Dept. of CIS, Temple Univ. CIS616/661 – Principles of Data Management

1. Dept. of CIS, Temple Univ.CIS616/661 – Principles of Data Management V. Megalooikonomou Integrity Constraints (based on slides by C. Faloutsos at CMU)

2. Constraints: Integrity constraints in the E-R Model: • Key • cardinalities of a Relationship

3. Overview • Domain; Ref. Integrity constraints • Assertions and Triggers • Functional dependencies

4. Domain Constraints • Domain Types, eg, SQL • Fixed Length characters • Int; float; (date) • null values eg., • create table student( ssn char(9) not null, ...)

5. Referential Integrity constraints ‘foreign keys’ - eg: create table takes( ssn char(9) notnull, c-id char(5) notnull, grade integer, primarykey(ssn, c-id), foreignkey ssn references student, foreignkey c-id references class)

6. Referential Integrity constraints … foreignkey ssn references student, foreignkey c-id references class) Effect: • expects that ssn exists in ‘student’ table • blocks ops that violate that - how?? • insertion? • deletion/update?

7. Referential Integrity constraints … foreignkey ssn references student on delete cascade on update cascade, ... • -> eliminate all student enrollments • other options (set to null, to default etc)

8. Weapons for IC: • assertions • create assertion <assertion-name> check <predicate> • triggers (~ assertions with ‘teeth’) • on operation, if condition, then action

9. Triggers - example definetrigger zerograde onupdate takes (if new takes.grade < 0 then takes.grade = 0)

10. Triggers - discussion more complicated: “managers have higher salaries than their subordinates” - a trigger can automatically boost mgrs salaries triggers: tricky (infinite loops…)

11. Overview • Domain; Ref. Integrity constraints • Assertions and Triggers • Functional dependencies • why • definition • Armstrong’s “axioms” • closure and cover

12. Functional dependencies motivation: ‘good’ tables takes1 (ssn, c-id, grade, name, address) ‘good’ or ‘bad’?

13. Functional dependencies takes1 (ssn, c-id, grade, name, address)

14. Functional dependencies ‘Bad’ - why?

15. Functional Dependencies • Redundancy • space • inconsistencies • insertion/deletion anomalies (later…) • What caused the problem?

16. Functional dependencies ‘name’ depends on the ‘ssn’ define ‘depends’

17. Functional dependencies Definition: ‘a’ functionally determines ‘b’

18. Functional dependencies Informally: ‘if you know ‘a’, there is only one ‘b’ to match’

19. Functional dependencies formally: if two tuples agree on the ‘X’ attribute, they *must* agree on the ‘Y’ attribute, too (eg., if ssn is the same, so should address) … a functional dependency is a generalization of the notion of a key

20. Functional dependencies ‘X’, ‘Y’ can be sets of attributes other examples??

21. Functional dependencies ssn -> name, address ssn, c-id -> grade

22. Functional dependencies K is a superkey for relation R iff K -> R K is a candidate key for relation R iff: K -> R for no a  K, a -> R

23. Functional dependencies Closure of a set of FD: all implied FDs - eg.: ssn -> name, address ssn, c-id -> grade imply ssn, c-id -> grade, name, address ssn, c-id -> ssn

24. FDs - Armstrong’s axioms Closure of a set of FD: all implied FDs - eg.: ssn -> name, address ssn, c-id -> grade how to find all the implied ones, systematically?

25. FDs - Armstrong’s axioms “Armstrong’s axioms” guarantee soundness and completeness: Reflexivity: eg., ssn, name -> ssn Augmentation eg., ssn->name then ssn,grade-> ssn,grade

26. FDs - Armstrong’s axioms Transitivity ssn->address address-> county-tax-rate THEN: ssn-> county-tax-rate

27. FDs - Armstrong’s axioms Reflexivity: Augmentation: Transitivity: ‘sound’ and ‘complete’

28. FDs – finding the closure F+ F+ = F repeatfor each functional dependency f in F+ apply reflexivity and augmentation rules on fadd the resulting functional dependencies to F+for each pair of functional dependencies f1and f2 in F+iff1 and f2 can be combined using transitivitythen add the resulting functional dependency to F+ until F+ does not change any further We can further simplify manual computation of F+ by using the following additional rules.

29. FDs - Armstrong’s axioms Additional rules: Union Decomposition Pseudo-transitivity

30. FDs - Armstrong’s axioms Prove ‘Union’ from three axioms:

31. FDs - Armstrong’s axioms Prove ‘Union’ from three axioms:

32. FDs - Armstrong’s axioms Prove Pseudo-transitivity:

33. FDs - Armstrong’s axioms Prove Decomposition

34. FDs - Closure F+ Given a set F of FD (on a schema) F+ is the set of all implied FD. Eg., takes(ssn, c-id, grade, name, address) ssn, c-id -> grade ssn-> name, address }F

35. FDs - Closure F+ ssn, c-id -> grade ssn-> name, address ssn-> ssn ssn, c-id-> address c-id, address-> c-id ... F+

36. FDs - Closure F+ R=(A,B,C,G,H,I) F= { A->B A->C CG->H CG->I B->H} Some members of F+: A->H AG->I CG->HI

37. FDs - Closure A+ Given a set F of FD (on a schema) A+ is the set of all attributes determined by A: takes(ssn, c-id, grade, name, address) ssn, c-id -> grade ssn-> name, address {ssn}+ =?? }F

38. FDs - Closure A+ takes(ssn, c-id, grade, name, address) ssn, c-id -> grade ssn-> name, address {ssn}+ ={ssn, name, address } }F

39. FDs - Closure A+ takes(ssn, c-id, grade, name, address) ssn, c-id -> grade ssn-> name, address {c-id}+ = ?? }F

40. FDs - Closure A+ takes(ssn, c-id, grade, name, address) ssn, c-id -> grade ssn-> name, address {c-id, ssn}+ = ?? }F

41. FDs - Closure A+ if A+ = {all attributes of table} then ‘A’ is a candidate key

42. FDs - Closure A+ Algorithm to compute a+, the closure of a under F result := a;while (changes to result) do for each in F do begin if  result then result := result end

43. FDs - Closure A+ (example) R = (A, B, C, G, H, I) F = {A B, A C, CG H, CG I, B H} (AG)+ 1. result = AG 2. result = ABCG (A C and A  B) 3. result = ABCGH (CG H and CG  AGBC) 4. result = ABCGHI (CG I and CG  AGBCH) Is AG a candidate key? Is AG a super key? Does AG R? Is any subset of AG a superkey? Does A+R? Does G+R?

44. FDs - A+ closure C A B Diagrams AB->C (1) A->BC (2) B->C (3) A->B (4)

45. FDs - ‘canonical cover’ Fc Given a set F of FD (on a schema) Fc is a minimal set of equivalent FD. Eg., takes(ssn, c-id, grade, name, address) ssn, c-id -> grade ssn-> name, address ssn,name-> name, address ssn, c-id-> grade, name F

46. FDs - ‘canonical cover’ Fc ssn, c-id -> grade ssn-> name, address ssn,name-> name, address ssn, c-id-> grade, name Fc F

47. FDs - ‘canonical cover’ Fc why do we need it? define it properly compute it efficiently

48. FDs - ‘canonical cover’ Fc why do we need it? easier to compute candidate keys define it properly compute it efficiently

49. FDs - ‘canonical cover’ Fc define it properly - three properties every FD a->b has no extraneous attributes on the RHS ditto for the LHS all LHS parts are unique

50. ‘extraneous’ attribute: if the closure is the same, before and after its elimination or if F-before implies F-after and vice-versa FDs - ‘canonical cover’ Fc