1. Temple University – CIS Dept. CIS331– Principles of Database Systems
V. Megalooikonomou
Functional Dependencies
(based on notes by Silberchatz,Korth, and Sudarshan and notes by C. Faloutsos at CMU)

2. General Overview Formal query languages Commercial query languages
rel algebra and calculi
Commercial query languages
SQL
QBE, (QUEL)
Integrity constraints
Functional Dependencies
Normalization - ‘good’ DB design

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

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

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

6. Functional dependencies
‘Bad’ - why?

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

8. Functional dependencies
… ‘name’ depends on ‘ssn’
define ‘depends’

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

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

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

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

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

14. 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

15. Functional dependencies
Closure of a set of FD: all implied FDs – e.g.:
ssn -> name, address
ssn, c-id -> grade
imply
ssn, c-id -> grade, name, address
ssn, c-id -> ssn

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

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

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

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

20. FDs – finding the closure F+
F+ = F
repeat for each functional dependency f in F+
apply reflexivity and augmentation rules on f add the resulting functional dependencies to F+ for each pair of functional dependencies f1and f2 in F if f1 and f2 can be combined using transitivity then 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 

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

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

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

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

25. FDs - Armstrong’s axioms
Prove Decomposition

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

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

28. 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

29. }F 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

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

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

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

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

34. 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

35. 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?

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

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

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

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

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

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

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

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

44. FDs - ‘canonical cover’ Fc
Algorithm:
examine each FD; drop extraneous LHS or RHS attributes
merge FDs with same LHS
repeat until no change

45. FDs - ‘canonical cover’ Fc
Trace algo for
AB->C (1)
A->BC (2)
B->C (3)
A->B (4)

46. FDs - ‘canonical cover’ Fc
Trace algo for
AB->C (1)
A->BC (2)
B->C (3)
A->B (4)
(4) and (2) merge:
AB->C (1)
A->BC (2)
B->C (3)

47. FDs - ‘canonical cover’ Fc
AB->C (1)
A->BC (2)
B->C (3)
in (2): ‘C’ is extr.
AB->C (1)
A->B (2’)
B->C (3)

48. FDs - ‘canonical cover’ Fc
AB->C (1)
A->B (2’)
B->C (3)
in (1): ‘A’ is extr.
B->C (1’)
A->B (2’)
B->C (3)

49. FDs - ‘canonical cover’ Fc
B->C (1’)
A->B (2’)
B->C (3)
(1’) and (3) merge
A->B (2’)
B->C (3)
nothing is extraneous:
‘canonical cover’

50. FDs - ‘canonical cover’ Fc
BEFORE
AB->C (1)
A->BC (2)
B->C (3)
A->B (4)
AFTER
A->B (2’)
B->C (3)

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