1. 1 Multi-valued Dependencies Salman Azhar Multi-valued Dependencies Fourth Normal Form These slides use some figures, definitions, and explanations from Elmasri-Navathe’s Fundamentals of Database Systems and Molina-Ullman-Widom’s Database Systems

2. 2 A New Form of Redundancy Multi-valued dependencies (MVDs) express a condition among tuples of a relation that exists when the table (relation) is trying to representMulti-valued dependencies (MVDs) express a condition among tuples of a relation that exists when the table (relation) is trying to represent –more than one many-many relationship. –then certain columns (attributes) become independent of one another –and their values must appear in all combinations.

3. 3 Example Customers(name, addr, phones, sodasLiked) A customer’s phones are independent of the sodas they like.A customer’s phones are independent of the sodas they like. Many-many relations:Many-many relations: –Customers  Phones Customers  Sodas Each phone appear with each soda in all combinations.Each phone appear with each soda in all combinations. –E.g., For 3 phones (Home, Work, Cell) and 10 sodas, we need 30 tuples Repetition is unlike FD redundancy.Repetition is unlike FD redundancy. There is only one FD: name  addrThere is only one FD: name  addr

4. 4 Tuples Implied by Independence If we have pink tuples, then Then the blue tuples must also be in the relation nameaddrphones sodasLiked sueap1 s1 sueap2 s2 sueap2 s1 sueap1 s2 name-phone and name-soda relations are independent: The green relationship implies existence of blue relations

5. 5 Definition of MVD A multi-valued dependency (MVD) X  Y :A multi-valued dependency (MVD) X  Y : –if an assertion that if two rows of a table agree on all the attributes of X, –then their components may be swapped in the set of attributes Y, –and the result will be two tuples that are also in the relation.

6. 6 Example The name-addr-phones-sodasLiked example illustrated two MVDsThe name-addr-phones-sodasLiked example illustrated two MVDs –name  phones –name  sodasLiked.

7. 7 Picture of MVD X  Y XY others equal exchange Other attributes get copied remain the same If there is an FD X  Y, then swapping Ys components doesn’t change anything. We must exchange all components of Ys (not just some) Every FD is an MVD.

8. 8 MVD Rules Every FD is an MVD.Every FD is an MVD. –If X  Y, then swapping Y ’s between two tuples that agree on X doesn’t change the tuples. –Therefore, the “new” tuples are surely in the relation, and we know X  Y. Complementation:Complementation: –If X  Y, and Z is all the other attributes (complement of X and Y) then X  Z. –Reason: Swapping Ys that agree on X is the same effect as swapping Zs. –Reason: Swapping Ys that agree on X is the same effect as swapping Zs.

9. 9 Splitting Doesn’t Hold Recall in FDs we could split right side, but could NOT split left sideRecall in FDs we could split right side, but could NOT split left side –Recall FD A, B  C does not imply A  C and B  C –Recall FD A  B, C does imply A  B and A  C MVDs, we cannot generally split the left sideMVDs, we cannot generally split the left side BUT, we cannot split the right side EITHER!!!BUT, we cannot split the right side EITHER!!! –E.g., name  areaCode phone in (408) 555-9999 and (812)-555-1111, you can’t swap area codesin (408) 555-9999 and (812)-555-1111, you can’t swap area codes –sometimes you have to deal with several attributes on the right side.

10. 10 Example Consider phone has been split into area code and number and adding manf giving the following relation:Consider phone has been split into area code and number and adding manf giving the following relation: Customers(name, areaCode, phone, sodasLiked, manf) We can claim:We can claim: –A customer can have several phones, the number divided between areaCode and phonethe number divided between areaCode and phone –A customer can like several sodas, each with its own manufacturer.each with its own manufacturer. –Phones are sodasLiked are independent, so we get all possible combinations of areaCode-phone pairs & sodasLiked-manf pairsall possible combinations of areaCode-phone pairs & sodasLiked-manf pairs

11. 11 Example, Continued Since the areaCode-phone pairs for a customer are independent of the sodasLiked-manf pairs, we expect the following MVDs:Since the areaCode-phone pairs for a customer are independent of the sodasLiked-manf pairs, we expect the following MVDs: name  areaCode phone name  sodasLiked manf Observe that we can’t split the right hand sides:Observe that we can’t split the right hand sides: –areaCode and phone are linked –sodasLiked and manf are linked in (408) 555-9999 and (812)-555-1111, you can’t swap area codes!!!in (408) 555-9999 and (812)-555-1111, you can’t swap area codes!!!

