Maximal Consistent Subsystems of Max-T Fuzzy Relational Equations

• URL: http://arxiv.org/abs/2311.03059v1
• Date: Mon, 6 Nov 2023 12:41:21 GMT
• Title: Maximal Consistent Subsystems of Max-T Fuzzy Relational Equations
• Authors: Isma\"il Baaj
• Abstract summary: We study the inconsistency of a system of $max-T$ fuzzy relational equations of the form $A Box_Tmax x = b$, where $T$ is a t-norm among $min$, the product or Lukasiewicz's t-norm. For an inconsistent $max-T$ system, we construct a canonical maximal consistent subsystem. We show how to iteratively get all its maximal consistent subsystems.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\Delta = \inf_{c \in \mathcal{C}} \Vert b - c \Vert$ 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.

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