Functional Dependencies

Basics of FDs Manipulating FDs Closures and Keys Minimal Bases

T. M. Murali

October 18, 25, 2010

T. M. Murali October 18, 25, 2010 CS 4604: Functional Dependencies

Midterm Review

I The database server set up is driving me NUTS!

I The class is slow (no 3NF yet). Why are we doing design beforelearning normalisation? It seems odd to teach relational models afterSQL.

I Homework grading is strict/uneven.

I Increase amount of examples/hands-on work.

I Make the lectures more interactive.

I Office hours change a lot.

I Wish the project was free form.

I It would be nice to have a table of RA symbols and E/R diagramshapes.

Midterm Review

Example

Students

NameName

AdvisorsAdvises

Favourite

I Convert to relations:

Students(Id, Name)

Advisors(Id, Name)

Advises(StudentId, AdvisorId)

Favourite(StudentId, AdvisorId)

I Suppose we perversely decide to convert Students, Advises, andFavourite into one relation.

Students(Id, Name, AdvisorId, AdvisorName,

FavouriteAdvisorId)

Example

Students

NameName

AdvisorsAdvises

Favourite

Students(Id, Name)

Advisors(Id, Name)

FavouriteAdvisorId)

Example

Students

NameName

AdvisorsAdvises

Favourite

Students(Id, Name)

Advisors(Id, Name)

FavouriteAdvisorId)

Example of a Bad Relation

FavouriteAdvisorId)

I If you know a student’s Id, can you determine the values of any otherattributes?

Name and FavouriteAdvisorId.

Id→ Name

Id→ FavouriteAdvisorId

AdvisorId→ AdvisorName

I Can we say Id→ AdvisorId? NO! Id is not a key for the Students

relation.

I What is the key for the Students? {Id, AdvisorId}.I Why is this relation “bad?” Parts of the key determine other

attributes.

FavouriteAdvisorId)

I If you know a student’s Id, can you determine the values of any otherattributes? Name and FavouriteAdvisorId.

Id→ Name

relation.

attributes.

FavouriteAdvisorId)

Id→ Name

relation.

attributes.

FavouriteAdvisorId)

Id→ Name

AdvisorId→

AdvisorName

relation.

attributes.

FavouriteAdvisorId)

Id→ Name

relation.

attributes.

FavouriteAdvisorId)

Id→ Name

I Can we say Id→ AdvisorId?

NO! Id is not a key for the Students

relation.

attributes.

FavouriteAdvisorId)

Id→ Name

relation.

attributes.

FavouriteAdvisorId)

Id→ Name

relation.

I What is the key for the Students?

{Id, AdvisorId}.I Why is this relation “bad?” Parts of the key determine other

attributes.

FavouriteAdvisorId)

Id→ Name

relation.

I What is the key for the Students? {Id, AdvisorId}.

I Why is this relation “bad?” Parts of the key determine otherattributes.

FavouriteAdvisorId)

Id→ Name

relation.

I What is the key for the Students? {Id, AdvisorId}.I Why is this relation “bad?”

Parts of the key determine otherattributes.

FavouriteAdvisorId)

Id→ Name

relation.

attributes.

Motivation for Functional Dependencies

I Reason about constraints on attributes in a relation.

I Procedurally determine the keys of a relation.

I Detect when a relation has redundant information.

I Improve database designs systematically using normalisation.

Definition of Functional Dependency

I If t is a tuple in a relation R and A is an attribute of R, then tA is thevalue of attribute A in tuple t.

I The FD AdvisorId→ AdvisorName holds in R if in every instanceof R, for every pair of tuples t and u

if tAdvisorId = uAdvisorId, then tAdvisorName = uAdvisorName.

Definition of Functional Dependencies

A functional dependency (FD) on a relation R is a statement

I If two tuples in R agree on attributes A1,A2, . . .An then they agreeon attribute B.

I Notation: A1A2 . . .An → B

I FD says that for every pair of tuples t and u in any instance of R, iftA1 = uA1 and tA2 = uA2 and . . . tAn = uAn , then tB = uB .

