Consider the schema R over the attributes KNLMDFG with the following functional dependencies:

Question

KN -> L
LM -> N
LMN -> D
D -> M
M -> FG

and the following multivalued dependencies:
R = NLK ./ NDMFG
R = KNMF ./ FLDG

Decompose this schema into 4NF using the following method: first obtain a BCNF decomposition using the FDs only. Then proceed to apply the MVDs to further normalize the schemas that are not yet in 4NF.

Answer 1

Solution:
The minimal cover is
KN -> L
LM -> N
LM -> D
D -> M
M -> F
M -> G

So 3NF is
R1 = (KNL; KN -> L)
R2 = (LMND; LM -> ND)
R3 = (DM; D -> M)
R4 = (MFG; M -> FG)
This is not yet lossless, since none of the attribute sets of Ri is a superkey of R. So, we need
to add a new schema whose attribute set is a superkey: R0 = (KNLD, f g). (Actually, this
schema has some implied FDs, such as KN->L.)

R2 is not in BCNF due to D -> M. Split into R21 = DM (same as R3) and R22 =DLN.
We now need to check which of the resulting schemas are still not in 4NF and decompose
them with respect to the MVDs.
R1 and R21 are in 4NF.
R22 is not in 4NF due to MVD 1. Splitting into NL and ND (with no FDs embedded into these
schemas).
R3 is in 4NF.
R4 is not in 4NF due to MVD 2. Splitting into MF and FG.
The resulting 4NF decomposition has schemas with these attribute sets: KNL, DM, NL, ND, MF,
and FG.