Abstract
Functional dependencies represent a fundamental concept in the design of a database since they are capable of capturing some semantics of the data strongly connected with the occurrence of redundancy in a database. The development of applications working on fuzzy and multimedia databases has required the extension of the functional dependency notion to these new types of data. Unfortunately, the concept of imprecise functional dependence or fuzzy functional dependence (IFD, for short) has not had a cogent and largely accepted definition yet. In fact, in attempt to capture different aspects of this notion of many proposals of IFD definition exist in literature, all having semantics and objectives somewhat unclear, especially with respect to the concern of redundancy (Bosc, et al., 1994, Cubero & Vila, 1994, Raju & Majumdar, 1988, Tyagi, et al., 2005, Wang, et al., 2000). Moreover, the debate on the definition of the concept of fuzzy functional dependency seems to be still in progress, as shown by the following question: “But the question remains: are these extended notions of functional dependency a natural generalization?” (Tyagi et al., 2005).
TopBackground
In the sequel, we use the following notation: t, t1, t2, etc are tuples; t[X] are the values of the tuple t corresponding to the set of attributes X (these values are also called a X-representation); Ak[t] is the k-th attribute value of the tuple t; Ak[ti, tj] stand for the couple (Ak[ti], Ak[tj]).
In traditional relational databases a functional dependency (FD, for short) is defined as a constraint between two sets of attributes from the database. Given two sets of attributes X and Y, a functional dependency between them is denoted by X → Y. The constraint says that, for any two tuples t1 and t2 having t1[X] = t2[X], then t1[Y] = t2[Y]. More precisely, given a table R,
Key Terms in this Chapter
Resemblance Relation: It is a relaxing of the equality on subset of attributes. Given a set of attributes X, and let t[X] be the values of the tuple t corresponding to the set of attributes X, the resemblance on X is denoted by RESX(t1[X], t2[X]).
Imprecise Functional Dependency: Given two sets of attributes X and Y, an imprecise functional dependency between them is denoted by X ? Y. The constraint says that, for any two tuples t1 and t2 having t1[X] similar to t2[X], then t1[Y] is similar to t2[Y]. More precisely, given a table R, IFD: X ? Y ? ?t1, t2 ? R.(RESX(t1[X], t2[X] ?f RESY(t1[Y], t2[Y]).
Triangular Co-Norm: A triangular co-norm g is a 2-ary aggregation function with the following properties: 1. g(1,1)= 1; g(a,0)=g(0,a)=a, (?-conservation), 2. g(x,y)= g(y,x), (commutativity), 3. g(x,y) = g(x’,y’) if x = x’ and y = y’, (monotonicity), 4. g(g(x,y),z) = g(x,g(y,z)), (associativity).
Functional dependency: Given two sets of attributes X and Y, a functional dependency between them is denoted by X ? Y. The constraint says that, for any two tuples t1 and t2 having t1[X] = t2[X], then t1[Y] = t2[Y]. More precisely, given a table R, X ? Y ? ?t1, t2 ? R.(t1[X] = t2[X] ? t1[Y] = t2[Y]).
Fuzzy Implication: It is an extension of the classical implication in which the two values involved are not necessarily true or false (1 or 0), but can be also two degrees of truth (belonging to [0,1]). The result is another degree of truth. More precisely, it is a function ?f: [0,1] ? [0,1].
Multimedia Functional Dependency: (Type-M Functional Dependency): Let R be a relation with attribute set U, and X, Y ? U. Xg1(t’) ? Yg2(t’’) is a type-M functional dependency (MFD) relation if and only if for any two tuples t1 and t2 in R that have t1[X] ?g1(t’) t2[X], then t1[Y] ?g2(t’’) t2[Y], where g1?TD(X) and g2?TD(Y), whereas t’ and t’’ ? [0,1] are thresholds.
Similarity Relation: A similarity on a set D is a fuzzy subset of the Cartesian product D × D: µS: D × D ? [0,1] with the properties of reflexivity (µS(x,x)= 1 for all x?S), symmetry (µS(x,y)= µS(y,x) for all x,y?S) and max-min transitivity (µS(x,z) = maxy?D{min(µS(x,y), µS(y,z))} for all x,y,z?S).