Galois connection, formal concepts and Galois lattice in real relations: application in a real classifier

Abstract

In this paper, we introduce the notion of a real set as an extension of a crisp and a fuzzy set by using sequences of intervals as membership degrees, instead of a single value in [0,1]. We also propose, to extend the notion of Galois connection in a real binary relation as well as the notions of rectangular relation, formal concept and Galois lattice. We present finally a real classifier based on this mathematical foundation.

Introduction

Traditional fuzzy sets use a real numbers in [0,1] as a membership degree of elements to a set (Zadeh, 1965). This description is not always adequate since an imprecision could be attached to this degree for several reasons. In fact, in real life, people are reluctant to describe their degrees of certainty by exact numbers. For example, it is difficult to distinguish between the degree of belief 0.7 and 0.71. Moreover, a major difficulty is encountered in some fuzzy systems, since an imprecision could be attached to the used membership functions when several experts define them differently, or when we obtain several representations by different learning processes. A more adequate way of describing each degree of certainty is by using intervals of possible values in place of exact values. Also, sequences of real intervals can be more adequate when representing several separate imprecise possibilities for the happening of an event. For example, we can specify the real intervals $\text{[7}\text{a}\text{.}\text{m}\text{.,9}\text{a}\text{.}\text{m}\text{.]}$ and $\text{[17}\text{p}\text{.}\text{m}\text{.,19}\text{p}\text{.}\text{m}\text{.]}$ as the traffic jam periods during a journey, or the real intervals $\text{[}\text{December}\text{15}\text{,}\text{January}\text{15}\text{]}$, $\text{[}\text{February}\text{20}\text{,}\text{March}\text{10}\text{]}$ and $\text{[}\text{May}\text{15}\text{,}\text{August}\text{15}\text{]}$ as the red periods for the touristic season. So, in this paper, we are introducing the notion of a real set (Jaoua and Elloumi, 2000) as an extension of a crisp and a fuzzy set by using sequences of real intervals and we will present the basic set theoretical foundation of real sets and real relations. We are also proposing to extend the notion of Galois connection in real binary relation, as well as the notions of rectangular relation, formal concept and Galois lattice (Riguet, 1948; Everett, 1944) by considering a strict as well as a large sense.

This mathematical foundation of the formal concept analysis has been used, with a high intensity, to resolve several structural aspects of software, data and information engineering (Ganter and Wille, 1999; Wille, 1982). More specifically, formal concept analysis is offering standard and uniform tools for data organization (Mineau and Godin, 1995; Wille, 1992), knowledge extraction (Stumme et al., 1998; Wille, 1992; Wille, 1989), automatic classification and decision making (Godin et al., 1993; Liquière and Mephu Nguifo, 1998). As a matter of fact, regular crisp binary relations study has exhibited maximal rectangles as the atomic concept for decomposing any binary relation (Belkhiter et al., 1994a). The study of regular fuzzy binary relation (Ounalli and Jaoua, 1996) has permitted to define a “fuzzy concept” (Belohlavek, 1998; Wolff, 1999; Jaoua et al., 2000) as the atomic relation to decompose any fuzzy relation, and can be successfully used for the fuzzy classifier and decision making (Elloumi and Jaoua, 2000). We finally discover that we can extend Galois connection to organize general n-ary relations or real databases (Jaoua and Elloumi, 2000) by considering either a strict or a large sense.

Hence, this paper is organized as follows. In Section 2, the fundamental operations and properties of real sets are defined. In Section 3, the mathematical definitions and properties of a classical Galois lattice structure are recalled. In Section 4, the notion of a real maximal rectangle is defined. Then this lattice structure is mathematically extended to real binary relations and the notion of real Galois connection is defined, using sequences of intervals which are associated to each property, relatively to some precision. In Section 5, application of real Galois connection to the decision making, using a real classifier, is given. More specifically, using some specific illustrations and experimental results, we justify the use of the strict Galois connection in case it improves the quality of the classification process. Section 6 concludes this paper and points out some future work.