I The set of attributes A1,A2, . . .An functionally determine B.

I An FD is a constraint on a single relation schema. It must hold onevery instance of the relation.

I You cannot deduce an FD from a relation instance.

I FD says that for every pair of tuples t and u in any instance of R,

iftA1 = uA1 and tA2 = uA2 and . . . tAn = uAn , then tB = uB .

I FD says that for every pair of tuples t and u in any instance of R, iftA1 = uA1 and tA2 = uA2 and . . . tAn = uAn , then tB = uB .

Examples of FDs

I What FDs can we assert for the relation

Courses(Number, DeptName, CourseName, Classroom, Enrollment)

Number DeptName CourseName Classroom Enrollment

4604 CS Databases TORG 1020 454604 Dance Tree Dancing Drillfield 454604 English The Basis of Data Williams 44 452604 CS Data Structures MCB 114 1002604 Physics Dark Matter Williams 44 100

Number DeptName→ CourseName

Number DeptName→ Classroom

Number DeptName→ Enrollment

Number DeptName→ CourseName Classroom Enrollment

I Is Number→ Enrollment an FD?

Examples of FDs

Where do FDs come from?

I “Keyness” of attributes.

I Domain and application constraints.

I Real world constraints, e.g., ProfessorID Time→ Classroom

Where do FDs come from?

I “Keyness” of attributes.

I Domain and application constraints.

I Real world constraints, e.g., ProfessorID Time→ Classroom

Definition of Keys

I FDs allow us to formally define keys.

I A set of attributes {A1,A2, . . .An} is a key for a relation R if

Uniqueness {A1,A2, . . .An} functionally determine all the otherattributes of R and

Minimality no proper subset of {A1,A2, . . .An} functionallydetermines all the other attributes of R.

I A superkey is a set of attributes that has the uniqueness property butis not necessarily minimal.

I If a relation has multiple keys, specify one to be the primary key.

I Convention: in a relational schema, underline the attributes of theprimary key.

I If a key has only one attribute A, we say that A rather than {A} is akey.

Definition of Keys

I FDs allow us to formally define keys.

I A set of attributes {A1,A2, . . .An} is a key for a relation R if

Uniqueness {A1,A2, . . .An} functionally determine all the otherattributes of R and

Minimality no proper subset of {A1,A2, . . .An} functionallydetermines all the other attributes of R.

I A superkey is a set of attributes that has the uniqueness property butis not necessarily minimal.

I If a relation has multiple keys, specify one to be the primary key.

I Convention: in a relational schema, underline the attributes of theprimary key.

I If a key has only one attribute A, we say that A rather than {A} is akey.

Example of Keys

I What is the key for

Courses(Number, DeptName, CourseName, Classroom,

Enrollment)?

I The key is {Number, DeptName}.I These attributes functionally determine every other attribute.I No proper subset of {Number, DeptName} has this property.

Teach(, , ProfessorName, Classroom)?

I The key is {Number, DepartmentName}. Why?

Example of Keys

Enrollment)?

Teach(, , ProfessorName, Classroom)?

Example of Keys

Enrollment)?

Teach(Number, DepartmentName, ProfessorName,

Classroom)?

Example of Keys

Enrollment)?

Teach(Number, DepartmentName, ProfessorName,

Classroom)?

Keys in the Conversion from E/R to RelationalDesigns

I If the relation comes from an entity set, the key attributes of therelation are precisely the key attributes of the entity set.

I If the relation comes from a binary relationship R between entity setsE and F :

I R is many-many: key attributes of the relation are the key attributes ofE and of F .

I R is many-one from E to F : key attributes of the relation are the keyattributes of E .

I R is one-one: key attributes of the relation are the key attributes of Eor of F .

I If the relationship R is multiway, we need to reason about the FDsthat R satisfies.

I There is no simple rule.I If R has an arrow towards entity set E , at least one key for the relation

for R excludes the key for E .

I R is many-many:

key attributes of the relation are the key attributes ofE and of F .

I R is many-one from E to F :

key attributes of the relation are the keyattributes of E .

