Quine-McClusky Minimization Method

1
Quine-McCluskyMinimization Method

• Module M4.3
• Section 5.3

2
Quine-McCluskey Method

• Tabular Representations
• Prime Implicants
• Essential Prime Implicants

3
Tabular Representations
YZ
00
01
11
10
WX
00
1
01
1
1
1
1
11
1
1
1
10
1
F X Y !W Y !Z W !Y Z !W
X
4
Prime Implicants
Each product term is an implicant
F X !Y Z !X !Z !X Y
A product term that cannot have any of
its variables removed and still imply the
logic function is called a prime implicant.
5
Prime Implicants
-10
1
1
1
1
1
1--
F Y !Z X
6
Prime Implicants
-10
Minterm X Y Z F 0 O O O 0 1 0 0 1
0 2 0 1 0 1 3 1 1 1 0 4 1 O O 1 5 1
0 1 1 6 1 1 0 1 7 1 1 1 1
1--
F Y !Z X
7
Finding Prime Implicants
Step 3
Step 1
Step 2
(2,6) - 1 0
(4,5,6,7) 1 - -
2 0 1 0 4 1 O 0 5 1 0 1 6 1 1 0 7 1 1 1
(4,5) 1 0 -
(4,6,5,7) 1 - -
(4,6) 1 - 0
(5,7) 1 - 1
(6,7) 1 1 -
All unchecked entries are Prime Implicants
-10 Y !Z 1-- X
8
Prime Implicants
-10
Minterm X Y Z F 0 O O O 0 1 0 0 1
0 2 0 1 0 1 3 1 1 1 0 4 1 O O 1 5 1
0 1 1 6 1 1 0 1 7 1 1 1 1
1--
F Y !Z X
9
Essential Prime Implicants
YZ
00
01
11
10
WX
Find the essential prime implicants using the
Q-M method.
1
1
00
1
1
01
1
1
11
1
1
10
1
1
10
Essential Prime Implicants
minterms
YZ
00
01
11
10
0 0000 1 0001 2 0010 8 1000 3 0011 5
0101 10 1010 7 0111 14 1110 15 1111
WX
1
1
00
1
1
01
1
1
11
1
1
10
1
1
11
Finding Prime Implicants
Step 1
Step 2
Step 3
(0,1) 000-
(0,1,2,3) 00--
0 0000 1 0001 2 0010 8 1000 3 0011 5
0101 10 1010 7 0111 14 1110 15 1111
(0,2) 00-0
(0,2,1,3) 00--
(0,8) -000
(0,2,8,10) -0-0
(1,3) 00-1
(1,5) 0-01
(0,8,2,10) -0-0
(2,3) 001-
(1,5,3,7) 0--1
(2,10) -010
(1,3,5,7) 0--1
(8,10) 10-0
(3,7) 0-11
6 Prime Implicants
(5,7) 01-1
1-10 -111 111-
00-- -0-0 0--1
(10,14) 1-10
(7,15) -111
(14,15) 111-
12
Find Essential Prime Implicants
Prime Implicant
Minterms
Covered minterms
0 1 2 3 5 7 8 10 14 15
10,14 7,15 14,15 0,1,2,3 0,2,8,10 1,3,5,7
X
X
1-10 -111 111- 00-- -0-0 0--1
X
X

X
X
X
X
X
X
X
X
X
X
X
X
X
X
13
3 Prime Implicants
F !W Z W X Y !X !Z
YZ
00
01
11
10
WX
1
1
00
1
1
0--1
01
1
1
!W Z
11
1
1
111-
10
1
1
!X !Z
W X Y
-0-0