1. Multivalued Dependency Prepared by Tomasz Kaciak CS157A

2. Not All Designs are Equally Good This is a poor schema design Is this one better? StudentInfo(sid, sname, crno, subj, cid, exp-grade) Student(sid, sname) Course(crno, cid) Subject(cid, subj) Takes(sid, crno, exp-grade)

3. Problems of first design –Redundancy, potential inconsistency –Insertion Anomalies –Deletion Anomalies Second design –Illustrates decompositions and normalization.

4. Functional Dependencies Def. Given a relation schema R and subsets X, Y of R: An instance r of R satisfies FD X  Y if, for any two tuples t1, t2 2 r, t1[X ] = t2[X] implies t1[Y] = t2[Y] For an FD to hold for schema R, it must hold for every possible instance of r

5. Functional Dependencies Describe “Key-Like” Relationships A key is a set of attributes where: If keys match, then the tuples match A functional dependency (FD) is a generalization: If an attribute set determines another, written X ! Y then if two tuples agree on attribute set X, they must agree on Y: sid ! sname

6. Armstrong’s Axioms: Inferring FDs –Reflexivity: If Y  X then X  Y (trivial dependency) sname, sid  sname –Augmentation: If X  Y then XW  YW crno  subj so crno, exp-grade  subj, exp-grade –Transitivity: If X  Y and Y  Z then X  Z crno  cid and cid  subj so crno  subj

7. Armstrong’s Axioms Lead to… Union: If X  Y and X  Z then X  YZ Pseudotransitivity: If X  Y and WY  Z then XW  Z Decomposition: If X  Y and Z  Y then X  Z

8. Armstrong’s axioms Armstrong’s axioms are sound and complete inference rules for FDs! –Sound: all the derived FDs (by using the axioms) are those logically implied by the given set –Complete: all the logically implied (by the given set) FDs can be derived by using the axioms.

9. Now we can check FDs closure Defn. Let F be a set of FD’s. Its closure, F+ (cover F+), is the set of all FD’s: {X  Y | X  Y is derivable from F by Armstrong’s Axioms}

10. Consider FDs Example F= {A→C, B→C, CD→E}, let show that AD→E: 1) A→C(given) 2) AD→CD( Augmentation ) 3) CD→E(given) 4) AD→E(2, 3 and Transitivity )

11. One more Ex. R(C S Z), CS→Z, Z →C What is the relation NF? Z->C (given) SZ->SC ( Augmentation ) SZ->Z ( Transitivity ) SZ->C ( Transitivity )  All attributes are PRIME (since CS and ZS are keys)  Therefore, in 3NF, but not in BCNF

12. Computing F + To compute the closure of a set of functional dependencies F: F + = F repeat for each functional dependency f in F + apply reflexivity and augmentation rules on f add the resulting functional dependencies to F + for each pair of functional dependencies f 1 and f 2 in F + if f 1 and f 2 can be combined using transitivity then add the resulting functional dependency to F + until F + does not change any further

13. Example of computing X + F={A  B, AC  D, AB  C} 1.result=A A  B 1.result=AB AB->C 2.result=ABC ABC->BD 3.result=ABCD X+ {ABCD}

14. Finding primary key Ex. R: ( A, B, C, D, E, F ) F={A  B, BC  A, D  EF, B  D, A  C} Task: Find the candidate key(s) and identify a primary key A->B B->D D->EF A->BDEF A->C A->BCDEF

15. FD Redundancy Sets of functional dependencies may have redundant dependencies that can be inferred from the others Eg: A  C is redundant in: {A  B, B  C, A  C} Parts of a functional dependency may be redundant E.g. on RHS: {A  B, B  C, A  CD} can be simplified to {A  B, B  C, A  D} E.g. on LHS: {A  B, B  C, AC  D} can be simplified to {A  B, B  C, A  D}

16. Materials Used Schema Refinement & Normalization Theory mason.gmu.edu/~brodsky/infs614/fall03/lecture12.pptSchema Refinement & Normalization Theory FDs and Normal Forms www.cs.wisc.edu/~cs784-1/lecs/784_lec_6.pptFDs and Normal Forms Schema Normalization, Concluded Zachary G. Ives