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Lecture Six Chapter 5: Quine-McCluskey Method
Lecture Six Chapter 5: Quine-McCluskey Method
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Computer Minimization Techniques
Computer Minimization Techniques
• Boolean Algebra
• Karnaugh Maps
• Quine-McCluskey Method
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Boolean AlgebraBoolean Algebra
Review of Boolean Postulates Review of Boolean Identities Example1 Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Review of Boolean Postulates
Review of Boolean Postulates
A & B = B & A A # B = B # A Commutative Laws
A & (B # C) = (A & B) # (A & C) A # (B & C) = (A # B) & (A # C) Distributive Laws(not like ordinary algebra)
1 & A = A 0 # A = A Identity Elements
A & !A = 0 A # !A = 1 Inverse Elements
A # A & B = A A & ( A # B ) = A Absorption
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Review Boolean IdentitiesReview Boolean Identities
0 & A = 0, A & 0 = 0Contradiction (always false)
A # 1 = 1, 1 # A = 1Tautology (always true)
A & A = A A # A = A Idempotence
A & (B & C) = (A & B) & C 0 # A = A Associative Laws
!(A & B) = !A # !B orA NAND B = !A OR !B
!(A # B) = !A & !B orA NOR B = !A AND !B
DeMorgan’s Theorem
!!A = A Involution
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Example1Example1A # A & B = A
Proof:
1. A # A & B = A & 1 # A & B Identity
2. = A & ( 1 # B ) Distribution
3. = A & 1 Identity
4. = A
(X # Y) & (!X # Y) = (X & !X) # (!X & Y) # (X & Y) # (Y & Y) = 0 # (!X & Y) # (X & Y) # Y = (!X # X) & Y # Y = 1 & Y = Y
Proof:
1. (X # Y) & (!X # Y) = !![(X # Y) & (!X # Y)]
2. = ![(!X & !Y) # (X & !Y)] DeMorgan’s
3. = ![(!X # X) & !Y] Distribution
4. = ![1 & !Y] Identity
5. = ![!Y] = Y Involution
Example2Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Karnaugh MapsKarnaugh Maps
A 2 Variable K - mapReview 3 Variable K - maps Example1 Example2 Review 4 Variable K - maps Example1 Example2 A 5 Variable K - map
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
2-Variable K -map2-Variable K -map
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
m0m1
m2 m3
0 1
0
1
X
Y
F(X,Y) = (0,1,2,3)
X
Y
3-Variable K -map3-Variable K -map
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
m0m1
m4m5
m3m2
m7m6
00 01 11 10
0
1
X
YZ
(0,1,2,3,4,5,6,7)
X
Y
Z
Example1Example1
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
F(X,Y,Z) = (1,3,4,5,6,7)
Example1Example1
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
11
00 01 11 10
0
1
X
YZ
1 1
1 1
F(X,Y,Z) = (1,3,4,5,6,7)
Example1Example1
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
11
00 01 11 10
0
1
X
YZ
1 1
1 1
F(X,Y,Z) = (1,3,4,5,6,7) = m1 # m3 # m4 # m5 # m6 # m7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z
Example1Example1
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
11
00 01 11 10
0
1
X
YZ
1 1
1 1
F(X,Y,Z) = (1,3,4,5,6,7) = m1 # m3 # m4 # m5 # m6 # m7 = !X&!Y&Z # !X&Y&Z # X&!Y&!Z # X&!Y&Z # X&Y&!Z # X&Y&Z
Example1Example1
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
11
00 01 11 10
0
1
X
YZ
1 1
1 1
F(X,Y,Z) = X # Z
Example2Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
F(X,Y,Z) = (0,2,4,6)
Example2Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
1
00 01 11 10
0
1
X
YZ
1
1 1
F(X,Y,Z) = (0,2,4,6)
Example2Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
1
00 01 11 10
0
1
X
YZ
1
1 1
F(X,Y,Z) =
Example2Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
1
00 01 11 10
0
1
X
YZ
1
1 1
F(X,Y,Z) = !Z
4-Variable K -map4-Variable K -map
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
m0 m1
m4 m5
m3 m2
m7 m6
00 01 11 10
00
WX
YZ
01
11