I R is one-one:

key attributes of the relation are the key attributes of Eor of F .

I If the relationship R is multiway

, we need to reason about the FDsthat R satisfies.

Rules for Manipulating FDs

I Learn how to reason about FDs.

I Define rules for deriving new FDs from a given set of FDs.

I Next class: use these rules to remove “anomalies” from relationaldesigns.

I Example: A relation R with attributes A, B, and C , satisfies the FDsA→ B and B → C . What other FDs does it satisfy? A→ C .

I What is the key for R? A, because A→ B and A→ C .

I Example: A relation R with attributes A, B, and C , satisfies the FDsA→ B and B → C . What other FDs does it satisfy?

A→ C .

I What is the key for R?

A, because A→ B and A→ C .

Equivalence of FDs

I An FD F follows from a set of FDs T if every relation instance thatsatisfies all the FDs in T also satisfies F .

I A→ C follows from T = {A→ B,B → C}.I Does T follow from S?

I Two sets of FDs S and T are equivalent if each FD in S follows fromT and each FD in T follows from S .

I S = {A→ B,B → C ,A→ C} and T = {A→ B,B → C} areequivalent.

I These notions are useful in deriving new FDs from a given set of FDs.

Equivalence of FDs

Splitting and Combining FDs

I The set of FDs

A1A2 . . .An → B1

A1A2 . . .An → B2

A1A2 . . .An → Bm

is equivalent to the FD

A1A2 . . .An → B1B2 . . .Bm.I This equivalence implies two rules.

I Splitting ruleI Combining ruleI These rules work because all the FDs in S and T have identical left

hand sides.

I Can we split and combine left hand sides of FDs?

I For the relation Courses, is the FD

equivalent to the set of FDs

{Number→ CourseName, DeptName→ CourseName}? NO!

{Number→ CourseName, DeptName→ CourseName}?

{Number→ CourseName, DeptName→ CourseName}? NO!

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

I trivial if the B’s are a subset of the A’s, i.e.,{B1,B2, . . .Bn} ⊆ {A1,A2, . . .An}.

I non-trivial if at least one B is not among the A’s, i.e.,{B1,B2, . . .Bn} − {A1,A2, . . .An} 6= ∅.

I completely non-trivial if none of the B’s are among the A’s, i.e.,{B1,B2, . . .Bn} ∩ {A1,A2, . . .An} = ∅.

I What good are trivial and non-trivial dependencies?I Trivial dependencies are always true.I They help simplify reasoning about FDs.

I Trivial dependency rule: The FD A1A2 . . .An → B1B2 . . .Bm isequivalent to the FD A1A2 . . .An → C1C2 . . .Ck , where the C ’s arethose B’s that are not A’s, i.e.,{C1,C2, . . . ,Ck} = {B1,B2, . . . ,Bm} − {A1,A2, . . . ,An}.

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

I trivial if the B’s are a subset of the A’s,

i.e.,{B1,B2, . . .Bn} ⊆ {A1,A2, . . .An}.

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

I non-trivial if at least one B is not among the A’s,

i.e.,{B1,B2, . . .Bn} − {A1,A2, . . .An} 6= ∅.

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

I completely non-trivial if none of the B’s are among the A’s,

i.e.,{B1,B2, . . .Bn} ∩ {A1,A2, . . .An} = ∅.

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

Triviality of FDs

An FD A1A2 . . .An → B1B2 . . .Bm is

Closures of Attributes: Example

Suppose a relation with attributes A,B,C ,D,E , and F satisfies the FDs

AB → C BC → AD D → E , CF → B

Given these FDs,

I what is the set X of attributes such that AB → X is true?

X = {A,B,C ,D,E}, i.e., AB → ABCDE .

I what is the set Y of attributes such that BCF → Y is true?Y = {A,B,C ,D,E ,F}, i.e., BCF → ABCDEF

I what is the set Z of attributes such that AF → Z is true?Z = {A,F}, i.e., AF → AF .

I X ,Y , and Z are the closures of {A,B}, {B,C ,F}, and {A,F},respectively.

I {B,C ,F} is a superkey.

