On a fuzzy bipolar relational algebra

Abstract

This paper presents an extension of relational algebra suitable for the handling of bipolar concepts. The type of queries considered involves two parts: a first one which expresses a (possibly flexible) constraint, and a second one that corresponds to a (possibly flexible) wish. The framework considered is that of bipolar fuzzy relations where each tuple is associated with a pair of satisfaction degrees.

Introduction

The idea of introducing preferences into queries is gaining more and more attention in the database community. In this paper, we focus on the fuzzy-set-based approach to preference queries, which relies on the use of fuzzy set membership functions that describe the preference profiles of the user on each attribute domain involved in the query. Then, satisfaction degrees associated with elementary conditions are combined using a panoply of fuzzy set connectives, which go much beyond conjunction and disjunction.

A complementary concept is that of bipolarity in general and its application to queries in the context of databases and information systems. Bipolarity refers to the propensity of the human mind to reason and make decisions on the basis of positive and negative affects [22], [23]. Positive information states what is possible, satisfactory, permitted, desired, or considered as being acceptable. On the other hand, negative statements express what is impossible, rejected, or forbidden. Negative preferences correspond to constraints, since they specify which values or objects have to be rejected (i.e., those that do not satisfy the constraints), while positive preferences correspond to wishes, as they specify which objects are more desirable than others (i.e., satisfy user wishes) without rejecting those that do not meet the wishes.

Three types of bipolarity have been pointed out [23]. The simplest type, called symmetric univariate bipolarity, uses a bipolar scale whose negative and positive parts are the mirror images of each other. The second type of bipolarity, termed symmetric bivariate, refers to the use of two separate unipolar scales (one for the positive affects, the other one for the negative affects) still pertaining to the same information, with generally a duality relation putting the scales in symmetric correspondence. The third type of bipolarity, called asymmetric, takes place when dealing with two unrelated kinds of information in parallel (see also [3] where this type of bipolarity is described and used). In the rest of this paper, asymmetric (also called heterogeneous) bipolarity is considered, and positive and negative poles are assumed to refer to potentially different notions (attributes). More precisely, we will deal with queries made of two poles, one meant as a constraint—denoted by C—and the other acting as a wish—denoted by W—, and a pair (C, W) is interpreted as: “C and if possible W”. In the situation considered later on, the two components of a query, although they can be assessed on a same scale (true/false, or the unit interval, or a qualitative scale), are not of the same nature and it is convenient to specify how any pair of elements (tuples or objects) are compared depending on their scores with respect to the constraint and the wish. Let us recall, however, that inside a complex constraint (resp. wish), the fuzzy-set-based approach requires the elementary preferences to be commensurable. A commonly made choice (see in particular [23]) for interpreting a bipolar condition consists in discriminating between two objects x and y using first the constraint, then if needed (i.e., if x and y are not distinguishable on the constraint) using the wish. In what follows, this point of view is chosen and a lexicographic order is used. If (C(x), W(x)) and (C(y), W(y)) denote the scores of x and y with respect to the constraint C and the wish W, one has:

$$x\succ y\iff (C(x)>C(y))\mathrm{or}(C(x)=C(y)\mathrm{and}W(x)>W(y))$$

where x ≻ y means that x is preferred to y. A consequence is the fact that an object which is beaten on the constraint cannot win even if it is significantly better on the wish.

In this paper—which is a much extended version of [14]—, our aim is to propose an extension of relational algebra in order to have a querying framework capable of handling bipolar fuzzy queries and relations. The set of operators we propose generalizes the “fuzzy relational algebra” that has been previously proposed to handle non-bipolar fuzzy queries and relations (see, e.g., [5]), which itself generalized classical relational algebra.

The rest of the paper is structured as follows. In Section 2, we recall some basic notions about database fuzzy querying, as well as the concept of bipolarity in this context. Section 3 is devoted to a presentation of the extended relational algebraic operators in the framework of bipolar fuzzy relations. Section 4 deals with query equivalences whereas Section 5 is devoted to implementation aspects. In Section 6, some related works are briefly discussed. Finally, Section 7 concludes the paper and outlines some perspectives for future work.