Codd's relational model describes just one possible world. To better cope with incomplete information, extended database models allow several possible worlds. Vague tables are one such convenient extended model where attributes accept sets of possible values (e.g., the manager is either Jill or Bob). However, conceptual database design in such cases remains an open problem. In particular, there is no canonical definition of functional dependencies (FDs) over possible worlds (e.g., each employee has just one manager). We identify several desirable properties that the semantics of such FDs should meet including Armstrong's axioms, the independence from irrelevant attributes, seamless satisfaction and implied by strong satisfaction. We show that we can define FDs such that they have all our desirable properties over vague tables. However, we also show that no notion of FD can satisfy all our desirable properties over a more general model (disjunctive tables). Our work formalizes a trade-off between having a general model and having well-behaved FDs.
Work on fuzzy databases is of a different nature, in that database values are not considered atomic entities, but they are fuzzy (membership) functions over some base set @cite_10 . For instance, given a domain , the fuzzy values young and infant can be seen as functions giving a degree of membership to each value in Age . These functions can even be modified, e.g., the function very young can be considered as modification (a translation of sorts) of young . Thus, when considering an FD @math , and two tuples @math in some table, we usually need to decide whether @math (for some @math ). In the case of fuzzy databases (when @math and @math can be fuzzy functions), we need to rely on fuzzy logic to determine their degree of similarity---in other words, we have no equality relation as a crisp, binary relation @cite_24 . The exact concept of FD depends on the underlying fuzzy logic being adopted @cite_1 .
@cite_24 @cite_1 @cite_10
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This paper deals with the application of fuzzy logic in a relational database environment with the objective of capturing more meaning of the data. It is shown that with suitable interpretations for the fuzzy membership functions, a fuzzy relational data model can be used to represent ambiguities in data values as well as impreciseness in the association among them. Relational operators for fuzzy relations have been studied, and applicability of fuzzy logic in capturing integrity constraints has been investigated. By introducing a fuzzy resemblance measure EQUAL for comparing domain values, the definition of classical functional dependency has been generalized to fuzzy functional dependency (ffd). The implication problem of ffds has been examined and a set of sound and complete inference axioms has been proposed. Next, the problem of lossless join decomposition of fuzzy relations for a given set of fuzzy functional dependencies is investigated. It is proved that with a suitable restriction on EQUAL, the design theory of a classical relational database with functional dependencies can be extended to fuzzy relations satisfying fuzzy functional dependencies. Abstract The paper contains an analysis of integrity constraints for -ary relations in fuzzy databases. Apart from dependencies between all attributes there may be also dependencies describing relationships of fewer attributes. However, there is no complete arbitrariness. Relationships comprising ( 1) attributes must not infringe integrity constraints of the -ary relation. The analysis is carried out using the theory of functional dependencies. In this paper, we assume that attribute values are represented by means of interval-valued possibility distributions. The notion of fuzzy functional dependency has been appropriately extended according to the representation of fuzzy data. The paper formulates the rules to which fuzzy functional dependencies between ( 1) attributes of the -ary relation must be subordinated.