Closing sets of FDs
Basis: set F of FDs that hold in R; F1 ≈ F
Minimal basis B for R:
1. B - basis
2.  f  B has singleton r.s.
3. If any FD is removed from B, the result is not a basis
4. If  f  B we remove at least one attr. from the l.s. of f, the result is not a basis
Min_basis(F)
{ B = F;
while (  ( f = X  Y )  B; | Y| >1) split( f );
while (  f  B | f  B – f) B=B-f;
while(  ( f = X  Y )  B | | X| >1, Z=X-xi , Z  Y  B) X  Y := Z  Y }
Projecting FDs
R(S), F, S1  S, R1 = πS1 R
F1 - projection of F: F1 follows from F and involves only attr. from S1
Proj_FD(S,F,S1)
{ F1=Ǿ
// Init
for (all X  S1) {compute X+F ; for(all f= XY nontrivial, Y  X+F , Y  S1) F1:=F1+f
}
F1:=min_basis(F1);}
RDB Design
Problem: redundancy; update, insertion, deletion anomalies
Solution: decomposition R(S), R1(S1), R2(S2), … , Rn(Sn)
S1, S2, … , Sn  S; S1  S2  …  Sn = S; R1 = π S1 R , R2 = π S2 R ,
Rn = π Sn R
Reason: dependencies
Normal forms: 1NF , … , 4NF
FD types: X Y
FFD (Full)
∄ 𝑋′  X
PFD (Partial)
∃ 𝑋′  X
TFD (Transitive) ∃ 𝑍  S
X’Y
X’Y
XZ, ZY
2NF: no PFD on the key
R(S), key K, PFD KY, K’  K, FFD K’Y
Dec2NF: S1 = (K’, Y), S2 = (K, S-K-Y)
Decomposition examples
Recovering of info from decomposition: natural join
Theorem. R(S), S = X  Y  Z, F = {Y  Z}
R = π X  Y R |X| π Y  Z R
Example
R(X,Y,Z), Y  Z
R(X,Y,Z), Y -/-> Z, Y -/-> X
Lossless-join dec.
Lossy-join dec.
X Y Z
x1 y z
x2 y z
X Y
x1 y
x2 y
X Y Z
x1 y z1
x2 y z2
Y Z
y z
X Y Z
x1 y z
x2 y z
Join - the only way to recover!
X Y
x1 y
x2 y
Y Z
y z1
y z2
X Y Z
x1 y z1
x1 y z2
x2 y z1
x2 y z2