Plan of the lecture. G53RDB: Theory of Relational Databases Lecture 9. Informal exercise from last lecture. Example (from Stanczyk et al.

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1 Plan of the lecture G53RDB: Theory of Relational Databases Lecture 9 Answers to informal coursework Multivalued dependencies 4NF Join dependencies 5NF Natasha Alechina School of Computer Science & IT nza@cs.nott.ac.uk Lecture 9 2 Informal coursework from the lecture on conditional tables Given a conditional table T below, define: π A T σ A = 3 T. T π A T σ A = 3 T A B A A B 0 y y 4 0 y 4 x 2 x = 3 x 2 x Informal exercise from last lecture Compute projections (given the COURSE-REGISTER example of 2NF) R1 = π, ST-Name, Address COURSE-REGISTER R2 = π Module#,MOD-Name COURSE-REGISTER R3 = π,module#, COURSE-REGISTER Check that COURSE-REGISTER = R1 >< R2 >< R3 Check that R1, R2, R3 are in 2NF. For the BCNF example, check that decomposition of TUTEES into STUDENTS and TUTORS given above is lossless. Lecture 9 3 Lecture 9 4 Example (from Stanczyk et al.) Example (from Stanczyk et al.) Relation COURSE-REGISTER Module# ST-Name Address MOD-Name 123 G51MCS John Lenton Math1 123 G5AIAI John Lenton Intro AI 456 G5BAIP Mary Beeston Prolog R1 = π, ST-Name, Address COURSE-REGISTER ST-Name Address 123 John Lenton 456 Mary Beeston Lecture 9 5 Lecture 9 6 1

2 Example (from Stanczyk et al.) Example (from Stanczyk et al.) R2 = π Module#,MOD-Name COURSE-REGISTER R3 = π,module#, COURSE-REGISTER Module# MOD-Name Module# G51MCS G5AIAI G5BAIP Math1 Intro AI Prolog 123 G51MCS 123 G5AIAI 456 G5BAIP Lecture 9 7 Lecture 9 8 Join R2 and R3: Join R2 and R3: Module# MOD-Name Module# Module# MOD-Name G51MCS Math1 123 G51MCS 123 G51MCS Math1 G5AIAI Intro AI 123 G5AIAI 123 G5AIAI Intro AI G5BAIP Prolog 456 G5BAIP 456 G5BAIP Prolog Lecture 9 9 Lecture 9 10 Join with R1: Join with R1: Get R back: ST-NameAddress Module# MOD-Name Module# ST-Name Address MOD-Name 123 John Lenton 456 Mary Beeston 123 G51MCS Math1 123 G5AIAI Intro AI 456 G5BAIP Prolog 123 G51MCS John Lenton Math1 123 G5AIAI John Lenton Intro AI 456 G5BAIP Mary Beeston Prolog Lecture 9 11 Lecture

3 Decomposition into BCNF: ST-Name Tutor-ID Tutor-Name 123 John xyz Peter 456 Mary xyz Peter 789 Jane abc Paul STUDENTS TUTORS ST-Name Tutor-ID Tutor-ID Tutor-Name 123 John xyz xyz Peter 456 Mary xyz abc Paul 789 Jane abc Lecture 9 13 Multivalued dependencies A multivalued dependency X Y holds in R if for every two tuples r,s R, if r(x) = s(x) then the sets of Y- values with which r and s occur in R are the same. {t(y): t R, t(x) = r(x), t(z) = r(z)} = {t(y): t R, t(x) = s(x), t(z) = s(z)} In other words, we fix the values of X and Z in r and look what other tuples with the same values of X and Z occur in R, and collect their Y values. Then we do the same with s. If the two sets of Y-values are always the same, it means that when X is fixed, then Y and Z are independent of each other. Any Y-value can occur with any Z-value (for a fixed X). Lecture 9 14 Multivalued dependencies: example Café Drinks, Café Food Cafe Drinks Food OneCafe Orange juice Salad OneCafe Orange juice Carrot cake OneCafe Apple juice Salad OneCafe Apple juice Carrot cake AnotherCafe Coffee Sandwich AnotherCafe Tea Sandwich Problems caused by mvds Multivalued dependencies (mvds) cause data redundancy and update anomalies (to change the drinks served at OneCafe, need to do this for every single item of food they serve). Essentially, unrelated data is kept in a single relation. Lecture 9 15 Lecture 9 16 Fourth normal form (4NF) A relation R is in 4NF if it is in 1NF and for every nontrivial multivalued dependency X Y (non-trivial means that it is not the case that X Y are all attributes of R), X is a key, that is for every attribute A in R, X A. Normalisation to 4NF: find a non-trivial mvd X Y where X is not a key; decompose the relation into two projections, π XY (R) and π XZ (R) where Z are the rest of R s attributes. An analogue of Heath s theorem can be proved for mvds. Lossless decomposition again Recall that decomposition of R into projections π XY (R) and π XZ (R) is lossless, or non-loss, if: R = π XY (R) >< X π XZ (R) Lecture 9 17 Lecture