10 m8 m9 m11 m10
m12 m13 m15 m14W
Y
X
ZF(W,X,Y,Z) = (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
F(W,X,Y,Z) = (5,7,9,11,13,15)
Example1Example1
Example1Example1
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
00 01 11 10
00
WX
YZ
01
11
10W
Y
X
Z
11
11
1 1
F(W,X,Y,Z) = (5,7,9,11,13,15)
Example1Example1
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
00 01 11 10
00
WX
YZ
01
11
10W
Y
X
Z
11
11
1 1
F(W,X,Y,Z) =X & Z # W & Z =(X # W) & Z
Example2Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
F(W,X,Y,Z) = (2,3,6,7,8,10,11,12,14,15)
Example2Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
00 01 11 10
00
WX
YZ
01
11
10W
Y
X
Z
1
1
1 1
11
1
1
1
1
F(W,X,Y,Z) =(2,3,6,7,8,10,11,12,14,15)
Example2Example2
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
00 01 11 10
00
WX
YZ
01
11
10W
Y
X
Z
1
1
1 1
11
1
1
1
1
F(W,X,Y,Z) = W & !Z # Y
5-Variable K -map5-Variable K -map
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
m0m1
m4 m5
m3m2
m7 m6
00
00
WX
YZ
01
11
10 m8m9 m11
m10
m12 m13 m15 m14W
Y
X
Z
01 11 10
m16m17
m20m21
m19m18
m23m22
00
00
WX
YZ
01
11
10 m24m25 m27
m26
m28 m29 m31 m30W
Y
X
Z
01 11 10
V=0 V=1
Quine-McCluskey MethodQuine-McCluskey Method
Prime Implicants Table3 or 4 steps
Essential Prime Implicants Table
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)
F(W,X,Y,Z) = (5,7,9,11,13,15)
Step 1 Step 2 Step 3
5
9
7
11
13
15
List minterms by the number of 1s it contains.
2
3
4
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)
F(W,X,Y,Z) = (5,7,9,11,13,15)
Step 1 Step 2 Step 3
5 0101
9 1001
7 0111
11 1011
13 1101
15 1111
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)
F(W,X,Y,Z) = (5,7,9,11,13,15)
Step 1 Step 2 Step 3
5 0101 5,7
9 1001 5,13
9,11
7 0111 9,13
11 1011
13 1101 7,15
11,15
15 1111 13,15
Enter combinations of minterms by the number of 1s it contains.
2
3
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)
F(W,X,Y,Z) = (5,7,9,11,13,15)
Step 1 Step 2 Step 3
5 0101 5,7 01-1
9 1001 5,13 -101
9,11 10-1
7 0111 9,13 1-01
11 1011
13 1101 7,15 -111
11,15 1-11
15 1111 13,15 11-1
Check off elements used from Step 1.
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)
F(W,X,Y,Z) = (5,7,9,11,13,15)
Step 1 Step 2 Step 3
5 0101 5,7 01-1 5,7,13,15 -1-1
9 1001 5,13 -101 5,13,7,15 -1-1
9,11 10-1 9,11,13,15 1- -1
7 0111 9,13 1-01 9,13,11,15 1- -1
11 1011
13 1101 7,15 -111
11,15 1-11
15 1111 13,15 11-1
Enter combinations of minterms by the number of 1s it contains.
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)
F(W,X,Y,Z) = (5,7,9,11,13,15)
Step 1 Step 2 Step 3
5 0101 5,7 01-1 5,7,13,15 -1-1
9 1001 5,13 -101 5,13,7,15 -1-1
9,11 10-1 9,11,13,15 1- -1
7 0111 9,13 1-01 9,13,11,15 1- -1
11 1011
13 1101 7,15 -111
11,15 1-11
15 1111 13,15 11-1
The entries left unchecked are Prime Implicants.
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 5 7 9 11 13 15
- 1 - 1 5,7,13,15
1 - - 1 9,13,11,15
Enter the Prime Implicants and their minterms.
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 5 7 9 11 13 15
- 1 - 1 5,7,13,15 X X X X
1 - - 1 9,13,11,15 X X X X
Enter Xs for the minterms covered.
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 5 7 9 11 13 15
- 1 - 1 5,7,13,15 X X X X
1 - - 1 9,13,11,15 X X X X
Circle Xs that are in a column singularly.
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 5 7 9 11 13 15
- 1 - 1 5,7,13,15 X X X X
1 - - 1 9,13,11,15 X X X X
The circled Xs are the Essential Prime Implicants,so we check them off.
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 5 7 9 11 13 15
- 1 - 1 5,7,13,15 X X X X
1 - - 1 9,13,11,15 X X X X
We check off the minterms covered by each of the EPIs.
