Granular variable precision L-fuzzy rough sets based on residuated lattices
Introduction
The concept of rough sets was introduced by Pawlak in 1982 as a mathematical approach to deal with uncertain, incomplete and vague information [20]. With the development of almost 30 years, rough set theory has been extended in many different aspects [4], [5], [16], [17], [19], [21], [22], [23], [25], [27], [29], [30], [31], [36]. A major adhibition of rough set theory is to reduce the redundant conditional attributes in databases thereby improving the performance of applications in a large amount of areas such as storage, accuracy and so forth [32]. In addition, it has been successfully applied in many areas including approximate classification, machine learning, conflict analysis, pattern recognition, data mining and automated knowledge acquisition. However, Pawlak gave the notion of rough sets on the basis of equivalence relations and crisp sets, which are shown to be so strict that it limits the applicability of Pawlak's rough set model.
In recent years, a further innovation development of rough set theory is the variable precision rough set model introduced by Ziarko [37], which is efficient to deal with indeterminate information by an error-tolerance capability. With the development of almost 20 years, the variable precision rough set theory has been expanded in various aspects, see, e.g., [10], [12]. By setting a confidence threshold value β, this model can not only solve classification problems with uncertain data, but also relax the strict boundary definition of Pawlak's rough set to make the model suitability [28], [32].
In 2011, Chen et al. [5] proposed a new method to investigate fuzzy rough sets by combining the granular computing [1], [7] with fuzzy rough sets. Meanwhile, in [32], Yao et al. pointed out that some of fuzzy rough sets (see, e.g., [13], [14], [35]) are still sensitive to mislabeled samples and others (see, e.g., [8], [18]) only take into account the relative errors. Thus, in order to amend these deficiencies, in 2014, Yao et al. [32] studied the variable precision -fuzzy rough sets based on fuzzy granules when the fuzzy relations are considered as fuzzy ⁎-similarity relations. However, from the mathematical point of view, the fuzzy ⁎-similarity relations seem such strict that they limit the discussion of the variable precision -fuzzy rough sets, although they are useful in attribute reduction based on variable precision -fuzzy rough sets. Thus, recently, Wang and Hu [24] studied the granular variable precision fuzzy rough sets with general fuzzy relations from the theoretical perspective. But, in [24], the truth values set was supposed to be the unit interval . In addition, it is well known that complete lattice is a useful and wider structure than the unit interval. Hence, in this paper, we discuss the granular variable precision L-fuzzy rough sets based on arbitrary residuated and co-residuated lattices, which can be regarded as the extension of the works of [24] and [32] from the theoretical point of view. Meanwhile, we obtain the equivalent expressions for the approximation operators on arbitrary L-fuzzy relations, which can make sure to calculate the approximation operators efficiently.
The rest of this paper is organized as follows. In Section 2, we recall some fundamental concepts and related properties about L-fuzzy implications and L-fuzzy coimplications based on complete residuated lattices and complete co-residuated lattices, respectively. In Section 3, we introduce the granular variable precision L-fuzzy rough sets based on residuated and co-residuated lattices with L-fuzzy relations and give equivalent expressions of the lower and upper approximation operators. In Section 4, we investigate some basic properties of the granular variable precision L-fuzzy rough sets on residuated and co-residuated lattices and characterize the granular variable precision L-fuzzy rough sets on different L-fuzzy relations. In the final section, we give the conclusions of our research.
Section snippets
Preliminaries
A complete residuated lattice [6], [26] is a pair subjecting to the following conditions:
- (1)
L is a complete lattice with a top element 1 and a bottom element 0;
- (2)
is a commutative monoid;
- (3)
for all and .
A complete co-residuated lattice is a pair
Granular variable precision L-fuzzy rough sets based on residuated and co-residuated lattice
In this section, firstly, we introduce the granular variable precision L-fuzzy rough sets based on residuated and co-residuated lattices. And then, we obtain the equivalent expressions of the lower and upper approximation operators with L-fuzzy implications and L-fuzzy coimplications, respectively. Finally, at the end of this section, we discuss the duality of the lower and upper approximation operators.
Definition 3.1 Let R be an L-fuzzy relation on X, and . Then two L-fuzzy
Some properties of the granular variable precision L-fuzzy rough sets based on residuated and co-residuated lattice
In this section, we use the equivalent expressions of the lower and upper approximation operators to characterize the granular variable precision L-fuzzy rough sets on diverse L-fuzzy relations, such as serial, inverse serial, reflexive and &-transitive ones.
By Remark 3.5, the two L-fuzzy sets and are pretty important to calculate generalized granular variable precision lower and upper approximation operators, respectively. Thus, we start this section with the discussion of the properties
Conclusions
In this paper, we generalize the granular variable precision fuzzy rough sets with general fuzzy relations to the granular variable precision L-fuzzy rough sets based on residuated and co-residuated lattices with the L-fuzzy relations and propose some properties of the approximation operators. In particular, we obtain the equivalent expressions of the approximation operators on arbitrary L-fuzzy relations. In such way, the lower and upper approximation operators can be efficiently calculated.
Acknowledgements
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant Nos. 11571010 and 61179038).
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