Maximal Consistent Subsystems of Max-T Fuzzy Relational Equations
This work addresses inconsistency issues in fuzzy relational equations, which is an incremental advancement in the domain of fuzzy logic and systems theory.
The authors tackled the problem of inconsistency in systems of max-T fuzzy relational equations by constructing a canonical maximal consistent subsystem and providing an efficient method to obtain all consistent subsystems for max-min systems, achieving results based on analytical formulas for Chebyshev distance.
In this article, we study the inconsistency of a system of $\max-T$ fuzzy relational equations of the form $A \Box_{T}^{\max} x = b$, where $T$ is a t-norm among $\min$, the product or Lukasiewicz's t-norm. For an inconsistent $\max-T$ system, we directly construct a canonical maximal consistent subsystem (w.r.t the inclusion order). The main tool used to obtain it is the analytical formula which compute the Chebyshev distance $Δ= \inf_{c \in \mathcal{C}} \Vert b - c \Vert$ associated to the inconsistent $\max-T$ system, where $\mathcal{C}$ is the set of second members of consistent systems defined with the same matrix $A$. Based on the same analytical formula, we give, for an inconsistent $\max-\min$ system, an efficient method to obtain all its consistent subsystems, and we show how to iteratively get all its maximal consistent subsystems.