Digital Electronics. Simplification of Boolean algebra using qunine Mcclusky method or Tabulation method

1. Quine-McCluskey
(Tabular) Minimization
1
Dr.S.Joshua Kumaresan M.E, M.S, Ph.D
Associate professor/ECE
R.M.K Engineering college

2. Quine-McCluskey (Tabular)
Minimization
 Two step process utilizing tabular listings to:
 Identify prime implicants (implicant tables)
 Identify minimal PI set (cover tables)
 All work is done in tabular form
 Number of variables is not a limitation
 Basis for many computer implementations
 Don’t cares are easily handled
 Proper organization and term identification are key
factors for correct results

3. EXAMPLE 1:
3
Simplify the Boolean Expression using Quine
McClusky method (Tabular Method)
F ( A, B, C, D) 
m(0,1,3,7,8,9,11,15)

4. CONVERT DECIMAL NUMBERS TO
BINARY NUMBERS
4
DECIMAL
NUMBER
EQUIVALENT
BINARY
NUMBER
MINTERMS
0 0000 m0
1 0001 m1
3 0011 m3
7 0111 m7
8 1000 m8
9 1001 m9
11 1011 m11
15 1111 m15

5. 5
STEP 1:
Arrange all Minterms according to number of 1
as shown in column 2
Minterm
No.
IN BINARY
A B C D
0 0 0 0 0
1 0 0 0 1
8 1 0 0 0
3 0 0 1 1
9 1 0 0 1
7 0 1 1 1
11 1 0 1 1
15 1 1 1 1
COLUMN : 2

6. 6
STEP: 2
Compare each minterm in group ‘n’ with each minterm in group (n+1) and
identify the match pairs. A match pair is a pair of minterms which differ only
in one variable. For the variables differ place (-) dash, as shown in column 3
Minterm
No.
IN BINARY
A B C D
0 0 0 0 0
1 0 0 0 1
8 1 0 0 0
3 0 0 1 1
9 1 0 0 1
7 0 1 1 1
11 1 0 1 1
15 1 1 1 1
column: 3
Minterm
No.
IN BINARY
A B C D
(0,1) 0 0 0 -
(0,8) - 0 0 0
(1,3) 0 0 - 1
(1,9) - 0 0 1
(8,9) 1 0 0 -
(3,7) 0 - 1 1
(3,11) - 0 1 1
(9,11) 1 0 - 1
Column: 2

7. Minterm
No. IN BINARY
A B C D
0 0 0 0 0
1 0 0 0 1
8 1 0 0 0
3 0 0 1 1
9 1 0 0 1
7 0 1 1 1
11 1 0 1 1
15 1 1 1 1
Column : 3
Minterm
No.
IN BINARY
A B C D
(0,1) 0 0 0 -
(0,8) - 0 0 0
(1,3) 0 0 - 1
(1,9) - 0 0 1
(8,9) 1 0 0 -
(3,7) 0 - 1 1
(3,11) - 0 1 1
(9,11) 1 0 - 1
(7,15) - 1 1 1
(11,15) 1 - 1 1
Minter
m
No.
IN BINARY
A B C D
0,1,8,9 - 0 0 -
0,8,1,9 - 0 0 -
1,3,9,11 - 0 - 1
1,9,3,11 - 0 - 1
3,7,11,15 - - 1 1
3,11,7,15 - - 1 1
7
Column : 2
Column : 4
STEP 3:
Now compare all the pairs of column 3 with those in the
minterms adjacent groups as shown in column 4

8. STEP: 4
Repeat the procedure for grouping. If can group the Quads of minterms in
the adjacent groups of column 4 to obtain groups of eight minterms.
There are no such matching.
Now prepare Prime Implicant Table as shown
Essential Prime Implicants areand CD
Required Output  CD
PI Minterms group
& Boolean
representation
GIVEN MINTERMS
0 1 3 7 8 9 11 15
√ (0,1,8,9) X X X X
(1,3,9,11) X X X X
√ (3,7,11,15) C D
X X X X
Prime Implicant Table
√ √ √ √ √ √ √ √

9. Example 2 Simplify
ƒ(A,B,C,D) = Σ m(4,5,6,8,9, 10,13) + Σ d(0,7,15)
In Binary
0000
0100
0101
0110
0111
1000
1001
1010
1101
1111
Minterms
0
4
5
6
7
8
9
10
13
15
Column I

10. Column II
Binary
A B C D
0 0 0 0 
0 1 0 0 
1 0 0 0 
0 1 0 1 
0 1 1 0 
1 0 0 1 
1 0 1 0 
0 1 1 1 
1 1 0 1 
1 1 1 1 
Column III
Binary
A B C D
0 – 0 0 *
- 0 0 0 *
0 1 0 - 
0 1 - 0 
1 0 0 - *
1 0 - 0 *
0 1 - 1 
- 1 0 1 
0 1 1 - 
1 – 0 1 *
- 1 1 1 
1 1 - 1 
Column IV
Binary
A B C D
0 1 - - *
0 1 - - *
- 1 - 1 *
- 1 - 1 *
Min
term
0
4
8
5
6
9
10
7
13
15
Min
term
0,4
0,8
4,5
4,6
8,9
8,10
5,7
5,13
6,7
9,13
7,15
13,15
Min term
4,5,6,7
4,6,5,7
5,7,13,15
5,13,7,15

11. PI Minterms group
and Boolean
expressions
0,4
0,8
8,9 A
8,10 A
9,13 A D
4,5,6,7 B
5,7,13,15BD
4
X
X
5
X
X
6
X
8
X
X
X
9
X
X
10
X
13
X
X
Prime Implicant Table
√
B
A
D
C
A
D
B
A
F 


√ √ √ √ √
√
√
√
√
√

12. Example 3 simplify
f(W,X,Y,Z)=∑m(2,6,8,9,10,11,14,15) Column1
Column 2
Group
Name
Min
terms
W X Y Z
GA1
2 0 0 1 0
8 1 0 0 0
GA2
6 0 1 1 0
9 1 0 0 1
10 1 0 1 0
GA3
11 1 0 1 1
14 1 1 1 0
GA4 15 1 1 1 1
12
Min
terms
W X Y Z
2 0 0 1 0
6 0 1 1 0
8 1 0 0 0
9 1 0 0 1
10 1 0 1 0
11 1 0 1 1
14 1 1 1 0
15 1 1 1 1

13. Column 3
Group
Name
Min
terms
W X Y Z
GB1
2,6 0 - 1 0
2,10 - 0 1 0
8,9 1 0 0 -
8,10 1 0 - 0
GB2
6,14 - 1 1 0
9,11 1 0 - 1
10,11 1 0 1 -
10,14 1 - 1 0
GB3
11,15 1 - 1 1
14,15 1 1 1 - 13

14. Column 4
Group Name Min terms W X Y Z
GB1
2,6,10,14 - - 1 0
2,10,6,14 - - 1 0
8,9,10,11 1 0 - -
8,10,9,11 1 0 - -
GB2
10,11,14,15 1 - 1 -
10,14,11,15 1 - 1 -
14

15. Prime Implicant Table
Min terms /
Prime
Implicants
2 6 8 9 10 11 14 15
2,6,10,14
YZ’
X X X X
8,9,10,11
WX’
X X X X
10,11,14,15
WY
X X X X
√ √ √ √ √ √ √
√
f = YZ’ + WX’ + WY.