# Theory I: Database Foundations

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1 Theory I: Database Foundations Theory I: Database Foundations

2 Theory I: Database Foundations 20. Formal Design : Formal Design We want to distinguish good from bad database design. What kind of additional information do we need? Can we transform a bad into a good design? By which cost? 2

3 Theory I: Database Foundations 20. Formal Design Motivation 20.1 Motivation Relations and anomalies Stadt SNr SName LCode LFläche 7 Freiburg D Berlin D Moscow RU St.Petersburg RU Kontinent KName LCode KFläche Prozent Europe D Europe RU Asia RU

4 Theory I: Database Foundations 20. Formal Design Motivation Having removed anomalies Stadt SNr SName LCode 7 Freiburg D 9 Berlin D 40 Moscow RU 43 St.Petersburg RU Lage LCode KName Prozent D Europe 100 RU Europe 20 RU Asia 80 LCode Land LFläche D 357 RU KName Kontinent KFläche Europe 3234 Asia

5 Theory I: Database Foundations 20. Formal Design Functional Dependencies 20.2 Functional Dependencies Definition Let a relation schema be given by its format V and let X, Y V. Let r Rel(V ). r fulfills a functional dependency (FD) X Y, if for all µ, ν r: µ[x ] = ν[x ] µ[y ] = ν[y ]. Let F a set of functional dependencies over V and X, Y V. The set of all relations r Rel(V ), which fulfill all FD s in F, is called Sat(V, F). 5

6 Theory I: Database Foundations 20. Formal Design Functional Dependencies Membership-Test The FD X Y, F = X Y is implied by F, if for each relation r, whenever r Sat(V, F) then r fulfills X Y. The set F + = {X Y F = X Y } is called closure of F. X Y F + is called Membership-Test. 6

7 Theory I: Database Foundations 20. Formal Design Functional Dependencies key Let V = {A 1,..., A n }. X V is called key of V (bzgl. F), if X A 1... A n F +, Y X Y A 1... A n / F +. 7

8 Theory I: Database Foundations 20. Formal Design Functional Dependencies Armstrong-Axiome Let r Sat(V, F). (A1) Reflexivity: If Y X V, then r fulfills FD X Y. (A2) Augmentation: If X Y F, Z V, then r fulfills FD XZ Y Z. (A3) Transitivity: If X Y, Y Z F, then r fulfills FD X Z. (A1): trivial FD s. 8

9 Theory I: Database Foundations 20. Formal Design Functional Dependencies Soundness and Completeness Every FD derivable by the Armstrong axioms is an element of the closure (soundness). Every FD in F + is derivable by the Armstrong axioms (completeness) To show completeness: If X Y not derivable by (A1) (A3), then X Y F +, i.e. r, r fulfills F, however does not X Y. 9

10 Theory I: Database Foundations 20. Formal Design Functional Dependencies Membership-Test Variante 1: Starting from F apply (A1) (A3)until X Y is derived, or F + is derived and X Y F +. Complexity? 10

11 Theory I: Database Foundations 20. Formal Design Functional Dependencies more axioms Let r Sat(V, F). Let X, Y, Z, W V und A V. (A4) Union: If X Y, X Z F, r fulfills FD X Y Z. (A5) Pseudotransitivity: If X Y, W Y Z F, r fulfills FD XW Z. (A6) Decomposition: If X Y F, Z Y, r fulfills FD X Z. (A7) Reflexivity: If X V, r fulfills FD X X. (A8) Accumulation: If X Y Z, Z AW F, r fullfills X Y ZA. Axiom systems {(A1), (A2), (A3)} and {(A6), (A7), (A8)} are equivalent. 11

