partially dependent components, while IFS cannot deal with these. Since there are many types of indeterminacies in our world, we can construct different approaches to various neutrosophic concepts. Also, Regret Theory, Grey System Theory, and Three-Ways Decision are particular cases of Neutrosophication and of Neutrosophic Probability. We extended for the first time the Three-Ways Decision to n-Ways Decision, and the Spherical Fuzzy Set to n-HyperSpherical Fuzzy Set and to n-HyperSpherical Neutrosophic Set.

math.GM↗
arXiv2019

Extended Nonstandard Neutrosophic Logic, Set, and Probability based on Extended Nonstandard Analysis

We extend for the second time the Nonstandard Analysis by adding the left monad closed to the right, and right monad closed to the left, while besides the pierced binad (we introduced in 1998) we add now the unpierced binad - all these in order to close the newly extended nonstandard space under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations. Then, we extend the Nonstandard Neutrosophic Logic, Nonstandard Neutrosophic Set, and Nonstandard Probability on this Extended Nonstandard Analysis space - that we prove it is a nonstandard neutrosophic lattice of first type (endowed with a nonstandard neutrosophic partial order) as well as a nonstandard neutrosophic lattice of second type (as algebraic structure, endowed with two binary neutrosophic laws, inf_N and sup_N). Many theorems, new terms introduced, and examples of nonstandard neutrosophic operations are given.

math.GM↗