CMPUT329 - Fall 2003
Topic 5: Quine-McCluskey Method
José Nelson Amaral
CMPUT 329 - Computer
Organization and Architecture II
1
Reading Assignment
Section 4.4
CMPUT 329 - Computer
Organization and Architecture II
2
Motivation
Karnaugh maps are very effective for the minimization
of expressions with up to 5 or 6 inputs. However they
are difficult to use and error prone for circuits with
many inputs.
Karnaugh maps depend on our ability to visually
identify prime implicants and select a set of prime
implicants that cover all minterms. They do not
provide a direct algorithm to be implemented in a
computer.
For larger systems, we need a programmable method!!
CMPUT 329 - Computer
Organization and Architecture II
3
Quine-McCluskey
Willard van Orman Quine 1908-2000,
Edgar Pierce Chair of Philosophy at
Harvard University.
http://members.aol.com/drquine/wv-quine.html
Quine, Willard, “The problem of simplifying truth functions.”
American Mathematical Monthly, vol. 59, 1952.
Quine, Willard, “A way to simplify truth functions.”
American Mathematical Monthly, vol. 62, 1955.
Edward J. McCluskey, Professor of Electrical
Engineering and Computer Science at Stanford
http://www-crc.stanford.edu/users/ejm/McCluskey_Edward.html
McCluskey Jr., Edward J. “Minimization of Boolean Functions.”
Bell Systems Technical Journal, vol. 35, pp. 1417-1444, 1956
CMPUT 329 - Computer
Organization and Architecture II
4
Outline of the QuineMcCluskey Method
1. Produce a minterm expansion (standard
sum-of-products form) for a function F
2. Eliminate as many literals as possible by
systematically applying XY + XY’ = X.
3. Use a prime implicant chart to select a
minimum set of prime implicants that
when ORed together produce F, and that
contains a minimum number of literals.
CMPUT 329 - Computer
Organization and Architecture II
5
Determination of Prime
Implicants
AB’CD’ + AB’CD = AB’C
(The dash indicates a missing variable)
1010 +1011=101-
We can combine the minterms above because they
differ by a single bit.
The minterms below won’t combine
A’BC’D + A’BCD’
0101 +0110
CMPUT 329 - Computer
Organization and Architecture II
6
Quine-McCluskey Method
An Example
1. Find all the prime implicants
f (a, b, c, d )   m(0,1,2,5,6,7,8,9,10,14)
group 0
0 0000
group 1
1 0001
2 0010
8 1000
group 2
5
6
9
10
group 3
7 0111
14 1110
0101
0110
1001
1010
Group the minterms
according to the number
of 1s in the minterm.
This way we only have to
compare minterms from
adjacent groups.
CMPUT 329 - Computer
Organization and Architecture II
7
Quine-McCluskey Method
An Example
Column I
Combining
group 0 and
group 1:
Column II
group 0
0 0000
group 1
1 0001
2 0010
8 1000
group 2
5
6
9
10
group 3
7 0111
14 1110
0101
0110
1001
1010
CMPUT 329 - Computer
Organization and Architecture II
8
Quine-McCluskey Method
An Example
Column I
Combining
group 0 and
group 1:
Column II
group 0
0 0000 
group 1
1 0001 
2 0010
8 1000
group 2
5
6
9
10
group 3
7 0111
14 1110
0,1 000-
0101
0110
1001
1010
CMPUT 329 - Computer
Organization and Architecture II
9
Quine-McCluskey Method
An Example
Column I
Combining
group 0 and
group 1:
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000
group 2
5
6
9
10
group 3
7 0111
14 1110
0,1 0000,2 00-0
0101
0110
1001
1010
CMPUT 329 - Computer
Organization and Architecture II
10
Quine-McCluskey Method
An Example
Column I
Does it make
group 0
sense to no
combine group 0
group 1
with group 2 or 3?
No, there are at
least two bits that
are different.
Column II
0 0000 
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
0,1 0000,2 00-0
0,8 -000
0101
0110
1001
1010
CMPUT 329 - Computer
Organization and Architecture II
11
Quine-McCluskey Method
An Example
Column I
Does it make
group 0
sense to no
combine group 0
group 1
with group 2 or 3?
No, there are at
least two bits that
are different.
Thus, next we
combine group 1
and group 2.
Column II
0 0000 
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
0,1 0000,2 00-0
0,8 -000
0101
0110
1001
1010
CMPUT 329 - Computer
Organization and Architecture II
12
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0,1
0,2
0,8
1,5
00000-0
-000
0-01
0101 
0110
1001
1010
CMPUT 329 - Computer
Organization and Architecture II
13
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0,1
0,2
0,8
1,5
00000-0
-000
0-01
0101 
0110
1001
1010
CMPUT 329 - Computer
Organization and Architecture II
14
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101 
0110
1001 
1010
CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
00000-0
-000
0-01
-001
15
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101 
0110
1001 
1010
CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
00000-0
-000
0-01
-001
16
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101 
0110
1001 
1010
CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
00000-0
-000
0-01
-001
17
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101 
0110
1001 
1010
CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
00000-0
-000
0-01
-001
18
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101 
0110 
1001 
1010
CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
00000-0
-000
0-01
-001
0-10
19
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101 
0110 
1001 
1010
CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
00000-0
-000
0-01
-001
0-10
20
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
00000-0
-000
0-01
-001
0-10
-010
21
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
00000-0
-000
0-01
-001
0-10
-010
22
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
00000-0
-000
0-01
-001
0-10
-010
23
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
Combine group 1
and group 2.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
00000-0
-000
0-01
-001
0-10
-010
100-
24
Quine-McCluskey Method
An Example
Column I
Again, there is
no need to try
to combine group
1 with group 2.
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111
14 1110
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
00000-0
-000
0-01
-001
0-10
-010
10010-0
25
Quine-McCluskey Method
An Example
Column I
Again, there is
no need to try
to combine group
1 with group 3.
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
Lets try to combine group 2
group 2 with
group 23.
group 3
5
6
9
10
0101
0110
1001
1010




