temple university – cis dept. cis616– principles of data management

Temple University – CIS Dept.CIS616– Principles of Data Management

V. Megalooikonomou

Functional Dependencies

(based on notes by Silberchatz,Korth, and Sudarshan and notes by C. Faloutsos at CMU)

General Overview

Formal query languages rel algebra and calculi

Commercial query languages SQL QBE, (QUEL)

Integrity constraints Functional Dependencies Normalization - ‘good’ DB design

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

why definition Armstrong’s “axioms” closure and cover

Functional dependencies

motivation: ‘good’ tables

takes1 (ssn, c-id, grade, name, address)

‘good’ or ‘bad’?

Functional dependencies

takes1 (ssn, c-id, grade, name, address)

Ssn c-id Grade Name Address

123 413 A smith Main

123 415 B smith Main

123 211 A smith Main

Functional dependencies

‘Bad’ - why?

Ssn c-id Grade Name Address

123 413 A smith Main

123 415 B smith Main

123 211 A smith Main

Functional Dependencies

Redundancy space inconsistencies insertion/deletion anomalies (later…)

What caused the problem?

Functional dependencies

… ‘name’ depends on ‘ssn’ define ‘depends’

Ssn c-id Grade Name Address

123 413 A smith Main

123 415 B smith Main

123 211 A smith Main

Functional dependencies

Definition: ‘a’ functionally determines ‘b’

Ssn c-id Grade Name Address

123 413 A smith Main

123 415 B smith Main

123 211 A smith Main

ba

Functional dependencies

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

Ssn c-id Grade Name Address

123 413 A smith Main

123 415 B smith Main

123 211 A smith Main

Functional dependenciesformally:

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

])[2][1][2][1( ytytxtxtYX

Functional dependencies

‘X’, ‘Y’ can be sets of attributes other examples??

Ssn c-id Grade Name Address

123 413 A smith Main

123 415 B smith Main

123 211 A smith Main

Functional dependencies

ssn -> name, address ssn, c-id -> grade

Ssn c-id Grade Name Address

123 413 A smith Main

123 415 B smith Main

123 211 A smith Main

Functional dependencies

K is a superkey for relation R iff K -> R

K is a candidate key for relation R iff:K -> Rfor no a K, a -> R

Functional dependencies

Closure of a set of FD: all implied FDs – e.g.:ssn -> name, addressssn, c-id -> grade

implyssn, c-id -> grade, name, addressssn, c-id -> ssn

FDs - Armstrong’s axioms

Closure of a set of FD: all implied FDs – e.g.:ssn -> name, addressssn, c-id -> grade

how to find all the implied ones, systematically?

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

YXXY

YWXWYX

FDs - Armstrong’s axioms

Transitivity

ssn->address address-> county-tax-rateTHEN:

ssn-> county-tax-rate

ZXZY

YX

FDs - Armstrong’s axiomsReflexivity:

Augmentation:

Transitivity:

ZXZY

YX

YXXY

YWXWYX

‘sound’ and ‘complete’

FDs – finding the closure F+

F+ = Frepeat

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

FDs - Armstrong’s axioms

Additional rules:

Union

Decomposition

Pseudo-transitivity

ZXWZYW

YX

ZX

YXYZX

YZXZX

YX

FDs - Armstrong’s axioms

Prove ‘Union’ from the three axioms:

YZXZX

YX

?

FDs - Armstrong’s axioms

Prove ‘Union’ from the three axioms:

YZXtytransitiviand

thusXisXXbut

XZXXXwaugm

YZXZZwaugm

ZX

YX

)4()3(

;

)4(/.)2(

)3(/.)1(

)2(

)1(

FDs - Armstrong’s axioms

Prove Pseudo-transitivity:

ZXWZYW

YX

?

ZXZY

YX

YXXY

YWXWYX

FDs - Armstrong’s axioms

Prove Decomposition

ZXZY

YX

YXXY

YWXWYX

ZX

YXYZX

?

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

FDs - Closure F+

ssn, c-id -> grade ssn-> name, address ssn-> ssn ssn, c-id-> address c-id, address-> c-id ...

F+

FDs - Closure F+

R=(A,B,C,G,H,I)

F= { A->BA->CCG->HCG->IB->H}

Some members of F+:A->HAG->ICG->HI

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

FDs - Closure A+

takes(ssn, c-id, grade, name, address)ssn, c-id -> grade

ssn-> name, address

{ssn}+ ={ssn, name, address }

}F

FDs - Closure A+

takes(ssn, c-id, grade, name, address)ssn, c-id -> grade

ssn-> name, address

{c-id}+ = ??

}F

FDs - Closure A+

takes(ssn, c-id, grade, name, address)ssn, c-id -> grade

ssn-> name, address

{c-id, ssn}+ = ??

}F

FDs - Closure A+

if A+ = {all attributes of table}then ‘A’ is a candidate key

FDs - Closure A+

Algorithm to compute +, the closure of under Fresult := ;while (changes to result) do

for each in F do begin

if result then result := result

end

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 = AG2.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? 1. Is AG a super key?

1. Does AG R? 2. Is any subset of AG a superkey?

1. Does A+ R?2. Does G+ R?

FDs - A+ closure

Diagrams

AB->C (1)A->BC (2)B->C (3)A->B (4)

CA

B

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

FDs - ‘canonical cover’ Fc

ssn, c-id -> grade ssn-> name, address ssn,name-> name, address ssn, c-id-> grade, name

FFc

FDs - ‘canonical cover’ Fc

why do we need it? define it properly compute it efficiently

FDs - ‘canonical cover’ Fc

why do we need it? easier to compute candidate keys

define it properly compute it efficiently

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

‘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

FDs - ‘canonical cover’ Fc

ssn, c-id -> grade ssn-> name, address ssn,name-> name, address ssn, c-id-> grade, name

F

FDs - ‘canonical cover’ Fc

Algorithm: examine each FD; drop extraneous

LHS or RHS attributes merge FDs with same LHS repeat until no change

FDs - ‘canonical cover’ Fc

Trace algo forAB->C (1)A->BC (2)B->C (3)A->B (4)

FDs - ‘canonical cover’ Fc

Trace algo forAB->C (1)A->BC (2)B->C (3)A->B (4) (4) and (2)

merge:

AB->C (1)A->BC (2)B->C (3)

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)

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)

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’

FDs - ‘canonical cover’ Fc

AFTER

A->B (2’)B->C (3)

BEFOREAB->C (1)A->BC (2)B->C (3)A->B (4)

Overview - conclusions

Domain; Ref. Integrity constraints Assertions and Triggers Functional dependencies

why definition Armstrong’s “axioms” closure and cover