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Reducing fuzzy relation equations via concept lattices

David Lobo, Víctor López-Marchante, Jesús Medina

TL;DR

This paper has taken into advantage the relationship between Fuzzy Relation Equations and Concept Lattices to introduce a procedure to reduce a FRE, without losing information, and introduces a novel method for computing approximate solutions of unsolvable FRE related to a (real) dataset with uncertainty/imprecision data.

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.

Reducing fuzzy relation equations via concept lattices

TL;DR

This paper has taken into advantage the relationship between Fuzzy Relation Equations and Concept Lattices to introduce a procedure to reduce a FRE, without losing information, and introduces a novel method for computing approximate solutions of unsolvable FRE related to a (real) dataset with uncertainty/imprecision data.

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.
Paper Structure (7 sections, 13 theorems, 70 equations, 4 figures)

This paper contains 7 sections, 13 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multi-adjoint FRE reduction
  4. Approximation of unsolvable multi-adjoint FRE
  5. Reduction and approximation in a dual multi-adjoint FRE
  6. Reducing and approximating a FRE with max-min composition
  7. Conclusions and future work

Key Result

Proposition 1

Let $(U,V,R,\sigma)$ be the multi-adjoint context associated with the multi-adjoint FRE $R\odot_\sigma X=T$. A fuzzy relation $X\in L_2^{V\times W}$ is a solution of the FRE if and only if we obtain that for all $w\in W$, where $X_w$ and $T_w$ are the columns of $X$ and $T$, respectively, that is, $X_w(v)=X(v,w)$ and $T_w(u)=T(u,w)$, for all $u\in U$, $v\in V$ and $w\in W$.

Figures (4)

  • Figure 1: Concept lattice associated with FRE \ref{['exp:reducSystem']}
  • Figure 2: Sublattice associated with FRE \ref{['exp:reducSystem']}
  • Figure 3: Sublattice associated with FRE $R_{Y_3}\odot_\sigma X=T_{Y_3}$
  • Figure 4: Part of the concept lattice associated with FRE \ref{['exp:example_approx_equation']}

Theorems & Definitions (36)

  • definition 1: Cornejo2021
  • definition 2: ins-medina
  • definition 3: ins-medina
  • definition 4: DaveyPriestley
  • definition 5: ar:ins:2015
  • definition 6: ar:ins:2015
  • definition 7: dm:mare
  • definition 8: dm:mare
  • definition 9: dm:mare
  • Proposition 1: dm:mare
  • ...and 26 more