mvds - Multivalued Dependencies Fourth Normal Form...

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Unformatted text preview: Multivalued Dependencies Fourth Normal Form Reasoning About FD’s + MVD’s 1 Definition of MVD x A multivalued dependency (MVD) on R, X ­>­>Y , says that if two tuples of R agree on all the attributes of X, then their components in Y may be swapped, and the result will be two tuples that are also in the relation. x i.e., for each value of X, the values of Y are independent of the values of R­X­Y. 2 Example: MVD Drinkers(name, addr, phones, beersLiked) x A drinker’s phones are independent of the beers they like. x Thus, each of a drinker’s phones appears with each of the beers they like in all combinations. x This repetition is unlike FD redundancy.  name­>addr is the only FD. 3  name­>­>phones and name ­>­>beersLiked. Tuples Implied by name­>­>phones If we have tuples: name sue sue sue sue addr a a a a phones beersLiked p1 b1 p2 b2 p2 b1 p1 b2 Then these tuples must also be in the relation. 4 Picture of MVD X ­>­>Y X equal exchange Y others 5 MVD Rules x Every FD is an MVD (promotion ).  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. x Complementation : If X ­>­>Y, and Z is all the other attributes, then X ­>­>Z. 6 Splitting Doesn’t Hold x Like FD’s, we cannot generally split the left side of an MVD. x But unlike FD’s, we cannot split the right side either ­­­ sometimes you have to leave several attributes on the right side. 7 Example: Multiattribute Right Sides Drinkers(name, areaCode, phone, beersLiked, manf) x A drinker can have several phones, with the number divided between areaCode and phone (last 7 digits). x A drinker can like several beers, each with its own manufacturer. 8 Example Continued x Since the areaCode­phone combinations for a drinker are independent of the beersLiked­manf combinations, we expect that the following MVD’s hold: name ­>­> areaCode phone name ­>­> beersLiked manf 9 Example Data Here is possible data satisfying these MVD’s: name Sue Sue Sue Sue areaCode 650 650 415 415 phone 555­1111 555­1111 555­9999 555­9999 beersLiked Bud WickedAle Bud WickedAle manf A.B. Pete’s A.B. Pete’s But we cannot swap area codes or phones by themselves. That is, neither name­>­>areaCode nor name­>­>phone holds for this relation. 10 Fourth Normal Form x The redundancy that comes from MVD’s is not removable by putting the database schema in BCNF. x There is a stronger normal form, called 4NF, that (intuitively) treats MVD’s as FD’s when it comes to decomposition, but not when determining keys of the relation. 11 4NF Definition x A relation R is in 4NF if: whenever X ­>­>Y is a nontrivial MVD, then X is a superkey.  1. Y is not a subset of X, and 2. X and Y are not, together, all the attributes. Nontrivial MVD means that:  Note that the definition of “superkey” still depends on FD’s only. 12 BCNF Versus 4NF x Remember that every FD X ­>Y is also an MVD, X ­>­>Y. x Thus, if R is in 4NF, it is certainly in BCNF. x But R could be in BCNF and not 4NF, because MVD’s are “invisible” to BCNF. 13  Because any BCNF violation is a 4NF violation (after conversion to an MVD). Decomposition and 4NF x 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. 14 Example: 4NF Decomposition Drinkers(name, addr, phones, beersLiked) FD: name ­> addr MVD’s: name ­>­> phones name ­>­> beersLiked x Key is {name, phones, beersLiked}. x All dependencies violate 4NF. 15 Example Continued x Decompose using name ­> addr: 1. Drinkers1(name, addr) 2. Drinkers2(name, phones, beersLiked) x In 4NF; only dependency is name ­> addr. x Not in 4NF. MVD’s name ­>­> phones and name ­>­> beersLiked apply. No FD’s, so all three attributes form the key. 16 Example: Decompose Drinkers2 x Either MVD name ­>­> phones or name ­>­> beersLiked tells us to decompose to:  Drinkers3(name, phones)  Drinkers4(name, beersLiked) 17 Reasoning About MVD’s + FD’s x Problem: given a set of MVD’s and/or FD’s that hold for a relation R, does a certain FD or MVD also hold in R ? x Solution: Use a tableau to explore all inferences from the given set, to see if you can prove the target dependency. 18 Why Do We Care? 1. 4NF technically requires an MVD violation.  2. When we decompose, we need to project FD’s + MVD’s. Need to infer MVD’s from given FD’s and MVD’s that may not be violations themselves. 19 Example: Chasing a Tableau With MVD’s and FD’s x To apply a FD, equate symbols, as before. x To apply an MVD, generate one or both of the tuples we know must also be in the relation represented by the tableau. x We’ll prove: if A­>­>BC and D­>C, then A­>C. 20 The Tableau for A­>C Goal: prove that c1 = c2. A a a a B b1 b2 b2 C c1 c2 c2 c2 D d1 d2 d1 Use A­>­>BC (first row’s D with second row’s BC ). Use D­>C (first and third row agree on D, 21 therefore agree on C ). Example: Transitive Law for MVD’s x If A­>­>B and B­>­>C, then A­>­>C.  Obvious from the complementation rule if the Schema is ABC.  But it holds no matter what the schema; we’ll assume ABCD. 22 The Tableau for A­>­>C Goal: derive tuple (a,b1,c2,d1). A a a a a B b1 b2 b1 b1 C c1 c2 c2 c2 D d1 d2 d2 d1 Use A­>­>B to swap B from the first row into the second. Use B­>­>C to swap C from the third row into the first.3 2 Rules for Inferring MVD’s + FD’s x Start with a tableau of two rows.  These rows agree on the attributes of the left side of the dependency to be inferred.  And they disagree on all other attributes.  Use unsubscripted variables where they agree, subscripts where they disagree. 24 Inference: Applying a FD x Apply a FD X­>Y by finding rows that agree on all attributes of X. Force the rows to agree on all attributes of Y.  Replace one variable by the other.  If the replaced variable is part of the goal tuple, replace it there too. 25 Inference: Applying a MVD x Apply a MVD X­>­>Y by finding two rows that agree in X.  Add to the tableau one or both rows that are formed by swapping the Y­components of these two rows. 26 Inference: Goals x To test whether U­>V holds, we succeed by inferring that the two variables in each column of V are actually the same. x If we are testing U­>­>V, we succeed if we infer in the tableau a row that is the original two rows with the components of V swapped. 27 Inference: Endgame x Apply all the given FD’s and MVD’s until we cannot change the tableau. x If we meet the goal, then the dependency is inferred. x If not, then the final tableau is a counterexample relation.  Satisfies all given dependencies.  Original two rows violate target dependency. 28 ...
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mvds - Multivalued Dependencies Fourth Normal Form...

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