12. 12 Example Data Here is possible data satisfying these MVDs: nameareaCodephonesodasLikedmanf Sue812555-1111PepsiPepsiCo Sue812555-1111SpriteCocaCola Sue408555-9999PepsiPepsiCo Sue408555-9999SpriteCocaCola We cannot swap area codes or phones by themselves. Swapping (812) And (408) gives us incorrect phone numbers  Note: neither name  areaCode  nor name  phone holds for this relation. Now all possible combinations are represented. Swapping any pairs does not yield new rows.

13. 13 Fourth Normal Form The redundancy caused by MVDs can’t be removed by transforming the database schema to BCNF.The redundancy caused by MVDs can’t be removed by transforming the database schema to BCNF. There is a stronger normal form, called 4 th Normal for (4NF), that (intuitively):There is a stronger normal form, called 4 th Normal for (4NF), that (intuitively): –treats MVDs as FDs when it comes to decomposition –but not when determining keys of the relation.

14. 14 4NF Definition A relation R is in 4NF if whenever X  Y is a nontrivial MVD, then X is a superkey.A relation R is in 4NF if whenever X  Y is a nontrivial MVD, then X is a superkey. –Nontrivial means that: 1.Y is not a subset of X (swapping components does not change tuples) 2.X and Y are not, together, all the attributes. (swapping components yields the same tuples) –Note that the definition of “superkey” still depends on FDs only.

15. 15 BCNF Versus 4NF Remember thatRemember that –Every FD X  Y is also an MVD, X  Y. Thus, if R is in 4NF, it is certainly in BCNF.Thus, if R is in 4NF, it is certainly in BCNF. –Because any BCNF violation is a 4NF violation. However, R could be in BCNF and not 4NF, because MVDs are “invisible” to BCNF.However, R could be in BCNF and not 4NF, because MVDs are “invisible” to BCNF.

16. 16 Decomposition and 4NF If X  Y is a 4NF violation for relation R, we can decompose R using the same technique as for BCNF.If X  Y is a 4NF violation for relation R, we can decompose R using the same technique as for BCNF. 1.XY is one of the decomposed relations. 2.All but (Y – X) is the other.

17. 17 Example Customers(name, addr, phones, sodasLiked) FD: name  addr MVDs: name  phones name  sodasLiked Key is {name, phones, sodasLiked}.Key is {name, phones, sodasLiked}. All dependencies violate 4NF because name is not a superkey.All dependencies violate 4NF because name is not a superkey.

18. 18 Example, Continued Decompose using name  addr:Decompose using name  addr: 1.Customers1(name, addr) uIn 4NF, only dependency is name  addr. 2.Customers2(name, phones, sodasLiked) uNot in 4NF because MVDs uname  phones and name  sodasLiked apply. uNo FD’s, so all three attributes form the key.

19. 19 Example: Decompose Customers2 RecallRecall –Customers2(name, phones, sodasLiked) Either MVDEither MVD –name  phones –name  sodasLiked tells us to decompose to:tells us to decompose to: –Customers3(name, phones) –Customers4(name, sodasLiked)

20. 20 Example: Decompose Customers2 4 th Normal Form Consists of:4 th Normal Form Consists of: –Customers1(name, addr) ContactContact –Customers3(name, phones) PhonePhone –Customers4(name, sodasLiked) LikesLikes The following FD and MVDs are satisfied:The following FD and MVDs are satisfied: FD: name  addr MVDs: name  phones name  sodasLiked

21. 21 Continue Normal Forms

22. 22 Salman Teaching 4 th Normal Form How would I look teaching 937 th Normal Form?How would I look teaching 937 th Normal Form?

23. 23 Salman Teaching 957 th Normal Form