Rough-set-driven approach for attribute reduction in fuzzy formal concept analysis

Abstract

The reduction of the set of attributes is an important preliminary challenge in order to obtain information from knowledge systems. Two remarkable formal tools for extracting such information are Rough Set Theory (RST) and Formal Concept Analysis (FCA), as well as their fuzzy generalizations. This work introduces a new method to reduce attributes in Fuzzy FCA considering the reduction philosophy given in RST and studies its main properties. This method allows us to carry out a deeper study of the relation between these two theories. Moreover, the proposed methodology has been compared with other existing reduction mechanisms.

Introduction

The study of the knowledge stored in databases is one of the most important goals in several research fields, which has produced the necessity of developing mathematical tools to manage the collected information. In addition, to deal with imprecise or incomplete information is crucial in many knowledge systems. Formal Concept Analysis (FCA) [50] and Rough Set Theory (RST) [47] are two widely studied mathematical theories, devoted to obtain information from relational databases that contain uncertainty.

Although it is easy to notice that both theories have several common aspects, the philosophy that underlies the information management strategy employed by each theories is well differentiated. On the one hand, FCA groups the knowledge in small pieces, called concepts, which contain a subset of objects, as well as, the subset of shared attributes by that objects. All these information pieces can be ordered, obtaining the algebraic structure of a concept lattice. On the other hand, RST was born with the intention of defining vague sets or sets which contain imprecisions, providing approximations of them.

Originally, these theories were developed in a crisp environment. Later, the notions and results given in FCA and RST were studied in more general frameworks which considered fuzzy environments. Particularly, these ideas were generalized considering the multi-adjoint framework [23], [25], [41], [44], embedding other fuzzy approaches, such as [4], [13], [24], [36], [42]. This framework presents interesting advantages such as the consideration of adjoint-triples as operators, which do not need to be commutative or associative. Another important benefit of the multi-adjoint framework is the capability of considering several adjoint-triples allowing to establish in a simple way different degrees of preference over the set of attributes and objects of the considered relational system.

On the other hand, the size of the database directly affects the efficiency in the management and extraction of information in RST and FCA, which reveals the relevance of the study and development of techniques that allow to reduce databases. In fact, this reduction is a widely study issue in both theories, separately [3], [7], [19], [39], [40]. In addition, in the literature, several papers can be found that establish the existing connections between these two mathematical tools, considering the classical framework [14], [45], [49].

This paper introduces a novel mechanism to reduce the set of attributes in the fuzzy general framework of multi-adjoint concept lattices, considering the RST philosophy with tolerance relations. This study extend the one introduced in [12] in a fuzzy case, shown that the same properties are not satisfied. Besides that, the proposed mechanism has been enriched with other interesting properties, showing that the new procedure also keeps important features. One of these properties is that the reduction is directly apply to the context and the whole concept lattice is not needed to be computed. Moreover, the main structure, based on the join-irreducible elements, is preserved including no new join-irreducible element after the reduction procedure. All these notions and results are illustrated with examples and they are related to other mechanisms. Therefore, we present a connection between a general version of rough set theory, considering tolerance relations and multi-adjoint formal concept analysis, from the point of view of the attribute reduction. This reduction can straightforwardly apply to other fuzzy frameworks such as [1], [2], [6], [13], [44].

This paper is organized as follows: some necessary definitions and results are recalled in Section 2. A new reduction mechanism in a fuzzy environment is presented in Section 3, together with properties and illustrative examples. A comparison among the introduced mechanism and other reduction mechanisms is included in Section 4. Finally, Section 5 shows some conclusions and challenges for future work that have arisen in view of the obtained results.