12 Theory I: Database Foundations 20. Formal Design Functional Dependencies Membership-Test Variante 2: (Attribut-)closure X + of X (w.r.t. F): X + = {A A V and X A is derivable by (A1) (A3)}. First compute X + by (A6) - (A8) and afterwards test whether Y X +. XPlus-Algorithm XPlus(X, Y, F) boolean { result := X ; WHILE (changes to result) DO FOR each X Y F DO IF (X result) THEN result := result Y ; end. IF (Y result) RETURN true ELSE false; } 12

13 Theory I: Database Foundations 20. Formal Design Functional Dependencies Example XPlus-Algorithm Let V = {A, B, C, D, E, F, G, H, I } and F = {AB E, BE I, E G, GI H}. AB GH F +? Axiom Anwendung result (A7) AB AB {A, B} Using XPlus-Algorithm we can, given V, F, compute a key. How? 13

14 Theory I: Database Foundations 20. Formal Design Functional Dependencies Minimal Cover Equivalence Let F, G sets of FD s. F, G are called equivalent, F G, if F + = G +. 14

15 Theory I: Database Foundations 20. Formal Design Functional Dependencies Left and right reduction A set F of FD s is called left-reduced, if the following condition is fulfilled. If X Y F, Z X, then F = (F \ {X Y }) {Z Y } not equivalent F. left-reduction: replace X Y in F by Z Y. It is called right-reduced, if X Y F, Z Y, then F = (F \ {X Y }) {X Z} not equivalent F. right-reduction: replace X Y in F by X Z. 15

16 Theory I: Database Foundations 20. Formal Design Functional Dependencies looking for possible reductions Let X Y be a FD in F and let Z Y, where Z X. We perform a left-reduction, if XPlus(Z, Y, F) is true. Let X Y a FD in F and let X Z, where Z Y. We perform a right-reduction, if XPlus(X, Y, F ) is true. 16

17 Theory I: Database Foundations 20. Formal Design Functional Dependencies Theorem Let F be a set of FD s and F be derived from F by left-, resp. right-reduction. F F. 17

18 Theory I: Database Foundations 20. Formal Design Functional Dependencies Example F 1 = {A B, B A, B C, A C, C A}. right-reduction? F 2 = {AB C, A B, B A}. left-reduction? 18

19 Theory I: Database Foundations 20. Formal Design Functional Dependencies minimal cover F min is a minimal cover of F, if it is derived from F by the following steps: Perform all possible left-reductions. Perform all possible right-reductions. Delete all trivial FD s of the form X. Compute the union of all FD s X Y 1,..., X Y n to derive X Y 1... Y n. A Minimal cover can be computed in polynomial time. How? F min is not unique, in general. Why? 19

20 Theory I: Database Foundations 20. Formal Design Decomposition 20.3 Decomposition Lossless Let ρ = {X 1,..., X k } a decomposition of V, F a set of FD s. Let r Sat(V, F) and let r i = π[x i ]r, 1 i k. ρ is called lossless, if for any r Sat(V, F) there holds: r = π[x 1 ]r... π[x k ]r. 20

21 Theory I: Database Foundations 20. Formal Design Decomposition Example V = {A, B, C} and F = {A B, A C}). r Sat(V, F): r = ρ 1 = {AB, BC} and ρ 2 = {AB, AC}. r r π[ab]r π[bc]r, π[ab]r π[ac]r. A B C a 1 b 1 c 1 a 2 b 1 c 2 21

22 Theory I: Database Foundations 20. Formal Design Decomposition Theorem Let a format V and set F of FD s. Let ρ = (X 1, X 2 ) be a decomposition of V. ρ is lossless, iff (X 1 X 2 ) (X 1 \ X 2 ) F +, oder (X 1 X 2 ) (X 2 \ X 1 ) F +. 22

23 Theory I: Database Foundations 20. Formal Design Decomposition Dependency Preserving Example V = {A, B, C, D}, ρ = {AB, BC}. F = {A B, B C, C A}. Is ρ dependency preserving w.r.t. F? ConsiderF = {A B, B C, C B, B A}. Is ρ dependency preserving w.r.t. F? 23

