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

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

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

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

Referential Integrity constraints ‘foreign keys’ - eg: create table takes( ssn char(9) not null, c-id char(5) not null, grade integer, primary key(ssn, c-id), foreign key ssn references student, foreign key c-id references class)

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

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

Weapons for IC: assertions create assertion check triggers (~ assertions with ‘teeth’) on operation, if condition, then action

Triggers - example define trigger zerograde on update takes (if new takes.grade < 0 then takes.grade = 0)

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

Overview Domain; Ref. Integrity constraints Assertions and Triggers 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 the ‘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 (eg., 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 - eg.: 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 - eg.: 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: eg., ssn, name -> ssn Augmentation eg., 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 f 1 and f 2 in F + if f 1 and f 2 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 three axioms:

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

FDs - Armstrong’s axioms Prove Pseudo-transitivity:

FDs - Armstrong’s axioms Prove Decomposition

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}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

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, c-id -> grade ssn-> name, address {ssn}+ ={ssn, name, address } }F}F

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

FDs - Closure A+ takes(ssn, c-id, grade, name, address) ssn, c-id -> grade ssn-> name, address {c-id, ssn}+ = ?? }F}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 F result :=  ; 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? 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) C A B

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

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

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 ditto 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

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 AFTER A->B (2’) B->C (3) BEFORE AB->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