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 5 7 9 11 13 15
- 1 - 1 5,7,13,15 X X X X
1 - - 1 9,13,11,15 X X X X
W X Y Z
- 1 - 1
1 - - 1
EPIs: F = X & Z # W & Z = (X # W) & Z
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)F(W,X,Y,Z) = (2,3,6,7,8,10,11,12,14,15)
Step 1 Step 2 Step 3 Step 4
2 0010
8 1000
3 0011
6 0110
10 1010
12 1100
7 0111
11 1011
14 1110
15 1111
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)F(W,X,Y,Z) = (2,3,6,7,8,10,11,12,14,15)
Step 1 Step 2 Step 3 Step 4
2 0010 2,3 001-
8 1000 2,6 0-10
2,10 -010
3 0011 8,10 10-0
6 0110 8,12 1-00
10 1010
12 1100 3,7 0-11
3,11 -011
7 0111 6,7 011-
11 1011 6,14 -110
14 1110 10,14 1-10
10,11 101-
15 1111 12,14 11-0
7,15 -111
11,15 1-11
14,15 111-
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)F(W,X,Y,Z) = (2,3,6,7,8,10,11,12,14,15)
Step 1 Step 2 Step 3 Step 4
2 0010 2,3 001- 2,3,6,7 0-1-
8 1000 2,6 0-10 2,6,3,7 0-1-
2,10 -010 2,3,10,11 -01-
3 0011 8,10 10-0 2,6,10,14 - - 10
6 0110 8,12 1-00 2,10,3,11 - 01-
10 1010 2,10,6,14 - - 10
12 1100 3,7 0-11 8,10,12,14 1 - - 0
3,11 -011 8,12,10,14 1 - - 0
7 0111 6,7 011-
11 1011 6,14 -110 3,7,11,15 - - 11
14 1110 10,14 1-10 3,11,7,15 - - 11
10,11 101- 6,7,14,15 - 11 -
15 1111 12,14 11-0 6,14,7,15 - 11 -
10,14,11,15 1 - 1 -
7,15 -111 10,11,14,15 1 - 1 -
11,15 1-11
14,15 111-
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Prime Implicants (PIs)
Finding Prime Implicants (PIs)F(W,X,Y,Z) = (2,3,6,7,8,10,11,12,14,15)
Step 1 Step 2 Step 3 Step 4
2 0010 2,3 001- 2,3,6,7 0-1- 2,3,6,7,10,14,11,15 - - 1 -
8 1000 2,6 0-10 2,6,3,7 0-1- 2,3,10,11,6,14,7,15 - - 1 -
2,10 -010 2,3,10,11 -01- 2,6,3,7,10,11,14,15 - - 1 -
3 0011 8,10 10-0 2,6,10,14 - - 10 2,6,10,14,3,7,11,15 - - 1 -
6 0110 8,12 1-00 2,10,3,11 - 01- 2,10,3,11,6,7,14,15 - - 1 -
10 1010 2,10,6,14 - - 10 2,10,6,14,3,11,7,15 - - 1 -
12 1100 3,7 0-11 8,10,12,14 1 - - 0
3,11 -011 8,12,10,14 1 - - 0
7 0111 6,7 011-
11 1011 6,14 -110 3,7,11,15 - - 11
14 1110 10,14 1-10 3,11,7,15 - - 11
10,11 101- 6,7,14,15 - 11 -
15 1111 12,14 11-0 6,14,7,15 - 11 -
10,14,11,15 1 - 1 -
7,15 -111 10,11,14,15 1 - 1 -
11,15 1-11
14,15 111-
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 2 3 6 7 8 10 11 12 14 15
1 - - 0 8,12,10,14
- - 1 - 2,3,6,7,10,11,14,15
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 2 3 6 7 8 10 11 12 14 15
1 - - 0 8,12,10,14 X X X X
- - 1 - 2,3,6,7,10,11,14,15 X X X X X X X X
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 2 3 6 7 8 10 11 12 14 15
1 - - 0 8,12,10,14 X X X X
- - 1 - 2,3,6,7,10,11,14,15 X X X X X X X X
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 2 3 6 7 8 10 11 12 14 15
1 - - 0 8,12,10,14 X X X X
- - 1 - 2,3,6,7,10,11,14,15 X X X X X X X X
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 2 3 6 7 8 10 11 12 14 15
1 - - 0 8,12,10,14 X X X X
- - 1 - 2,3,6,7,10,11,14,15 X X X X X X X X
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370
Finding Essential Prime Implicants (EPIs)
Finding Essential Prime Implicants (EPIs)
Prime Implicants Covered Minterms Minterms 2 3 6 7 8 10 11 12 14 15
1 - - 0 8,12,10,14 X X X X
- - 1 - 2,3,6,7,10,11,14,15 X X X X X X X X
W X Y Z
1 - - 0
- - 1 -
EPIs: F = (W & !Z) # Y
Dr. S.V. ProvidenceDr. S.V. ProvidenceCOMP 370COMP 370