Abstract
This article considers the bipolar approach to define database queries expressing users’ preferences (flexible queries). An algebraic framework for the definition of flexible queries of relational databases using fuzzy bipolar conditions of type and-if-possible and or-else has been considered. This paper defines some qualitative calibrations of such queries to specify a minimal quality of answers and to reduce their number. Different operators (extended \(\alpha \)-cuts) are defined and studied in this article. They can apply on the set of answers to express a qualitative calibrations of bipolar fuzzy queries. Some properties of these extended \(\alpha \)-cuts are pointed out and some of their applications for query evaluation are shown.
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Notes
- 1.
A fuzzy set A defined on an universe U is a set whose membership is gradual and belongs to [0, 1]. Membership degrees are given by a membership function \(\mu _A\) (\(\forall x \in U\), the more \(\mu _A(x)\) is high, the more x belongs to A, when \(\mu _A(x)=1\), x fully belongs to A, when \(\mu _A(x)=0\), x does not belong to A).
- 2.
A fuzzy condition is defined by a fuzzy set. The membership degree expresses the preference (the more it is high for a given element, the more the element is preferred).
- 3.
The inclusion of two fuzzy sets A and B from the same universe U is defined by: \(A \subseteq B \Leftrightarrow \forall x \in U, \mu _A(x) \le \mu _B(x)\).
- 4.
Both types of conditions are different because it is possible to show [14] that the same values of satisfaction for w and c does not lead to a same ordering whether it is an and-if-possible condition or an or-else condition.
- 5.
The \(\alpha \)-cut of a fuzzy set F defined on the universe U is the regular set \(F_{\alpha }\) made of its elements having a membership degree greater than (or equal to) the level \(\alpha \): \(F_{\alpha } = \{ t \in U | \mu _{F}(t) \ge \alpha \}\).
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Akaichi, J., Liétard, L., Rocacher, D., Slama, O. (2015). On the Qualitative Calibration of Bipolar Queries. In: Esposito, F., Pivert, O., Hacid, MS., Rás, Z., Ferilli, S. (eds) Foundations of Intelligent Systems. ISMIS 2015. Lecture Notes in Computer Science(), vol 9384. Springer, Cham. https://doi.org/10.1007/978-3-319-25252-0_10
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