[Paper Review] Interval Neutrosophic Sets and Logic: Theory and Applications in Computing

Haibin Wang, Smarandache, Florentin+2 more|ArXiv.org|May 6, 2005

Advanced Algebra and Logic82 references454 citations

TL;DR

This paper introduces Interval Neutrosophic Sets (INS) and Interval Neutrosophic Logic (INL) as an extension of fuzzy and intuitionistic fuzzy sets, explicitly modeling truth, indeterminacy, and falsity membership with interval values. It proposes a formal framework for handling uncertainty, inconsistency, and indeterminacy in information fusion, expert systems, and semantic web services, demonstrating its application through a neutrosophic neural network with genetic algorithms achieving a 19% total prediction error on a 150-sample test set.

ABSTRACT

This book presents the advancements and applications of neutrosophics. Chapter 1 first introduces the interval neutrosophic sets which is an instance of neutrosophic sets. In this chapter, the definition of interval neutrosophic sets and set-theoretic operators are given and various properties of interval neutrosophic set are proved. Chapter 2 defines the interval neutrosophic logic based on interval neutrosophic sets including the syntax and semantics of first order interval neutrosophic propositional logic and first order interval neutrosophic predicate logic. The interval neutrosophic logic can reason and model fuzzy, incomplete and inconsistent information. In this chapter, we also design an interval neutrosophic inference system based on first order interval neutrosophic predicate logic. The interval neutrosophic inference system can be applied to decision making. Chapter 3 gives one application of interval neutrosophic sets and logic in the field of relational databases. Neutrosophic data model is the generalization of fuzzy data model and paraconsistent data model. Here, we generalize various set-theoretic and relation-theoretic operations of fuzzy data model to neutrosophic data model. Chapter 4 gives another application of interval neutrosophic logic. A soft semantic Web Services agent framework is proposed to faciliate the registration and discovery of high quality semantic Web Services agent. The intelligent inference engine module of soft Semantic Web Services agent is implemented using interval neutrosophic logic.

Motivation & Objective

• To address the limitations of fuzzy and intuitionistic fuzzy sets in modeling indeterminate and inconsistent information.
• To formalize a framework for neutrosophic sets using interval-valued memberships for truth, indeterminacy, and falsity.
• To develop Interval Neutrosophic Logic for propositional and predicate calculus with proof theory and semantics.
• To apply the INS framework to relational databases and semantic web services for enhanced data modeling and discovery.
• To integrate neutrosophic logic with soft computing techniques for intelligent inference and service discovery.

Proposed method

• Defining Interval Neutrosophic Sets (INS) using three interval-valued memberships: truth, indeterminacy, and falsity, with independent ranges in [0,1].
• Establishing set-theoretic operators (union, intersection, complement) and proving their properties under interval-valued operations.
• Introducing Interval Neutrosophic Propositional and Predicate Calculus with syntax, semantics, and proof theory for logical reasoning under uncertainty.
• Designing a Neutrosophic Relational Data Model supporting generalized algebraic operations on neutrosophic relations and infinite-valued tuple relational calculus.
• Developing a Soft Semantic Web Services (SWS) agent using neutrosophic rule bases, deneutrosophication, and genetic algorithms for QoS-based service discovery.
• Implementing a neutrosophic neural network with interval-valued inputs and outputs, trained via genetic algorithms to minimize prediction error.

Experimental results

Research questions

• RQ1 How can neutrosophic sets be formalized using interval-valued memberships to represent truth, indeterminacy, and falsity independently?
• RQ2 What are the logical properties and operations (e.g., union, intersection) of Interval Neutrosophic Sets, and how do they generalize fuzzy and intuitionistic fuzzy logic?
• RQ3 How can Interval Neutrosophic Logic be extended to propositional and predicate calculus with sound semantics and proof systems?
• RQ4 Can the INS framework be effectively applied to relational databases and semantic web services for handling uncertainty and inconsistency?
• RQ5 To what extent can neutrosophic logic and soft computing enhance intelligent service discovery and prediction accuracy in real-world applications?

Key findings

• The paper establishes a comprehensive theoretical foundation for Interval Neutrosophic Sets, proving key properties such as convexity and closure under set-theoretic operations.
• Interval Neutrosophic Logic supports a complete proof system for propositional and predicate calculus, enabling formal reasoning under indeterminacy.
• The Neutrosophic Relational Data Model enables handling of uncertain and inconsistent data in databases using infinite-valued logic and generalized algebraic operations.
• The Soft SWS agent architecture successfully supports both capability-based and QoS-based discovery of semantic web services, outperforming systems like MWSDI that ignore non-functional properties.
• A neutrosophic neural network trained with genetic algorithms achieved a total prediction error of 19% across 150 test entries, with a maximum error of 1.64, demonstrating feasibility in real-world prediction tasks.
• The study shows that designing application-specific neutrosophic membership functions and selecting domain-relevant training data can significantly reduce prediction error.