Properties of Armstrong’s Axioms

•Soundness• All dependencies generated by the Axioms are correct

•Completeness• Repeatedly applying these rules can generate all

correct dependency (i.e., any FDs in F+ be generated)

Closure of Attribute Set X

Let F be a set of functional dependencies on a set of attributes U and let X U. We define X+ to be the set of all attributes that are dependent on X (under F).

}|{ AXAX

X+ enables us to tell at a glance whether a dependency XA follows from F.

Algorithm to compute X+ under F // initialization1. X+:= X; // reflexive rule

// find new attribute2. For each function dependency YZ in F, if Y X+ and Z has an attribute not in X+, add the new attribute to X+; //transitive rule

3. Repeat Step 2 until no new attribute can be found

R(A,B,C,D,E,F), F={A->D, B->E, DB, C->F}, What is {A}+?

Example 1

Closure of Attribute Set: Example 2EMP_PROJ(SSN,PNUMBER,HOURS,ENAME,PNAME,PLOCATION)

F={ {SSN}->{ENAME}, {PNUMBER}->{PNAME,PLOCATION}, {SSN,PNUMBER}{HOURS} }

Initialization:{SSN}+={SSN}1st iteration: SSN->ENAME, ENAME to is a new attribute {SSN}+={SSN,ENAME}2nd iteration: {SSN}+={SSN,ENAME}, cannot find any new attribute

(1) Compute {SSN}+

(2) Compute {PNUMBER}+

Initialization:{PNUMBER}+={PNUMBER}1st iteration: {PNUMBER} -> {PNAME,PLOCATION} {PNUMBER}+={PNUMBER,PNAME,PLOCATION}2nd iteration: cannot find any new attribute {PNUMBER}+={PNUMBER,PNAME,PLOCATION}

(3) Compute {SSN, PNUMBER}+

F={ {SSN}->{ENAME}, {PNUMBER}->{PNAME,PLOCATION}, {SSN,PNUMBER}{HOURS} }

• Steps of checking if an FD X Y is in the closure of a set of FDs F :1. Compute X+ wrt F.2. Check if Y is in X+.3. Y X+ XY is in F+

• Does F = {AB, BC, CDE } imply AE?• i.e, is A E in F+? Equivalently, is E in A+?• A+ (w.r.t. F)={A,B,C}• E is not in A+, thus, AE is not in F+.

Use of X+ 1: Checking if XY

Set K:=RFor each attribute A in K

Compute (K-A)+ w.r.t. FIf (K-A)+ contains all the attributes in R then set K:= K-A

• This algorithm returns only one key out of the possible candidate keys for R.

• The key returned depends on the order in which attributes are removed from R.

Examples: (1) R={A,B,C,D} F={AB,BC,ABD}; find a key of R.(2) R={A,B,C,D,E,F} F= {A->C, A->D, B->C, E->F}; find a

key of R

Use of X+ 2: Finding a key KLet R be the set of attributes for a schema and F

be its functional dependency set

Given a set of functional dependencies F, we define F+ to be the set of all functional dependencies that can be inferred from F.

Use of X+ 3: Compute F+

1. F+ ={};2. For each attribute set A in R, computing A+

3. For each XY implied by A+, add XY to F+

Equivalence of Sets of Functional DependenciesLet E&F be two sets of functional dependencies.

• F covers E if E F+.• E and F are equivalent if E+=F+.• E+=F+ iff E covers F and F covers E.

Note: Equivalence means that every FD in E can be inferred from F, and every FD in F can be inferred from E.

Determine whether F covers E:For each FD XY in E, calculate X+ with respect to F, then check whether X+ Y.

EXAMPLE: Check whether or not F is equivalent to G.

F={AC, ACD, EAD,EH}G={ACD, EAH}

• Prove that F is covered by G.{A}+={A,C,D} (wrt to G). Since {C} A+, AC can be inferred from G.{AC}+={A,C,D} (wrt to G). Since {D} {AC}+, ACD is covered by G. {E}+={E,A,H,C,D} (wrt G), Since {AD} {E}+, EAD is covered by G.Since {H} {E}+, EH is covered by G.

