Reducing fuzzy relation equations via concept lattices

TL;DR

This paper leverages the relationship between fuzzy relation equations and concept lattices to develop a method for reducing equations and efficiently computing solutions, including approximate solutions for unsolvable cases with uncertain data.

Contribution

It introduces a novel reduction procedure for fuzzy relation equations using concept lattices and a new method for approximate solutions in uncertain datasets.

Findings

Reduces the computation of solution sets for solvable FREs

Detects redundant equations using attribute and object-oriented lattices

Provides a new approach for approximate solutions in uncertain data scenarios

Abstract

This paper has taken into advantage the relationship between Fuzzy Relation Equations (FRE) and Concept Lattices in order to introduce a procedure to reduce a FRE, without losing information. Specifically, attribute reduction theory in property-oriented and object-oriented concept lattices has been considered in order to present a mechanism for detecting redundant equations. As a first consequence, the computation of the whole solution set of a solvable FRE is reduced. Moreover, we will also introduce a novel method for computing approximate solutions of unsolvable FRE related to a (real) dataset with uncertainty/imprecision data.