0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
00000-0
-000
0-01
-001
0-10
-010
10010-0
7 0111
14 1110
CMPUT 329 - Computer
Organization and Architecture II
26
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110
Combine group 2
and group 3.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
5,7
00000-0
-000
0-01
-001
0-10
-010
10010-0
01-1
27
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110
Combine group 2
and group 3.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
5,7
00000-0
-000
0-01
-001
0-10
-010
10010-0
01-1
28
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110
Combine group 2
and group 3.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
5,7
6,7
00000-0
-000
0-01
-001
0-10
-010
10010-0
01-1
011-
29
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
Combine group 2
and group 3.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
5,7
6,7
6,14
00000-0
-000
0-01
-001
0-10
-010
10010-0
01-1
011-110
30
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
Combine group 2
and group 3.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
5,7
6,7
6,14
00000-0
-000
0-01
-001
0-10
-010
10010-0
01-1
011-110
31
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
Combine group 2
and group 3.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
5,7
6,7
6,14
00000-0
-000
0-01
-001
0-10
-010
10010-0
01-1
011-110
32
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
Combine group 2
and group 3.
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1
0,2
0,8
1,5
1,9
2,6
2,10
8,9
8,10
5,7
6,7
6,14
00000-0
-000
0-01
-001
0-10
-010
10010-0
01-1
011-110
33
Quine-McCluskey Method
An Example
Column I
We have now
completed the
first step. All
minterms in
column I were
included.
We can divide
column II into
groups.
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
34
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




CMPUT 329 - Computer
Organization and Architecture II
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
35
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




Column III
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
36
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




Column III
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
37
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




Column III
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
38
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




Column III
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
39
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




Column III
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
40
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




0,1 000- 
0,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
Column III
0,1,8,9 -00-
41
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




0,1 000- 
0,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
Column III
0,1,8,9 -00-
42
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




0,1 000- 
0,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
Column III
0,1,8,9 -00-
43
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




0,1 000- 
0,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
Column III
0,1,8,9 -00-
44
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




0,1 000- 
0,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
Column III
0,1,8,9 -00-
45
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




0,1 000- 
0,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
Column III
0,1,8,9 -00-
46
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




0,1 000- 
0,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
CMPUT 329 - Computer
Organization and Architecture II
Column III
0,1,8,9 -00-
47
Quine-McCluskey Method
An Example
Column I
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




Column II
Column III
0,1 000- 
0,2 00-0 
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0 
5,7 01-1
6,7 0116,14 -110
10,14 1-10
0,1,8,9 -000,2,8,10 -0-0
CMPUT 329 - Computer
Organization and Architecture II
48
Quine-McCluskey Method
An Example
Column I
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
5
6
9
10
group 3
7 0111 
14 1110 
0101
0110
1001
1010




Column II
Column III
0,1 000- 
0,2 00-0 
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 100- 
8,10 10-0 
5,7 01-1
6,7 0116,14 -110
10,14 1-10
0,1,8,9 -000,2,8,10 -0-0
CMPUT 329 - Computer
Organization and Architecture II
49
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III



0,1,8,9 -000,2,8,10 -0-0
0,8,1,9 -00-



CMPUT 329 - Computer
Organization and Architecture II
50
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III



0,1,8,9 -000,2,8,10 -0-0
0,8,1,9 -00-



CMPUT 329 - Computer
Organization and Architecture II
51
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III




0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
-00-0-0
-00-0-0



CMPUT 329 - Computer
Organization and Architecture II
52
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III




0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
-00-0-0
-00-0-0



CMPUT 329 - Computer
Organization and Architecture II
53
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III




