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)

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

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)

Functional dependencies ‘Bad’ - why?

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

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

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

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

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

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

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

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

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

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?

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

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

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

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 

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

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

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

FDs - Armstrong’s axioms Prove Pseudo-transitivity:

FDs - Armstrong’s axioms Prove Decomposition

}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

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->B A->C CG->H CG->I B->H} Some members of F+: A->H AG->I CG->HI

}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

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

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

}F FDs - Closure A+ takes(ssn, c-id, grade, name, address) 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 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

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?

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

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 Fc F

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

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

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

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)

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

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