4 Heath s theorem for mvds Heath s theorem A relation R(X,Y,Z) that satisfies a multivalued dependency X Y can always be non-loss decomposed into its projections R1=π XY (R) and R2=π XZ (R). Proof. R π XY (R) >< X π XZ (R) holds for any R (we showed it in the proof of Heath s theorem for functional dependencies). Now we show that π XY (R) >< X π XZ (R) R. For this we need X Y. Assume r π XY (R) >< X π XZ (R). So there is a tuple t in R such that t(x,y) = r(x,y) and a tuple s in R such that s(x,z) = r(x,z). We need to show that r itself is in R, that is, a tuple where X has value r(x) (same as t(x) and s(x)), Y has the same value as t(y) and Z has the same value as s(z). But this holds because t(x)=s(x) and R satisfies X Y (so there must be a tuple with t s value for Y and s s value for Z). Lecture 9 19 Lecture 9 20 Generalising dependencies further Join dependencies Functional dependencies are a special kind of multivalued dependencies. If R satisfies functional or multivalued dependency between sets of attributes X and Y, a lossless decomposition always exists. You could also say: if a relation is non-loss decomposable, it must be storing unrelated data together. Let R be a relation and X 1, X 2,, X n be sets of attributes of R. A join dependency JD*(X 1, X 2,, X n ) holds in R if and only if R is equal to the join of its projections on X 1, X 2,, X n : R = π X1 (R) >< π X2 (R) ><... >< π Xn (R) (join is on common attributes). Lecture 9 21 Lecture 9 22 Multivalued and join dependencies Join dependency: example Multivalued dependency X Y holds for a relation R(X,Y,Z) if, and only if, join dependency JD*(XY,XZ) holds for R. Multivalued dependency is a special case of join dependency where only two attribute sets are involved. Example from Stanczyk et al: JD*(University, Discipline, Degree) University Discipline Degree Old Town Computing BSc Old Town Mathematics PhD New City Computing PhD Old Town Computing PhD Lecture 9 23 Lecture

5 Decomposition (non-loss!): Problems caused by join dependencies University Discipline Old Town Computing Old Town Mathematics New City Computing Discipline Degree Computing BSc Computing PhD Mathematics PhD University Degree Old Town BSc Old Town PhD New City PhD Lecture 9 25 Similarly to multivalued dependencies, join dependencies cause data redundancy and update anomalies. They arise when unrelated data is kept in a single relation. Lecture 9 26 Fifth normal form (5NF) Fagin s Theorem A join dependency is called trivial if one of X i includes all attributes (so projection on X i is R itself). A relation R is in 5NF if for all non-trivial join dependencies JD*(X 1,,X n ) that hold for R, every X i is a superkey for R. Any relation can be non-loss decomposed into an equivalent collection of 5NF relations. Lecture 9 27 Lecture 9 28 Informal coursework Informal coursework Consider the following relation PR2 which describes arrangements for a first year programming course. Attributes are: StudentId (e.g. xyz01u), StudentName, TutorId (e.g. gjm), TutorName (e.g. Graham), Date (of the tutorial), Place (of the tutorial, e.g. B53), Assignment (e.g. SimpleGUI), Mark (assuming each student gets a mark for each assignment). Each student has one tutor, tutor may have multiple tutees and give multiple tutorials on different times, there is one tutorial per room, one mark per assignment, one assignment discussed and marked at a tutorial. Tutorial place may change. Determine candidate keys in relation PR2. Normalize to BCNF, 4NF, 5NF. Lecture 9 29 Lecture

6 Recommended reading Stanczyk, Champion, Leyton: chapter 7. Ullman, Widom, chapter Lecture