On the use of irreducible elements for reducing multi-adjoint concept lattices☆
Introduction
Real databases are usually very large and give rise to complex concept lattices, from which extracting conclusions is a really difficult task. This fact highlights the importance of obtaining new procedures which lets us reduce the size of concept lattices and preserve the most relevant information of the database. Specifically, the survey of these strategies have become a key research topic in (fuzzy) Formal Concept Analysis (FCA) [20].
In the literature, we can find different mechanisms to achieve this aim. However, most of them alter the original concepts, such as the use of hedges [2], [14]. Other methodologies change the original context (granular computing [13]) or consider a restrictive setting, for instance, they do not use fuzzy subsets of objects and attributes but a crisp subset of objects and a fuzzy subset of attributes, as in [17].
With the idea of providing a general framework in which the different approaches stated above could conveniently be accommodated, a new approach of formal concept analysis in which the philosophy of the multi-adjoint framework was applied and the multi-adjoint concept lattices were introduced in [18], [19]. In this frame, adjoint triples [7] are the main building blocks of a multi-adjoint concept lattice and so, are used as basic operators to carry out the calculus. The flexibility provided by these triples allows us to consider a general non-commutative and non-associative environment. Moreover, different degrees of preferences related to the set of objects and attributes can easily be established in this general concept lattice framework.
On the other hand, the fuzzy notion of attributes – fuzzy-attributes – which are fuzzy subsets of attributes, are very important in representation theory of fuzzy formal concept analysis. In [9], a characterization of the meet-irreducible elements of a concept lattice from the fuzzy-attributes was presented. This characterization will play a fundamental role in the results shown in this paper, since the irreducible elements form the basic information of a relational system and consequently, they must be considered in the reduction procedures.
This paper introduces new properties about the fuzzy-attributes and presents a mechanism in order to reduce the size of multi-adjoint concept lattices from their meet-irreducible elements and a threshold given by the user. This threshold represents the least truth-value of a fuzzy-attribute in order to be considered in the computation of the concept lattice. From this reduction, a sublattice of the original concept lattice is obtained (called irreducible α-cut concept lattice), which implies that the original concepts are not modified and the most representative knowledge is preserved. However, in order to obtain the concepts of the reduced concept lattice from the initial decomposition in meet-irreducible elements and the threshold, the original concept lattice needs to be distributive. Although this is a small restriction, since other efficient possibilities exist to obtain the concepts of the reduced concept lattice, we have introduced another reduction mechanism, in which all the fuzzy-attributes are considered instead of the meet-irreducible elements, and the previous restriction disappears. Finally, a comparison with related strategies is presented.
The paper is organized as follows: Section 2 recalls preliminary notions and results, together with the multi-adjoint concept lattice framework; Section 3 presents several properties about fuzzy-attributes that generate meet-irreducible elements of a concept lattice. A new reduction mechanism based on meet-irreducible elements and a given threshold, together with several properties, is introduced in Section 4. Section 5 presents another strategy to reduce the size of multi-adjoint concept lattices providing the α-cut concept lattices. Lastly, a comparison with other related works is presented in Section 6. The paper finishes with several conclusions and future challenges.
Section snippets
Preliminaries
This first section provides some necessary definitions and results in order to make the paper self-contained. Definition 1 Given a lattice , such that are the meet and the join operators, and an element verifying If L has a top element , then . If , then or , for all .[10]
we call x meet-irreducible (-irreducible) element of L. Condition (2) is equivalent to
- .
If and , then , for all .
Hence, if x is -irreducible, then it cannot be represented as the infimum of the
Fuzzy-attributes generating -irreducible elements
Several properties about the meet-irreducible elements of a multi-adjoint concept lattice generated from a fuzzy-attribute will be presented in this section.
The following technical property will play an important role in the proofs of the main results of this paper. Moreover, it will be used to prove other interesting properties in this section. Specifically, it shows how the extensions of the concepts can be obtained from the fuzzy-attributes. Proposition 21 Let g be the extension of a concept of such that
Reducing the size of multi-adjoint concept lattices from -irreducible elements
One of the most important problems in FCA is to decrease the size of the concept lattices [2], [13], [15], [16], [17]. Nevertheless, several of the existing mechanisms modify the information given by the concepts. This section uses the previous characterization in order to provide a new procedure to reduce the size of the multi-adjoint concept lattices, without modifying the information given by the context. This reduction begins from the fuzzy-attributes that really can represent an attribute,
The α-cut concept lattices
This section provides another new mechanism in order to reduce the size of a multi-adjoint concept lattice, conserving the information given in the considered database. In this case, instead of considering , the procedure considers all the fuzzy attributes with a representative value for the attribute a, that is, they might not be -irreducible elements.
Therefore, given a threshold α, we will consider the concepts whose fuzzy-attributes satisfy that , that is, we regard the
Related works
There exist several mechanisms that reduce the size of a concept lattice following different philosophies, however almost all of them modify the concept-forming operators and so, the original concepts. For example, in [2], [14] the proposed mechanisms consider hedges and so, the reduced concept lattice has new concepts, that is, it is not a sublattice of the original one. The same applies in [13], which considers granular computing.
Recently, a reduction in multi-adjoint concept lattices has
Conclusions and future work
Based on the characterization of the -irreducible elements of a multi-adjoint concept lattice using fuzzy-attributes, we have introduced two strategies in order to reduce the size of multi-adjoint concept lattices, with advantageous properties. The main one is that each reduced concept lattice is a sublattice of the original one and so, the initial information given by the concepts is neither altered nor modified and the most relevant knowledge is considered.
Moreover, we have shown that, to
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Partially supported by the Spanish Science Ministry Project TIN2012-39353-C04-04.