Characterization and reduction of concept lattices through matroid theory

doi:10.1016/j.ins.2014.05.044

Information Sciences

Volume 281, 10 October 2014, Pages 338-354

https://doi.org/10.1016/j.ins.2014.05.044 Get rights and content

Abstract

Concept lattice theory is an efficient tool for data analysis. Reduction of attributes and objects for a context is an important issue in concept lattice theory. We explore the relationships between matroidal spaces and concepts for a given context. With the assistance of matroid theory, we conduct the attribute reduction and object reduction for a context and formulate the concepts for a given context. As a generalization of the linear independence in vector spaces, matroids provide well established platforms for some algorithms such as greedy algorithm.

In this paper, we mainly propose a notion of matroidal space by the family of circuits of a matroid. These relationships characterize the constructions of not only concepts, but also concept lattices. Meanwhile, we demonstrate that reduction of attributes and objects proposed in this paper preserve the original hierarchy order in the concept lattice for a given context.

Introduction

The theory of concept lattices, a method of data analysis, was first proposed by Wille [22] and further embellished by Carpineto and Romano [3], Ganter et al. [4], [5] and Wei et al. [19]. A concept lattice is an ordered hierarchy that is defined by a binary relationship between objects and attributes in a data set, and as such, incorporates certain general and specific concepts. As an efficient tool of data analysis and knowledge processing, the concept lattice theory has been applied to fields such as knowledge engineering, data mining, information searches, and software engineering (see Carpineto and Romano [3] and Ganter et al. [4]).

Gross data reduction is needed for knowledge discovery and data mining and, towards this end, concept lattice analysis has achieved recent and abundant results in both these fields. Reducing the size of many concept lattices is necessary to gain useful results. Ganter and Wille [5] proposed reducible attributes and reducible objects from the viewpoint of shortening lines or rows. By searching for a minimal subset of attributes so that the new concept lattice is isomorphism with the raw concept lattice, Zhang et al. [23] presented another way to reduce attributes. Relationships between reducts and concept lattices are discussed also such as [4], [5], [19], [23]. Based on the above Ganter et al. and Zhang et al. for reducts of concept analysis respectively, there are many other methods to reduce the size of the concept lattices using various mathematical theories such as fuzzy sets [7], [9], Boolean algebra [13], Galois connection [10], matrix [16], and rough sets [1], [8].

After analyzing the methods in [5], [7], [8], [9], [10], [13], [16], [23], we find that every method of them comes from the combination of a mathematical theory and concept lattices. Hence, if we hope to develop the reducts of concept lattices, we should explore a mathematical structure, which is different from the known in the study on the reducts of concept lattices, to apply in the research. To pursuit this aim, we notice the following points (1.1) and (1.2):

(1.1)
In combinatorics, a branch of mathematics, matroid theory was introduced by Whitney [21]. This theory provides a unifying abstract treatment of dependence in linear algebra and graph theory. Matroids also play a fundamental role in combinatorial optimization and are widely used in greedy algorithms (see Oxley [14] and Welsh [20]). Mao [11] deals with the relationships between matroids and concept lattices. This same author utilizes some properties of concept lattices to research the axioms of matroids [12].
(1.2)
Pawlak [15] presents the definition of an ‘information system’. Comparing his definition to Ganter and Wille’s [5] definition of a ‘context’, we find that a context is a particular information system. Tang et al. [17] and Wang et al. [18] studied information systems and both used attribute reduction with matroid theory. We can see that many algorithms favored for the structures of concept lattices and attribute reductions have the same general ideas associated with greedy algorithms. These results imply that matroid theory may enlighten us on the use of attribute reduction when addressing concept lattices.

The above (1.1) and (1.2) shows that the use of matroids to achieve context reduction, based on attribute or object reduction, would be very useful in the study of concept lattices. To date however, this research topic is unexplored. If we could use matroids to reduce concept lattices, then the corresponding results between knowledge reduction in concept lattices and matroid theory could also be interchangeable. We can confirm that this interchangeability will be good for both concept lattices and matroids, and equally useful in searching methods of concepts in a context.

