1. Rensselaer Polytechnic Institute CSCI-4380 – Database Systems David Goldschmidt, Ph.D.

2.  Given a set of attributes X, the closure X+ is the set of attributes functionally determined by X  Given a relation R and a set F of functional dependencies, we need a way to find whether a functional dependency X  Y is true with respect to F

3.  Given relation R with attributes A, B, C, D, E and A  BC, CD  E, BE  C  AE  _____?  From reflexivity, AE+ = { A, E }  From A  BC, AE+ = { A, B, C, E }  No other rules are applicable or add to AE+  We conclude that AE  ABCE or simply AE  BC  Or AE  A, AE  B, AE  C, and AE  E

4.  Given a set F of functional dependencies, the closure X+ of a set of attributes X is determined by the following algorithm:  Initialize X+ to X  Repeat until X+ does not change: ▪ Find any unapplied functional dependency Y  Z in F such that Y  X+ ▪ Set X+ = X+  Z

5.  A key K for a given relation R is a minimal set of attributes A 1, A 2,..., A n such that closure {A 1, A 2,..., A n }+ is the set of all attributes of R  MusicGroup(name, artist, genre, dateformed, datefirstjoined)  name  genre dateformed  name artist  datefirstjoined  K must be (name, artist) because K+ = {name, artist, genre, dateformed, datefirstjoined}

6.  Given a set F of functional dependencies, closure F+ is the set of all functional dependencies implied by F  F+ can be found using the set of inference rules (reflexivity, transitivity, augmentation, etc.)  Sets F 1 and F 2 of functional dependencies are considered equal if they have the same closure (i.e. F 1 + = F 2 +)

7.  In addition to determining closure F+ for set F of functional dependencies, we can also derive any functional dependency that follows from F via Armstrong’s axioms:  Reflexivity (if Y  X, then X  Y)  Augmentation (if X  Y, then XZ  YZ)  Transitivity (if X  Y and Y  Z, then X  Z)

8.  Given a set F of functional dependencies, any set of functional dependencies that’s equivalent to F is called a basis  We limit the possibilities by requiring that each dependency has a single attribute on the right-hand side  How many bases are there for a relation R with n functional dependencies in F?

9.  A minimal basis for a relation R is a basis B that satisfies the following conditions:  All functional dependencies in B have singleton right-hand sides  If any functional dependency is removed from B, the result is no longer a basis  If any left-hand side attribute is removed from a functional dependency of B, the result is no longer a basis

10.  Given basis B, we can determine whether it is a minimal basis via the algorithm below:  For each functional dependency X  Y in B, check if B – { X  Y } still implies X  Y ▪ if so, remove X  Y  For each functional dependency XW  Y in B, check if X+ is the same with respect to F and ( F – { XW  Y } )  { X  Y } ▪ if so, replace XW  Y with X  Y

11.  Given relation R( A, B, C, D ) and functional dependencies AB  C, C  D, and D  A  What are the keys of R?  What are the superkeys of R that are not keys?  Is the given set of functional dependencies a basis? Is it a minimal basis?

12.  What’s wrong with the relation below?  MusicGroup( name, artist, genre, dateformed, datefirstjoined )  i.e. how can tuples become corrupted or incorrect?

13.  Without normalization, problems with relations include:  Unnecessary redundancy  Insert anomalies  Update anomalies  Delete anomalies

14.  Splitting a relation into two (more specific) relations is called decomposition  The objective is to have each resulting relation be atomic  i.e. each relation should contain only information related to the key

15.  A given relation R with set F of functional dependencies is in BCNF if and only if all functional dependencies X  Y in F are:  either trivial (i.e. Y  X)  or X is a superkey of R  If relation R is not in BCNF, it is possible to use decomposition to transform R to BCNF

16.  Given a set F of functional dependencies for relation R( A 1, A 2,..., A n ) that is not in BCNF:  Convert F to a minimal basis  Find an X  Y that violates BCNF  Compute closure X+  Decompose R into: ▪ R 1 containing all attributes of X+ ▪ R 2 containing { A 1, A 2,..., A n } – ( X+ – X )  Project functional dependencies onto R 1 and R 2 Repeat!

17.  Given relation R( A, B, C, D, E ) and functional dependencies AB  AC, CE  D, B  A, and D  AE  What are the keys of R?  What are the superkeys of R that are not keys?  Is the given set of functional dependencies a basis? Is it a minimal basis?  Is relation R in BCNF? If not, decompose R such that it is in BCNF