24 Theory I: Database Foundations 20. Formal Design Decomposition Definition Let R = (V, F) and Z V. Define the projection of F on Z π[z]f = {X Y F + X Y Z}. A decomposition ρ = {X 1,..., X k } of V is called dependency preserving w.r.t. F, if k π[x i ]F F. i=1 24

25 Theory I: Database Foundations 20. Formal Design Decomposition There exist lossless decompositions which are not dependency preserving! R = (V, F), where V = {Stadt, Adresse, PLZ}, F ={Stadt Adresse PLZ, PLZ Stadt}. ρ = {X 1, X 2 }: X 1 = {Adresse, PLZ} und X 2 = {Stadt, PLZ}. ρ is lossless, as (X 1 X 2 ) (X 2 \ X 1 ) F. ρ is not dependency preserving. What are the keys! 25

26 Theory I: Database Foundations 20. Formal Design Normalform 20.4 Normalform Let R = (V, F). We are looking for a decomposition ρ = (X 1,..., X k ) of R with the following properties: each R i = (X i, π[x i ]F), 1 i k is in normalform, ρ is lossless and, if possible, dependency preserving. k minimal. 26

27 Theory I: Database Foundations 20. Formal Design Normalform Terminology Let X key of R and X Y V, then Y Superkey of R. If A X for any key X of R, then A Keyattribute (KA) of R; if A / X for every key X, then A Non-Keyattribute (NKA). 27

28 Theory I: Database Foundations 20. Formal Design Normalform 3rd Normalform Schema R = (V, F) is in 3rd Normalform (3NF), if for any NKA A V there holds: If X A F, A X, then X Superkey. 28

29 Theory I: Database Foundations 20. Formal Design Normalform 3NF? Stadt SNr SName LCode LFläche 7 Freiburg D Berlin D Moscow RU St.Petersburg RU Kontinent KName LCode KFläche Prozent Europe D Europe RU Asia RU NF? Stadt SNr SName LCode 7 Freiburg D 9 Berlin D 40 Moscow RU 43 St.Petersburg RU Lage LCode KName Prozent D Europe 100 RU Europe 20 RU Asia 80 Land LCode LFläche D 357 RU Kontinent KName KFläche Europe 3234 Asia

30 Theory I: Database Foundations 20. Formal Design Normalform Boyce-Codd-Normalform Schema R = (V, F) is in Boyce-Codd-Normalform (BCNF), if the following holds. If X A F, A X, then X superkey. BCNF implies 3NF. Consider R = (V, F), where V = { Stadt, Adresse, PLZ }, and F ={ Stadt Adresse PLZ, PLZ Stadt}. R is in 3NF, however not in BCNF. Let ρ = {Adresse PLZ, Stadt PLZ} a decomposition, then ρ is in BCNF, lossless and not dependency preserving. 30

31 Theory I: Database Foundations 20. Formal Design Normalization Algorithm 20.5 Normalization Algorithm BCNF-Analysis: lossless and not dependency-preserving Let R = (V, F) a schema. 1 Let X V, A V and X A F a FD, which violates BCNF. Let V = V \ {A}. Decompose R in R 1 = (V, π[v ]F), R 2 = (XA, π[xa]f). 2 Test for BCNF w.r.t. R 1 and R 2 and proceed recursively. 31

32 Theory I: Database Foundations 20. Formal Design Normalization Algorithm 3NF-Analysis: lossless and dependency-preserving Let R = (V, F) a schema and let ρ = (X 1,..., X k ) a decomposition of V, such that the Schemata R 1 = (X 1, π[x 1 ]F),..., R k = (X k, π[x k ]F) in BCNF. 1 Let F min a minimal cover of F. 2 Identify the set F F min of those FD s, which are not dependency preserving. 3 For any such FA, X A extend ρ by XA, resp. schema (XA, π[xa]f). 32

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