1. 5NF and other normal forms

2. Outline n-decomposability 3D constraint join dependency 5NF
non-5NF - update anomalies
problems in bringing a relation to 5NF
other normal forms

3. Always two projections?
so far every relation was non-loss decomposable into two projections
is this always possible?
n-decomposable relations

4. Courses - tutors - levels (CTL)

5. CTL - 2 attribute projectionsCL

6. CTL - 3-decomposable the join of any two projections is not CTL; e.g:
join(CT, TL)
Extra!

7. Constraint 3D Let R be a degree 3 relation. IF (a, b, x)  R
AND (a, y, c)  R
AND (z, b, c)  R
THEN (a, b, c)  R

8. Constraint 3D illustrated on the CTL relation
IF tutor t1 teaches subject s1
AND level l1 studies subject s1
AND tutor t1 teaches level l1
THEN tutor t1 teaches subject s1 for
level l1
note: this constraint is not expressed in CTL

9. Constraint 3D and Join Dependency
4NF does not express the constraint 3D
the constraint 3D is a facet of a more general constraint: join dependency

10. Join dependency
Let R be a relation. Let A, B, ..., Z be arbitrary subsets of R’s attributes. R satisfies the JD
 ( A, B, ..., Z )
if and only if R is equal to the join of its projections on A, B, ..., Z

11. 5 NF
R is in 5NF if and only if every join dependency in R is implied by the candidate keys of R
5NF is always achievable

12. Explanation
a join dependency,  (A, B, …, Z), is implied by the candidate keys, K1, …, Km of R if
the fact that K1, …, Km are candidate keys for R determine the fact that R has the JD  (A, B, …, Z)

13. Illustration - positive example
consider R (S_id, S_name, Status, City) with S_id and S_name candidate keys
 ({S_id, S_name, Status}, {S_id, City}) is a JD because S_id is a candidate key in R
 ({S_id, S_name}, {S_id, Status}, {S_name, City}) is a JD because S_id and S_name are both candidate keys in R

14. Illustration - negative example
consider CTL (Course, Tutor, Level) with (Course, Tutor, Level) - candidate key (and an extra constraint : constraint 3D)
({Course, Tutor}, {Course, Level}, {Tutor, Level}) is a JD, but this is not due to the CK, but to the constraint 3D
if CTL had not had constraint 3D, would it have been in 5NF?

15. Not 5NF - update anomalies
CTL satisfies
 ( {Course, Tutor}, {Tutor, Level}, {Course, Level} )
insert (Programming, M. Ursu, Level2)
what else must be done?

16. Not 5NF - update anomalies
CTL satisfies the same JD as before
delete (Databases, M. Ursu, Level2)
what else must be done?

17. JDs and MVDs Fagin’s theorem restated
R ( A, B, C ) satisfies  ( AB, AC ) if and only if it satisfies the MVDs A B | C
JD is the most general form of dependency (read as determination) possible between the attributes of a relation (in the relational model)

18. Activity
Is 4NF subsumed by 5NF? Can you prove this using Fagin’s theorem and the definitions for 4 and 5 NF?

19. Problems in bringing a relation to 5NF
check whether all JDs are implied by the candidate keys
Fagin : provides an algorithm for doing this for any relation, given all its JDs and all candidate keys
discover all JDs for a given relation
they do not have that intuitive meaning as FDs or MVDs

20. Concluding remarks
5NF is the ultimate normal form with respect to projection / join
5NF is guaranteed to be free of all anomalies that can be eliminated via projections
determining whether a relation is in 4NF but not in 5NF is still fuzzy
very rare in practice

21. Recap JD - a more general constraint than MD
a relation can be in 4NF and have un-expressed JDs
this results in update anomalies
such a relation can be decomposed (via projection) into an equivalent set of 5NF relations
a relation is 5NF if all its JDs are deducible from its candidate keys
for a relation in 4NF but not in 5NF, an unexpressed JD is a possible decomposition (towards 5NF)

22. Other normal forms FDs, MVDs or JDs are not used
domain-key normal form
R is in DK/NF if and only if every constraint of R is a logical consequence of domain constraints and (candidate) key constraints
restriction-union normal form
decomposing operator: restriction
abusing the language it can be said that: this normalisation theory is orthogonal on the “projection” normalisation theory