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Functional Dependencies in rdbms with examples
- 1. 1 Lecture 6: Schema refinement:Functional dependencies www.cl.cam.ac.uk/Teaching/current/Databases/
- 2. 2 Recall: Database design lifecycle •Requirements analysis – User needs; what must database do? • Conceptual design – High-level description; often using E/R model • Logical design – Translate E/R model into relational schema • Schema refinement – Check schema for redundancies and anomalies • Physical design/tuning – Consider typical workloads, and further optimise Next two lectures
- 3. 3 Today’s lecture • Whyare some designs bad? • What’s a functional dependency? • What’s the theory of functional dependencies? • (Next lecture: How can we use this theory to classify redundancy in relation design?)
- 4. 4 Not all designsare equally good • Why is this design bad? Data(sid,sname,address,cid,cname,grade) • Why is this one preferable? Student(sid,sname,address) Course(cid,cname) Enrolled(sid,cid,grade)
- 5. 5 An instance ofour bad design sid sname addres s ci d cname grade 124 Britney USA 206 Database A++ 204 Victoria Essex 202 Semantics C 124 Britney USA 201 S/Eng I A+ 206 Emma London 206 Database B- 124 Britney USA 202 Semantics B+
- 6. 6 Evils of redundancy •Redundancy is the root of many problems associated with relational schemas – Redundant storage – Update anomalies – Insertion anomalies – Deletion anomalies – LOW TRANSACTION THROUGHPUT • In general, with higher redundancy, if transactions are correct (no anomalies), then they have to lock more objects thus causing greater contention and lower throughput • (Aside: Could having a dummy value, NULL, help?)
- 7. 7 Decomposition • We removeanomalies by replacing the schema Data(sid,sname,address,cid,cname,grade) with Student(sid,sname,address) Course(cid,cname) Enrolled(sid,cid,grade) • Note the implicit extra cost here • Two immediate questions: 1. Do we need to decompose a relation? 2. What problems might result from a decomposition?
- 8. 8 Functional dependencies • Recall: –A key is a set of fields where if a pair of tuples agree on a key, they agree everywhere • In our bad design, if two tuples agree on sid, then they also agree on address, even though the rest of the tuples may not agree
- 9. 9 Functional dependencies cont. • Wecan say that sid determines address – We’ll write this sid address • This is called a functional dependency (FD) • (Note: An FD is just another integrity constraint)
- 10. 10 Functional dependencies cont. • We’dexpect the following functional dependencies to hold in our Student database – sid sname,address – cid cname – sid,cid grade • A functional dependency X Y is simply a pair of sets (of field names) – Note: the sloppy notation A,B C,D rather than {A,B} {C,D}
- 11. 11 Formalities • Given arelation R=R(A1:1, …, An:n), and X, Y ({A1, …, An}), an instance r of R satisfies XY, if – For any two tuples t1, t2 in R, if t1.X=t2.X then t1.Y=t2.Y • Note: This is a semantic assertion. We can not look at an instance to determine which FDs hold (although we can tell if the instance does not satisfy an FD!)