In this paper, we attempt to make a contribution to the study of concept lattices by using matroids. As we know, it is somewhat difficult to understand matroids directly. And this difficulty also arises when we consider the combination of matroids and concept lattices. So, in order to give an intuitive interpretation to this combination, we will first study it from the connection between matroids and contexts. By applying the axioms of circuits of matroids, we present the notion of a-bi-circuit and formulate the concepts in a context based on this notion. Next, we provide the notion of matroidal space based on the circuit axioms of a matroid and the notion of concepts for a context. Additionally, with the assistance of matroidal spaces, we characterize every concept in a context. Finally, for the two definitions of attribute reduction, as per Ganter and Zhang respectively, we characterize attribute reduction and object reduction based on matroidal spaces.

The rest of this paper is organized in five sections as follows. In Section 2, we review some basic facts relative to concept lattices and matroids. In Section 3, we discuss the relationships between matroids and concept lattices and provide a notion of matroidal space. Using these relationships, with the assistance of matroidal spaces, we present some characterizations of concept lattices. In Section 4, according to the definition of reduction in a context cf. Ganter and Zhang respectively, we deal with the reduction of concept lattices with matroidal spaces. The final Section 5, concludes the paper.

We declare that in this paper,

(1)
all the discussions are finite;
(2)
all the notions and properties for lattice theory are referred to Birkhoff [2] and Grätzer [6];
(3)
if two lattices $L_{1}$ and $L_{2}$ are isomorphic, then it is in notation $L_{1} ≅ L_{2}$ .

Section snippets

Preliminaries

As stated above, this section introduces some basic facts of concept lattices and matroids.

Matroidal spaces

This section will discuss the relationships between a context and matroids, provide the notion of matroidal spaces, and present some characterizations of concept lattices with matroidal spaces.

Through an example, we show the process of discovering the concept lattice for a context.

Example 3.1

A context $K = (O, P, I)$ is provided in which $O = {a_{1}, a_{2}, a_{3}, a_{4}}, P = {b_{1}, b_{2}, b_{3}, b_{4}, b_{5}}$ and the incidence relation shown in Table 1.

According to Table 1, for every $A \subseteq O$ and $B \subseteq P$ , using Definition 2.1.1, we may learn $A^{'}$ and $B^{'}$ . We

Reducing a context based on matroidal spaces

Reducing a context is a very important issue in concept lattices. The basic facts of reducing have been introduced in Subsection 2.1. Based on the two famous definitions for reducing—Ganter’s and Zhang’s, we will consider reducing a context with the assistance of matroidal spaces.

Conclusion

The matroidal space is proposed. The issues for the structure of concept lattice and attribute reductions are discussed with the matroidal approach. We review, in order, the construction of a concept lattice, two ways for defining attribute reduction and finally, object reduction. Next, we outline the above three under a matroid framework. The notation of a-bi-circuit is defined in order to convert a context into several matroids directly. Thus, the reduct of a context is described through

Acknowledgements

This research is supported by NSF of China (61073121, 11101115, 61202178) and NSF of Hebei Province (A2013201119, A2014201033, F2012402037).

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Characterization and reduction of concept lattices through matroid theory

Abstract

Introduction

Section snippets

Preliminaries

Matroidal spaces

Reducing a context based on matroidal spaces

Conclusion

Acknowledgements

Euro. J. Oper. Res.

Euro. J. Oper. Res.

Knowl.-Based Syst.

Appl. Soft Comput.

Knowl.-Based Syst.

Knowl.-Based Syst.

Lattice Theory

Concept Data Analysis: Theory and Applications

Formal Concept Analysis: Foundations and Applications

Formal Concept Analysis: Mathematical Foundations