Vincenzo Deufemia (Università di Salerno, Italy), Giuseppe Polese (Università di Salerno, Italy) and Mario Vacca (Università di Salerno, Italy)

Source Title: Handbook of Research on Innovations in Database Technologies and Applications: Current and Future Trends

Copyright: © 2009
|Pages: 9
DOI: 10.4018/978-1-60566-242-8.ch022

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TopIn the sequel, we use the following notation: *t*, *t _{1}*,

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 *t*_{1} and *t*_{2} having *t*_{1}[*X*] = *t*_{2}[*X*], then t_{1}[*Y*] = *t*_{2}[*Y*]. More precisely, given a table *R*,

*X*→*Y*⇔ ∀*t*_{1},*t*_{2}∈*R*.(*t*_{1}[*X*] =*t*_{2}[*X*] ⇒*t*_{1}[*Y*] =*t*_{2}[*Y*]).

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).

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