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).

X  {A | X  A}
X+ enables us to tell at a glance whether a
dependency XA follows from F.
Algorithm to compute X+ under F
// initialization
1. X+:= X; // reflexive rule
// find new attribute
2. For each function dependency YZ 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
Example 1
R(A,B,C,D,E,F), F={A->D, B->E, DB, C->F},
What is {A}+?
Closure of Attribute Set: Example 2
EMP_PROJ(SSN,PNUMBER,HOURS,ENAME,PNAME,PLOCATION)
F={ {SSN}->{ENAME},
{PNUMBER}->{PNAME,PLOCATION},
{SSN,PNUMBER}{HOURS} }
(1) Compute {SSN}+
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
F={ {SSN}->{ENAME},
{PNUMBER}->{PNAME,PLOCATION},
{SSN,PNUMBER}{HOURS} }
(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}+
Use of X+ 1: Checking if XY
•
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+  XY is in F+
•
Does F = {AB, BC, CDE } imply AE?
•
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, AE is not in F+.
Use of X+ 2: Finding a key K
Let R be the set of attributes for a schema and F
be its functional dependency set
Set K:=R
For each attribute A in K
Compute (K-A)+ w.r.t. F
If (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={AB,BC,ABD}; 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+ 3: Compute F+
Given a set of functional dependencies F, we define F+
to be the set of all functional dependencies that can
be inferred from F.
1.
2.
3.
F+ ={};
For each attribute set A in R, computing A+
For each XY implied by A+, add XY to F+
Equivalence of Sets of Functional Dependencies
Let 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 XY in E, calculate X with respect
to F, then check whether X+  Y.
EXAMPLE:
Check whether or not F is equivalent to G.
F={AC, ACD, EAD,EH}
G={ACD, EAH}
•Prove that F is covered by G.
+
+
{A} ={A,C,D} (wrt to G). Since {C}  A , AC can be
inferred from G.
+
{AC} ={A,C,D} (wrt to G). Since {D}  {AC}+, ACD is
covered by G.
{E}+={E,A,H,C,D} (wrt G), Since {AD}  {E}+, EAD is
covered by G.
Since {H}  {E}+, EH is covered by G.
F={AC, ACD, EAD,EH}
G={ACD, EAH}
Prove that G is covered by F:
•{A}+={A,C,D} (wrt F),
Since {CD}  {A}+, ACD is covered by F.
•{E}+={E,A,D,H,C} (with respect to F)
{A,H}  {E}+, EAH is covered by F.
Since F covers G and G covers F, F and G are equivalent.
Minimal Cover of Functional Dependencies
A 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 XA in F with a
dependency YA, 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 Cover
Definition: 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 XA1,A2,…,An in Fmin by the n FDs XA1,…,XAn.
/* minimize the left side of each FD, i.e., every attribute is needed */
3. For each FD XA in Fmin
•
For each B X,
•
Let T=(Fmin- {XA})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 XA in Fmin
Let T=Fmin-{XA}
Check whether T is equivalent to Fmin.
(2)
If (2) is true, set Fmin = T.
Minimal Cover
F={X1Y1, X2Y2, … XnYn} is a minimum cover
1) Any Yi is a single attribute
2) For any XiYi, it is impossible that X’Y and X’ is
a subset of X
3) No XiYi can be taken out
• F’ = F - {XiYi} is not equivalent to F
Example:
Find the minimal cover of the set
F={ABCDE,ED,AB,ACD}.
Example:
Find the minimal cover of the set
F={ABCDE,ED,AB,ACD}.
Step 2: Fmin={ABCDE,ED,AB,ACD}
Step 3:
Replace ACD with AD;
T={ABCDE,ED,AB,AD}
Compute {A}+ wrt to F, {A}+={A,B}
Compute {A}+ wrt to T, {A}+={A,B,D}
Cannot replace Fmin with T.
Is T equivalent to Fmin?
Can Fmin Cover T?
No, keep ACD
Replace ABCDE with ACDE,
T={ACDE,ED,AB,ACD}
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.
Step 3: (cont’d)
Replace ACDE with ACE,
T={ACE,ED,AB,ACD}
Compute {AC}+ wrt to F, {AC}+={ACDBE}
Compute {AC}+ wrt to T,{AC}+={ACEDB}
Replace Fmin with T.
Step 4:
Consider T={ACE,ED,ACD,AB} (take out ACD)
{AC}+={A,C,E,D,B} with respect to T;
{D}  {AC}+; we don’t have to include ACD
The minimal cover of F is:
{ACE,ED,AB}
Some Important Concepts
•
X+ : Closure of an attribute set X
–
•
K: a key
–
•
minimum set of attributes that determines all
attributes
F+ : Closure of a dependency set F
–
•
The set of all attributes that are determined by
X
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