A MATLAB Toolbox for Fuzzy Relational Calculus in a Variety of Fuzzy Algebras

This paper presents a comprehensive and up-to-date description of a mature software framework for fuzzy relational calculus, developed and extended over more than a decade. The contribution of the paper lies in the unified presentation of theoretical foundations, solution algorithms, and their software implementation, which have not previously been documented in a single coherent form. The presented MATLAB software package is designed to model and solve a broad class of problems in fuzzy relational calculus, including inverse problems for fuzzy linear systems of equations and inequalities, behavior analysis, reduction and minimization of finite fuzzy machines, and optimization of linear objective functions under fuzzy linear system constraints. The implemented algorithms can be applied in areas such as data and software security, modeling and verification of access control policies, anomaly detection, and diagnostics. The software supports a variety of fuzzy algebras, including max–min, min–max, max–product, Goguen, Gödel, and Łukasiewicz algebras, providing tools for both direct and inverse problem resolution. The implementation is based on well-established theoretical results in fuzzy relational calculus, ensuring a robust foundation for exact and efficient computation. Several illustrative examples are provided to demonstrate the applicability of the software in different problem settings.