Incremental controlled relaxation of failing flexible queries

Abstract

In this paper, we discuss an approach for relaxing a failing query in the context of flexible (or fuzzy) querying. The approach relies on the notion of a parameterized proximity relation which is defined in a relative way. We show how such a proximity relation allows for transforming a gradual predicate into an enlarged one. The resulting predicate is semantically close to the original one and it is obtained by a simple fuzzy arithmetic operation. Such a transformation provides the basis for a flexible query relaxation which can be controlled in a non-empirical rigorous way without requiring any additional information from the user. We also show how the search for a non-failing relaxed query over the lattice of relaxed queries can be improved by exploiting the notion of Minimal Failing Sub-queries derived from the failing query.

Appendix

Fuzzy sets theory was introduced by Zadeh (1965) for dealing with the representation of classes or sets whose boundaries are not quite defined. Then, there is a gradual rather crisp transition between the full membership and the full mismatch. Formally, a fuzzy set F on the referential U is characterized by a membership function

$$\mu _F :U\to \left[ {0,\;1} \right],$$

where μ _F (u) represents the grade of membership of u in F. In particular, μ _F (u)=1 reflects full membership of u in F, while μ _F (u) = 0 expresses absolute non-membership in F. When 0 < μ _F (u) < 1, one speaks of partial membership. Two crisp sets are of particular interest when defining a fuzzy set F: the core $\mathcal{C}(F) = \{ u \in U/\mu _F (u) = 1\}$ (it gathers the prototypes of F) and the support $\mathcal{S}(F) = \{ u \in U/\mu _F (u) > 0\}$ (it contains elements which belong to some extent to F). In practice, the membership function associated to F is often of trapezoidal form. Then, F is expressed by the quadruplet (A, B, a, b) where $\mathcal{C}(F) = [A,\,\,B]$ and $\mathcal{S}(F) = [A - a,\,\,B + b]$. For instance, if F represents the fuzzy set of Young persons then, F = (0, 25, 0, 15), see Fig. 3.

Let F and G be two fuzzy sets on the universe U, we say that $F \subseteq G$ iff μ _F (u) ≤ μ _G (u), ∀ u ∈ U. The complement of F, denoted F ^c, is defined by $\mu _{F^c } (u) = 1 - \mu _F (u)$. Furthermore, F ∩ G (resp. F ∪ G) is defined such that $\mu _{F\cap G} \left( u \right)=$ $\mbox{min}\left( {\mu _F \left( u \right)}\right.\!\!,\left.{\mu _G \left( u \right)} \right)$ (resp. $\mu _{F\cup G} \left( u \right)=\mbox{max}\left( {\mu _F \left( u \right)\!,\;\mu _G \left( u \right)} \right))$. Fuzzy intervals (or fuzzy numbers) are fuzzy sets on the real line. Finally, let us recall that if U and V are two referential, a fuzzy relation R from U to V is a fuzzy set on UxV. See Dubois and Prade (2000) for more details about this theory.