A Comprehensive Survey of Fuzzy Implication Functions
Raquel Fernandez-Peralta
TL;DR
This paper provides a comprehensive catalog of fuzzy implication functions, arguing that the literature contains far more families than the classic six. It organizes families by construction method, enumerates numerous subfamilies (including those built from negations, t-norms, copulas, and unary generators), and discusses a wide range of additional properties and generalized constructions. The work aims to help researchers avoid redundancy, facilitate comparisons, and assist practitioners in selecting operators for specific applications. By compiling motivations, properties, and construction techniques, it supports both theoretical analysis and practical deployment of fuzzy implication functions.
Abstract
Fuzzy implication functions are a key area of study in fuzzy logic, extending the classical logical conditional to handle truth degrees in the interval $[0,1]$. While existing literature often focuses on a limited number of families, in the last ten years many new families have been introduced, each defined by specific construction methods and having different key properties. This survey aims to provide a comprehensive and structured overview of the diverse families of fuzzy implication functions, emphasizing their motivations, properties, and potential applications. By organizing the information schematically, this document serves as a valuable resource for both theoretical researchers seeking to avoid redundancy and practitioners looking to select appropriate operators for specific applications.