0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
-00-0-0
-00-0-0



CMPUT 329 - Computer
Organization and Architecture II
54
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III




0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
-00-0-0
-00-0-0



CMPUT 329 - Computer
Organization and Architecture II
55
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III




0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
-00-0-0
-00-0-0



CMPUT 329 - Computer
Organization and Architecture II
56
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
2,6,10,14
-00-0-0
-00-0-0
--10

CMPUT 329 - Computer
Organization and Architecture II
57
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
2,6,10,14
2,10,6,14
-00-0-0
-00-0-0
--10
--10


CMPUT 329 - Computer
Organization and Architecture II
58
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
2,6,10,14
2,10,6,14
-00-0-0
-00-0-0
--10
--10


CMPUT 329 - Computer
Organization and Architecture II
59
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
0101
0110
1001
1010




group 2
5
6
9
10
group 3
7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
2,6,10,14
2,10,6,14
-00-0-0
-00-0-0
--10
--10


CMPUT 329 - Computer
Organization and Architecture II
60
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
group 3
5
6
9
10
0101
0110
1001
1010




7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III










CMPUT 329 - Computer
Organization and Architecture II
0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
2,6,10,14
2,10,6,14
-00-0-0
-00-0-0
--10
--10
No more combinations
are possible, thus we
stop here.
61
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
group 3
5
6
9
10
0101
0110
1001
1010




7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9
0,2,8,10
0,8,1,9
0,8,2,10
2,6,10,14
2,10,6,14
-00-0-0
-00-0-0
--10
--10
We can eliminate repeated
combinations


CMPUT 329 - Computer
Organization and Architecture II
62
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
group 3
5
6
9
10
0101
0110
1001
1010




7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9 -000,2,8,10 -0-0
2,6,10,14 --10
Now we form f with the
terms not checked
f = a’c’d


CMPUT 329 - Computer
Organization and Architecture II
63
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
group 3
5
6
9
10
0101
0110
1001
1010




7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9 -000,2,8,10 -0-0
2,6,10,14 --10
f = a’c’d + a’bd


CMPUT 329 - Computer
Organization and Architecture II
64
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
group 3
5
6
9
10
0101
0110
1001
1010




7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9 -000,2,8,10 -0-0
2,6,10,14 --10
f = a’c’d + a’bd + a’bc


CMPUT 329 - Computer
Organization and Architecture II
65
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
group 3
5
6
9
10
0101
0110
1001
1010




7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III



0,1,8,9 -000,2,8,10 -0-0
2,6,10,14 --10





f = a’c’d + a’bd + a’bc + b’c’


CMPUT 329 - Computer
Organization and Architecture II
66
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
group 3
5
6
9
10
0101
0110
1001
1010




7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9 -000,2,8,10 -0-0
2,6,10,14 --10
f = a’c’d + a’bd + a’bc + b’c’
+ b’d’


CMPUT 329 - Computer
Organization and Architecture II
67
Quine-McCluskey Method
An Example
Column I
Column II
group 0
0 0000 
group 1
1 0001 
2 0010 
8 1000 
group 2
group 3
5
6
9
10
0101
0110
1001
1010




7 0111 
14 1110 
0,1 0000,2 00-0
0,8 -000
1,5 0-01
1,9 -001
2,6 0-10
2,10 -010
8,9 1008,10 10-0
5,7 01-1
6,7 0116,14 -110
10,14 1-10
Column III








0,1,8,9 -000,2,8,10 -0-0
2,6,10,14 --10
f = a’c’d + a’bd + a’bc + b’c’
+ b’d’ + cd’


