Flexible queries in relational databases - The example of the division operator

Allowing for flexible queries enables database users to express preferences inside minimal requirements, and, if necessary, priorities inside compound queries. In other words, clear-cat properties can be refined by ordering the interpretations compatible with them, according to user's preferences. Often the representation of these preferences can be viewed as modelling linguistic-like terms in requests. In this paper, the theoretical issues raised by the introduction of flexible queries are studied in the case of the division operator, in the framework of fuzzy sets and possibility theory. The notion of division is well-known in the context of regular relations and the extension of this operation to fuzzy relations (induced by the flexible queries) is investigated. Several types of extended divisions can be envisaged, depending on the meaning of the grades attached to the tuples of the fuzzy relations (degree of fulfillment of gradual properties, level of importance of components in a query, or uncertainty pervading data). We focus on the first two meanings which are associated with different multiple-valued logic implications and we examine their properties and their expression in the framework of an extended SQL-like language where no specific construct for the division is available.