Lec8 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Quinn-McCluskey Method

ECE2030
Introduction to Computer Engineering
Lecture 8: Quine-McCluskey Method
Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean Lee
School of Electrical and Computer EngineeringSchool of Electrical and Computer Engineering
Georgia TechGeorgia Tech

Quine-McCluskey Method
• A systematic solution to K-Map when more
complex function with more literals is given
• In principle, can be applied to an arbitrary
large number of inputs, i.e. works for BBnn
where
nn can be arbitrarily large
• One can translate Quine-McCluskey method
into a computer program to perform
minimization

Quine-McCluskey Method
• Two basic steps
– Finding all prime implicants of a given Boolean
function
– Select a minimal set of prime implicants that
cover this function

Q-M Method (I)
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Transform the given Boolean function into a canonical
SOP function
• Convert each Minterm into binary format
• Arrange each binary minterm in groups
– All the minterms in one group contain the same number of “1”

Q-M Method: Grouping minterms
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
(29) 1 1 1 0 1
(30) 1 1 1 1 0
(2) 0 0 0 1 0
(4) 0 0 1 0 0
(8) 0 1 0 0 0
(16) 1 0 0 0 0
A B C D E
(0) 0 0 0 0 0
(6) 0 0 1 1 0
(10) 0 1 0 1 0
(12) 0 1 1 0 0
(18) 1 0 0 1 0
(7) 0 0 1 1 1
(11) 0 1 0 1 1
(13) 0 1 1 0 1
(14) 0 1 1 1 0
(19) 1 0 0 1 1

Q-M Method (II)
• Combine terms with Hamming distance=1
from adjacent groups
• Check (√√) the terms being combined
– The checked terms are “covered” by the
combined new term
• Keep doing this till no combination is
possible between adjacent groups

∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
(29) 1 1 1 0 1
(30) 1 1 1 1 0
(2) 0 0 0 1 0
(4) 0 0 1 0 0
(8) 0 1 0 0 0
(16) 1 0 0 0 0
A B C D E
(0) 0 0 0 0 0
(6) 0 0 1 1 0
(10) 0 1 0 1 0
(12) 0 1 1 0 0
(18) 1 0 0 1 0
(7) 0 0 1 1 1
(11) 0 1 0 1 1
(13) 0 1 1 0 1
(14) 0 1 1 1 0
(19) 1 0 0 1 1
A B C D E
√
√
(0,2) 0 0 0 – 0
√
(0,4) 0 0 - 0 0
√
(0,8) 0 - 0 0 0
√
(0,16) - 0 0 0 0
√
(2,6) 0 0 - 1 0
√
(2,10) 0 - 0 1 0
√
(2,18) - 0 0 1 0
√
(4,6) 0 0 1 - 0
(4,12) 0 - 1 0 0
(8,10) 0 1 0 - 0
(8,12) 0 1 - 0 0
(16,18) 1 0 0 - 0
(6,7) 0 0 1 1 -
√
(6,14) 0 - 1 1 0√
(10,11) 0 1 0 1 -
√
(10,14) 0 1 - 1 0
(12,13) 0 1 1 0 -
√
(12,14) 0 1 1 - 0
(18,19) 1 0 0 1 -
√
A B C D E
(13,29) - 1 1 0 1
√
(14,30) - 1 1 1 0
√

A B C D E
√
√
(0,2) 0 0 0 – 0
√
(0,4) 0 0 - 0 0
(0,8) 0 - 0 0 0
√
(0,16) - 0 0 0 0
√
(2,6) 0 0 - 1 0 √
(2,10) 0 - 0 1 0
√
(2,18) - 0 0 1 0
(4,6) 0 0 1 - 0
(4,12) 0 - 1 0 0
(8,10) 0 1 0 - 0
(8,12) 0 1 - 0 0
(16,18) 1 0 0 - 0
(6,7) 0 0 1 1 -
(6,14) 0 - 1 1 0
(10,11) 0 1 0 1 -
(10,14) 0 1 - 1 0
(12,13) 0 1 1 0 -
(12,14) 0 1 1 - 0
(18,19) 1 0 0 1 -
(13,29) - 1 1 0 1
(14,30) - 1 1 1 0
A B C D E
(0,2,4,6) 0 0 - – 0
√
(0,2,8,10) 0 - 0 – 0
√
(0,2,16,18) - 0 0 – 0
√
(0,4,8,12) 0 - - 0 0
√
√
(2,6,10,14) 0 - - 1 0
√
(4,6,12,14) 0 - 1 - 0
√
(8,10,12,14) 0 1 - - 0
√
A B C D E
(0,2,4,6 0 - - - 0
8,10,12,14)
√
√
√
√
√
√
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,

Prime Implicants
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
(6,7) 0 0 1 1 -
(10,11) 0 1 0 1 -
(12,13) 0 1 1 0 -
(18,19) 1 0 0 1 -
(13,29) - 1 1 0 1
(14,30) - 1 1 1 0
(0,2,16,18) - 0 0 – 0
(0,2,4,6 0 - - - 0
8,10,12,14)
A B C D E
• Unchecked terms are prime implicants

Prime Implicants
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
CDBA=
DCBA=
(6,7) 0 0 1 1 -
(10,11) 0 1 0 1 -
(12,13) 0 1 1 0 -
(18,19) 1 0 0 1 -
(13,29) - 1 1 0 1
(14,30) - 1 1 1 0
(0,2,16,18) - 0 0 – 0
(0,2,4,6 0 - - - 0
8,10,12,14)
A B C D E
• Unchecked terms are prime implicants
DBCA=
DCBA=
EDBC=
EBCD=
ECB=
EA=

Q-M Method (III)
• Form a Prime Implicant Table
– X-axis: the minterm
– Y-axis: prime implicants
• An ×× is placed at the intersection of a row
and column if the corresponding prime
implicant includes the corresponding product
(term)

Q-M Method: Prime Implicant Table
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14) XX XX XX XX XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,

Q-M Method (IV)
• Locate the essential row from the table
– These are essential prime implicants
– The row consists of minterms covered by a single “×”
• Mark all minterms covered by the essential
prime implicants
• Find non-essential prime implicants to cover
the rest of minterms
• Form the SOP function with the prime
implicants selected, which is the minimal
representation

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14)

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7)

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11)

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29), (14,30)
• Now all the minterms are covered by selected prime implicants !

