Handling the inconsistency of systems of $\min\rightarrow$ fuzzy relational equations

TL;DR

This paper analyzes the inconsistency of systems of min-implication fuzzy relational equations, providing formulas for Chebyshev distances to measure inconsistency and exploring their properties across different residual implicators.

Contribution

It introduces analytical formulas for Chebyshev distances in fuzzy relational systems and characterizes their bounds and attainability for various residual implicators.

Findings

Chebyshev distance is the lower bound of solutions for these systems.

For Godel implication, the Chebyshev distance may be an infimum.

For Goguen and Lukasiewicz implications, the Chebyshev distance is always a minimum.

Abstract

In this article, we study the inconsistency of systems of $\min-\rightarrow$ fuzzy relational equations. We give analytical formulas for computing the Chebyshev distances $\nabla = \inf_{d \in \mathcal{D}} \Vert \beta - d \Vert$ associated to systems of $\min-\rightarrow$ fuzzy relational equations of the form $\Gamma \Box_{\rightarrow}^{\min} x = \beta$, where $\rightarrow$ is a residual implicator among the G\"odel implication $\rightarrow_G$, the Goguen implication $\rightarrow_{GG}$ or Lukasiewicz's implication $\rightarrow_L$ and $\mathcal{D}$ is the set of second members of consistent systems defined with the same matrix $\Gamma$. The main preliminary result that allows us to obtain these formulas is that the Chebyshev distance $\nabla$ is the lower bound of the solutions of a vector inequality, whatever the residual implicator used. Finally, we show that, in the case of the $\min-\rightarrow_{G}$ system, the Chebyshev distance $\nabla$ may be an infimum, while it is always a minimum for $\min-\rightarrow_{GG}$ and $\min-\rightarrow_{L}$ systems.