- 12. 12 Properties of FDs •Assume that X Y and Y Z are known to hold in R. It’s clear that X Z holds too. • We shall say that an FD set F logically implies X Y, and write F [X Y – e.g. {X Y, Y Z} [ X Z • The closure of F is the set of all FDs logically implied by F, i.e. F+ @ {XY | F [ XY} • The set F+ can be big, even if F is small
- 13. 13 Closure of aset of FDs • Which of the following are in the closure of our Student FDs? – addressaddress – cidcname – cidcname,sname – cid,sidcname,sname
- 14. 14 Candidate keys andFDs • If R=R(A1:1, …, An:n) with FDs F and X{A1, …, An}, then X is a candidate key for R if – X A1, …,An F+ – For no proper subset YX is Y A1, …,An F+
- 15. 15 Armstrong’s axioms • Reflexivity:If YX then F XY – (This is called a trivial dependency) – Example: sname,addressaddress • Augmentation: If F XY then F X,WY,W – Example: As cidcname then cid,sidcname,sid • Transitivity: If F XY and F YZ then F XZ – Example: As sid,cidcid and cidcname, then sid,cidcname
- 16. 16 Consequences of Armstrong’s axioms •Union: If F XY and F XZ then F XY,Z • Pseudo-transitivity: If F XY and F W,YZ then F X,WZ • Decomposition: If F XY and ZY then F XZ Exercise: Prove that these are consequences of Armstrong’s axioms
- 17. 17 Proof of UnionRule Suppose that F XY and F XZ. By augmentation we have F XX,Y since X U X = X. Also by augmentation F X,YZ,Y Therefore, by transitivity we have F XZ,Y QED
- 18. 18 Functional Dependencies Canbe useful in Algebraic Reasoning Suppose R(A,B,C) is a relation schema with dependency AB, then ) ( , R R B A π = ) ( , R C A π A (This is called Heath’s rule.)
- 19. 19 Proof of Heath’sRule ) ( , R C A π A First show that Suppose then and Since we have ) ( , R C A π A
- 20. 20 Proof of Heath’sRule (cont.) A In the other direction, we must show that Suppose Then there must exist records and There must also exist Therefore, we have so that QED But the functional dependency tells us that
- 21. 21 Equivalence • Two setsof FDs, F and G, are said to be equivalent if F+ =G+ • For example: {(A,BC), (AB)} and {(AC), (AB)} are equivalent • F+ can be huge – we’d prefer to look for small equivalent FD sets
- 22. 22 Minimal cover • AnFD set, F, is said to be minimal if 1. Every FD in F is of the form XA, where A is a single attribute 2. For no XA in F is F-{XA} equivalent to F 3. For no XA in F and ZX is (F-{XA}){ZA} equivalent to F 1. For example, {(AC), (AB)} is a minimal cover for {(A,BC), (AB)}
- 23. 23 More on closures •FACT: If F is an FD set, and XYF+ then there exists an attribute AY such that XAF+
- 24. 24 Why Armstrong’s axioms? •Soundness – If F XY is deduced using the rules, then XY is true in any relation in which the dependencies of F are true • Completeness – If XY is is true in any relation in which the dependencies of F are true, then F XY can be deduced using the rules
- 25. 25 Soundness • Consider theAugmentation rule: – We have XY, i.e. if t1.X=t2.X then t1.Y=t2.Y – If in addition t1.W=t2.W then it is clear that t1. (Y,W)=t2.(Y,W)
- 26. 26 Soundness cont. Consider theTransitivity rule: – We have XY, i.e. if t1.X=t2.X then t1.Y=t2.Y (*) – We have YZ, i.e. if t1.Y=t2.Y then t1.Z=t2.Z (**) – Take two tuples s1 and s2 such that s1.X=s2.X then from (*) s1.Y=s2.Y and then from (**) s1.Z=s2.Z
- 27. 27 Completeness • Exercise – (Youmay need the fact from slide 23)
- 28. 28 Attribute closure • Ifwe want to check whether XY is in a closure of the set F, could compute F+ and check – but expensive • Cheaper: We can instead compute the attribute closure, X+ , using the following algorithm: • Then F XY iff Y is a subset of X+ Try this with sid,snamecname,grade closure:= X; repeat until no change{ if UVF, where Uclosure then closure:=closureV };
- 29. 29 Preview of nextlecture: Goals of normalisation • Decide whether a relation is in “good form” • If it is not, then we will “decompose” it into a set of relations such that – Each relation is in “good form” – The decomposition has not lost any information that was present in the original relation • The theory of this process and the notion of “good form” is based on FDs
- 30. 30 Summary You should nowunderstand: • Redundancy and various forms of anomalies • Functional dependencies • Armstrong’s axioms Next lecture: Schema refinement: Normalisation