Given these FDs,

I what is the set X of attributes such that AB → X is true?X = {A,B,C ,D,E}, i.e., AB → ABCDE .

Given these FDs,

I what is the set Y of attributes such that BCF → Y is true?

Y = {A,B,C ,D,E ,F}, i.e., BCF → ABCDEF

Given these FDs,

I what is the set Z of attributes such that AF → Z is true?

Z = {A,F}, i.e., AF → AF .

Given these FDs,

I {B,C ,F} is a

superkey.

Given these FDs,

Closures of Attributes: Definition

I a set of attributes {A1,A2, . . . ,An} and

I a set of FDs S ,

the closure of {A1,A2, . . . ,An} under the FDs in S is

I the set of attributes {B1,B2, . . . ,Bm} such that for 1 ≤ i ≤ m, theFD A1A2 . . .An → Bi follows from S .

I the closure is denoted by {A1,A2, . . . ,An}+.

I Which attributes must {A1,A2, . . . ,An}+ contain at a minimum?{A1,A2, . . . ,An}. Why? A1A2 . . .An → Ai is a trivial FD.

I a set of FDs S ,

I Which attributes must {A1,A2, . . . ,An}+ contain at a minimum?

{A1,A2, . . . ,An}. Why? A1A2 . . .An → Ai is a trivial FD.

I a set of FDs S ,

I Which attributes must {A1,A2, . . . ,An}+ contain at a minimum?{A1,A2, . . . ,An}. Why?

A1A2 . . .An → Ai is a trivial FD.

I a set of FDs S ,

I Which attributes must {A1,A2, . . . ,An}+ contain at a minimum?{A1,A2, . . . ,An}. Why? A1A2 . . .An → Ai is a trivial FD.

Closures of Attributes: Algorithm

I a set of attributes {A1,A2, . . . ,An} andI a set of FDs S ,I compute X = {A1,A2, . . . ,An}+.

1. Use the splitting rule so that each FD in S has one attribute on theright.

2. Set X ← {A1,A2, . . . ,An}.3. Find an FD B1B2 . . .Bk → C in S such that {B1,B2, . . .Bk} ⊆ X but

C 6∈ X .

4. Add C to X .

5. Repeat the last two steps until you cannot find such an attribute C .

6. The final value of X is the desired closure.

I Why does the algorithm compute the closure correctly? Read Chapter3.2.5 of the textbook.

C 6∈ X .

4. Add C to X .

C 6∈ X .

4. Add C to X .

Why is the Concept of Closures Useful?

1. Prove correctness of rules for manipulating FDs.

For example,I Transitive rule: if

A1A2 . . .An → B1B2 . . .Bm

andB1B2 . . .Bm → C1C2 . . .Ck

thenA1A2 . . .An → C1C2 . . .Ck .

I To prove this rule, simply check if{C1,C2, . . . ,Ck} ⊆ {A1,A2, . . . ,An}+.

2. Check if a “new” FD A1A2 . . .An → B follows from a set of FDs S :simply check if B is in {A1,A2, . . .An}+ under S .

3. Procedurally define keys. A set of attributes X is a key for a relationR if and only if

I {X}+ is the set of all attributes of R andI for no attribute A ∈ X is {X − {A}}+ the set of all attributes of R.

1. Prove correctness of rules for manipulating FDs. For example,I Transitive rule: if

A1A2 . . .An → B1B2 . . .Bm

andB1B2 . . .Bm → C1C2 . . .Ck

thenA1A2 . . .An → C1C2 . . .Ck .

A1A2 . . .An → B1B2 . . .Bm

andB1B2 . . .Bm → C1C2 . . .Ck

thenA1A2 . . .An → C1C2 . . .Ck .

A1A2 . . .An → B1B2 . . .Bm

andB1B2 . . .Bm → C1C2 . . .Ck

thenA1A2 . . .An → C1C2 . . .Ck .

A1A2 . . .An → B1B2 . . .Bm

andB1B2 . . .Bm → C1C2 . . .Ck

thenA1A2 . . .An → C1C2 . . .Ck .