Prove that G is covered by F:

•{A}+={A,C,D} (wrt F), Since {CD} {A}+, ACD is covered by F.•{E}+={E,A,D,H,C} (with respect to F){A,H} {E}+, EAH is covered by F.

Since F covers G and G covers F, F and G are equivalent.

F={AC, ACD, EAD,EH}G={ACD, EAH}

Minimal Cover of Functional DependenciesA set of functional dependencies F is minimal if it satisfies the following three conditions:

•Every FD in F has a single attribute for its right-hand side.

(This is a standard form, not a requirement.)

•We cannot replace any dependency XA in F with a dependency YA, where Y is a proper subset of X, and still have a set of dependencies that is equivalent to F.

•We cannot remove any dependency from F and still have a set of dependencies that is equivalent to F.

There can be several minimal covers for a set of functional dependencies!

Minimal CoverDefinition: A minimal cover of a set of FDs F is a minimal set of functional dependencies Fmin that is equivalent to F./* The following procedure finds one minimal cover of F. */

Procedure: Find a minimal cover Fmin for F.1. Set Fmin=F/*put every FD in a standard form, i.e., it has a single attribute as its right-hand

side*/2. Replace each FD XA1,A2,…,An in Fmin by the n FDs XA1,…,XAn./* minimize the left side of each FD, i.e., every attribute is needed */3. For each FD XA in Fmin

• For each B X,• Let T=(Fmin- {XA})U{(X-{B})A}• Check whether T is equivalent to Fmin (1)• If (1) is true, then set Fmin = T.

/* delete redundant FDs, i.e., No redundant FDs remain in Fmin. */4. For each FD XA in Fmin

Let T=Fmin-{XA}Check whether T is equivalent to Fmin. (2)If (2) is true, set Fmin = T.

Minimal Cover

F={X1Y1, X2Y2, … XnYn} is a minimum cover

1) Any Yi is a single attribute2) For any XiYi, it is impossible that X’Y and X’

is a subset of X3) No XiYi can be taken out

• F’ = F - {XiYi} is not equivalent to F

Example:Find the minimal cover of the set F={ABCDE,ED,AB,ACD}.

Example:Find the minimal cover of the set F={ABCDE,ED,AB,ACD}.Step 2: Fmin={ABCDE,ED,AB,ACD}

Step 3: Replace ACD with AD; T={ABCDE,ED,AB,AD}

Compute {A}+ wrt to F, {A}+={A,B}Compute {A}+ wrt to T, {A}+={A,B,D}

Cannot replace Fmin with T.

Replace ABCDE with ACDE,T={ACDE,ED,AB,ACD}Compute {ACD}+ wrt to F, {ACD}+={A,C,D,B,E}Compute {ACD}+ wrt to T,{ACD}+={A,C,D,E,B}Replace Fmin with T.

Is T equivalent to Fmin?Can Fmin Cover T?No, keep ACD

Step 3: (cont’d)Replace ACDE with ACE,T={ACE,ED,AB,ACD}Compute {AC}+ wrt to F, {AC}+={ACDBE}Compute {AC}+ wrt to T,{AC}+={ACEDB}Replace Fmin with T.

Step 4:Consider T={ACE,ED,ACD,AB} (take out ACD){AC}+={A,C,E,D,B} with respect to T;{D} {AC}+; we don’t have to include ACD

The minimal cover of F is:{ACE,ED,AB}

Some Important Concepts• X+ : Closure of an attribute set X

– The set of all attributes that are determined by X

• K: a key – minimum set of attributes that determines

all attributes • F+ : Closure of a dependency set F

– The set of all dependencies that are implied from F

• Fmin: a minimum cover of a dependency set F

– a minimum set of FDs that is equivalent to F