Q-M Method
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
∑= 30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29), (14,30)
• Now all the minterms are covered by selected prime implicants !
• Note that (12,13), a non-essential prime implicant, is not needed

Q-M Method Result
0 2 4 6 7 8 1
0
1
1
1
2
1
3
1
4
1
6
1
8
1
9
2
9
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
EAECBEBCDEDBCDCBADCBACDBA
,10,12,14)(0,2,4,6,8
)(0,2,16,18(14,30)(13,29)(18,19)(10,11)(6,7)
30)29,19,18,16,14,13,12,11,10,8,7,6,4,2,m(0,E)D,C,B,F(A,
++++++=
+
+++++=
= ∑

Q-M Method Example 2
• Sometimes,
simplification by K-map
method could be less
than optimal due to
human error
• Quine-McCluskey
method can guarantee
an optimal answer
d(5,7,14)12)10,9,8,6,4,1,m(0,F += ∑
00 01 11 10
00 1 1 0 0
01 1 X X 1
11 1 0 0 X
10 1 1 0 1
AB
CD

Grouping minterms
(1) 0 0 0 1
(4) 0 1 0 0
(8) 1 0 0 0
A B C D
(0) 0 0 0 0
(5) 0 1 0 1
(6) 0 1 1 0
(9) 1 0 0 1
(10) 1 0 1 0
(12) 1 1 0 0
(7) 0 1 1 1
(14) 1 1 1 0
d(5,7,14)12)10,9,8,6,4,1,m(0,F += ∑
A B C D
(0,1) 0 0 0 -
(0,4) 0 - 0 0
(0,8) - 0 0 0√
√
√
√
(1,5) 0 - 0 1
(1,9) - 0 0 1
(4,5) 0 1 0 –
(4,6) 0 1 – 0
(4,12) – 1 0 0
(8,9) 1 0 0 –
(8,10) 1 0 – 0
(8,12) 1 – 0 0
√
√
√
√
√
(5,7) 0 1 - 1
(6,7) 0 1 1 -
(6,14) - 1 1 0
(10,14) 1 – 1 0
(12,14) 1 1 - 0
√
√
A B C D
(4,5,6,7) 0 1 - -
(4,6,12,14) - 1 - 0
(8,10,12,14) 1 - - 0
√
√
√
√
√
√
√
(0,1,4,5) 0 - 0 -
(0,1,8,9) - 0 0 –
(0,4,8,12) - - 0 0
√
√
√
√
√
√
√
√
√

Prime Implicants
d(5,7,14)12)10,9,8,6,4,1,m(0,F += ∑
A B C D
(4,5,6,7) 0 1 - -
(4,6,12,14) - 1 - 0
(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -
(0,1,8,9) - 0 0 –
(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care

Prime Implicants
d(5,7,14)12)10,9,8,6,4,1,m(0,F += ∑
A B C D
(4,5,6,7) 0 1 - -
(4,6,12,14) - 1 - 0
(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -
(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –
(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care

Prime Implicants
d(5,7,14)12)10,9,8,6,4,1,m(0,F += ∑
A B C D
(4,5,6,7) 0 1 - -
(4,6,12,14) - 1 - 0
(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -
(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –
(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care

Prime Implicants
d(5,7,14)12)10,9,8,6,4,1,m(0,F += ∑
A B C D
(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -
(4,6,12,14) - 1 - 0
(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -
(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –
(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care
Essential PI
Essential PI
Non-Essential PI

Q-M Method Solution
0 1 4 6 8 9 1
0
1
2
5 7 1
4
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
A B C D
(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -
(4,6,12,14) - 1 - 0
(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -
(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –
(0,4,8,12) - - 0 0
Don’t Care
BADACB
d(5,7,14)12)10,9,8,6,4,1,m(0,F
++=
+= ∑

Yet Another Q-M Method Solution
0 1 4 6 8 9 1
0
1
2
5 7 1
4
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
A B C D
(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -
(4,6,12,14) - 1 - 0(4,6,12,14) - 1 - 0
(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -
(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –
(0,4,8,12) - - 0 0
Don’t Care
DBDACB
d(5,7,14)12)10,9,8,6,4,1,m(0,F
++=
+= ∑

To Get the Same Answer w/ K-Map
BADACB
d(5,7,14)12)10,9,8,6,4,1,m(0,F
++=
+= ∑
00 01 11 10
00 1 1 0 0
01 1 X X 1
11 1 0 0 X
10 1 1 0 1
AB
CD
00 01 11 10
00 1 1 0 0
01 1 X X 1
11 1 0 0 X
10 1 1 0 1
AB
CD
DBDACB
d(5,7,14)12)10,9,8,6,4,1,m(0,F
++=
+= ∑

Lec8 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Quinn-McCluskey Method

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Lec8 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Quinn-McCluskey Method (20)

More from Hsien-Hsin Sean Lee, Ph.D. (14)

Recently uploaded (20)

Lec8 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Quinn-McCluskey Method