3. Procedurally define keys.

A set of attributes X is a key for a relationR if and only if

A1A2 . . .An → B1B2 . . .Bm

andB1B2 . . .Bm → C1C2 . . .Ck

thenA1A2 . . .An → C1C2 . . .Ck .

Examples of Closure Computations

I Consider the “bad” relation Students(Id, Name, AdvisorId,

AdvisorName, FavouriteAdvisorId).

I What are the FDs that hold in this relation?

Id→ Name

I To compute the key for this relation,

1. Compute the closures for all sets of attributes.2. Find the minimal set of attributes whose closure is the set of all

attributes.

Examples of Closure Computations

I Consider the “bad” relation Students(Id, Name, AdvisorId,

AdvisorName, FavouriteAdvisorId).

I What are the FDs that hold in this relation?

Id→ Name

I To compute the key for this relation,

1. Compute the closures for all sets of attributes.2. Find the minimal set of attributes whose closure is the set of all

attributes.

Algorithm for Computing Keys

I a relation R(A1,A2, . . . ,An) and

I the set of all FDs S that hold in R,

I compute all the keys of R.

1. For every subset K ⊆ {A1,A2, . . . ,An}, compute {K}+.

2. If {K}+ = {A1,A2, . . . ,An} and for every attribute A,{K − {A}}+ 6= {A1,A2, . . . ,An}, then output K as a key.

I Running time of the algorithm is exponential in the number ofattributes.

Armstrong’s Axioms

I We can use closures of attributes to determine if any FD follows froma given set of FDs.

I Armstrong’s axioms: complete set of inference rules from which it ispossible to derive every FD that follows from a given set:

Reflexivity If {B1,B2, . . . ,Bm} ⊆ {A1,A2, . . . ,An}, thenA1A2 . . .Am → B1B2 . . .Bm (Trivial FDs).

Augmentation If A1A2 . . .Am → B1B2 . . .Bm, thenA1A2 . . .AmC1C2 . . .Ck → B1B2 . . .BmC1C2 . . .Ck , forany set of attributes {C1,C2, . . . ,Ck}.

Transitivity If A1A2 . . .An → B1B2 . . .Bm andB1B2 . . .Bm → C1C2 . . .Ck thenA1A2 . . .An → C1C2 . . .Ck .

Differences in Notation

Relation schema R(A1,A2,A3): parentheses surround attributes,attributes separated by commas.

Set of attributes {A1,A2,A3}: curly braces surround attributes, attributesseparated by commas.

FD A1A2 → A3: no parentheses or curly braces, attributesseparated by spaces, arrows separates left hand side and righthand side.

Set of FDs {A1A2 → A3,A2 → A1}: curly braces surround FDs, FDsseparated by commas.

Minimal Basis

I A relation may have a large set of equivalent sets of FDs.

I If we are given a set S of FDs, then any set of FDs that is equivalentto S is called a basis of S .

I We will limit ourselves to bases in which each FD has only oneattribute on the right hand side.

I A set B of FDs is a minimal basis for a relation R if

1. Every FD in B has one attribute on the right hand side.2. If we remove any FD from B, then the result is not a basis.3. If for any FD in B, we remove one or more attributes from the left

hand side of the FD, then the result is not a basis.

Minimal Basis

Example of Minimal Basis

I R(A,B,C ) is a relation such that each attribute functionallydetermines the other two attributes.

I What are the FDs that hold in R and what are the minimal bases?(Assume only one attribute on the right-hand side, only non-trivialFDs)

I FDs: A→ B, A→ C , B → A, B → C , C → A, C → B, AB → C ,BC → A, AC → B.

I Minimal bases: {A→ B,B → A,B → C ,C → B},{A→ B,B → C ,C → A}, and others.

I FDs:

A→ B, A→ C , B → A, B → C , C → A, C → B, AB → C ,BC → A, AC → B.

I FDs: A→ B, A→ C , B → A, B → C , C → A, C → B,

AB → C ,BC → A, AC → B.

I Minimal bases:

{A→ B,B → A,B → C ,C → B},{A→ B,B → C ,C → A}, and others.

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