CMPUT 329 - Computer
Organization and Architecture II
68
Quine-McCluskey Method
An Example
But, the form below is not minimized, using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
1
d
c
b
CMPUT 329 - Computer
Organization and Architecture II
69
Quine-McCluskey Method
An Example
But, the form below is not minimized, using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
1
d
1
c
b
CMPUT 329 - Computer
Organization and Architecture II
70
Quine-McCluskey Method
An Example
But, the form below is not minimized, using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
1
d
1
c
1
b
CMPUT 329 - Computer
Organization and Architecture II
71
Quine-McCluskey Method
An Example
But, the form below is not minimized, using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
1
1
1
1
d
1
c
1
b
CMPUT 329 - Computer
Organization and Architecture II
72
Quine-McCluskey Method
An Example
But, the form below is not minimized, using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
1
1
1
1
d
1
c
1
1
1
b
CMPUT 329 - Computer
Organization and Architecture II
73
Quine-McCluskey Method
An Example
But, the form below is not minimized. Using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
1
1
1
1
d
1
c
1
1
1
1
b
CMPUT 329 - Computer
Organization and Architecture II
74
Quine-McCluskey Method
An Example
But, the form below is not minimized, using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
F = a’bd
1
1
1
1
d
1
c
1
1
1
1
b
CMPUT 329 - Computer
Organization and Architecture II
75
Quine-McCluskey Method
An Example
But, the form below is not minimized, using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
F = a’bd + cd’
1
1
1
1
d
1
c
1
1
1
1
b
CMPUT 329 - Computer
Organization and Architecture II
76
Quine-McCluskey Method
An Example
But, the form below is not minimized, using a
Karnaugh map we can obtain:
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
F = a’bd + cd’ + b’c’
1
1
1
1
d
1
c
1
1
1
1
b
CMPUT 329 - Computer
Organization and Architecture II
77
Quine-McCluskey Method
An Example
What are the extra terms in the solution obtained
with the Quine-McCluskey method?
f = a’c’d + a’bd + a’bc + b’c’ + b’d’ + cd’
a
1
F = a’bd + cd’ + b’c’
1
Thus, we need a method to
c
eliminate this redundant terms
from the Quine-McCluskey solution.
1
1
1
d
1
1
1
1
1
b
CMPUT 329 - Computer
Organization and Architecture II
78
The Prime Implicant Chart
The prime implicant chart is the second part of
the Quine-McCluskey procedure.
It is used to select a minimum set of prime
implicants.
Similar to the Karnaugh map, we first select
the essential prime implicants, and then we
select enough prime implicants to cover all
the minterms of the function.
CMPUT 329 - Computer
Organization and Architecture II
79
Prime Implicants
Prime Implicant Chart
(Example)
minterms
0 1 2 5 6 7 8 9 10 14
X X
(0,1,8,9)
b’c’ X X
X
X
X
(0,2,8,10) b’d’ X
X
X
X X
(2,6,10,14) cd’
X
X
(1,5)
a’c’d
X
X
(5,7)
a’bd
X X
(6,7)
a’bc
Question: Given the prime implicant chart above,
how can we identify the essential prime
implicants of the function?
CMPUT 329 - Computer
Organization and Architecture II
80
Prime Implicants
Prime Implicant Chart
(Example)
minterms
0 1 2 5 6 7 8 9 10 14
X X
(0,1,8,9)
b’c’ X X
X
X
X
(0,2,8,10) b’d’ X
X
X
X X
(2,6,10,14) cd’
X
X
(1,5)
a’c’d
X
X
(5,7)
a’bd
X X
(6,7)
a’bc
Similar to the Karnaugh map, all we have to do is
to look for minterms that are covered by a single
term.
CMPUT 329 - Computer
Organization and Architecture II
81
Prime Implicants
Prime Implicant Chart
(Example)
minterms
0 1 2 5 6 7 8 9 10 14
X X
(0,1,8,9)
b’c’ X X
X
X
X
(0,2,8,10) b’d’ X
X
X
X X
(2,6,10,14) cd’
X
X
(1,5)
a’c’d
X
X
(5,7)
a’bd
X X
(6,7)
a’bc
Once a term is included in the solution, all the
minterms covered by that term are covered.
Therefore we may now mark the covered minterms
and find terms that are no longer useful.
CMPUT 329 - Computer
Organization and Architecture II
82
Prime Implicants
Prime Implicant Chart
(Example)
minterms
0 1 2 5 6 7 8 9 10 14
X X
(0,1,8,9)
b’c’ X X
X
X
X
(0,2,8,10) b’d’ X
X
X
X X
(2,6,10,14) cd’
X
X
(1,5)
a’c’d
X
X
(5,7)
a’bd
X X
(6,7)
a’bc
CMPUT 329 - Computer
Organization and Architecture II
83
Prime Implicants
Prime Implicant Chart
(Example)
minterms
0 1 2 5 6 7 8 9 10 14
X X
(0,1,8,9)
b’c’ X X
X
X
X
(0,2,8,10) b’d’ X
X
X
X X
(2,6,10,14) cd’
X
X
(1,5)
a’c’d
X
X
(5,7)
a’bd
X X
(6,7)
a’bc
As we have not covered all the minterms with
essential prime implicants, we must choose
enough non-essential prime implicants to
cover the remaining minterms.
CMPUT 329 - Computer
Organization and Architecture II
84
Prime Implicants
Prime Implicant Chart
(Example)
minterms
0 1 2 5 6 7 8 9 10 14
X X
(0,1,8,9)
b’c’ X X
X
X
X
(0,2,8,10) b’d’ X
X
X
X X
(2,6,10,14) cd’
X
X
(1,5)
a’c’d
X
X
(5,7)
a’bd
X X
(6,7)
a’bc
What strategy should we use to find a minimum
cover for the remaining minterms?
CMPUT 329 - Computer
Organization and Architecture II
85
Prime Implicants
Prime Implicant Chart
(Example)
minterms
0 1 2 5 6 7 8 9 10 14
X X
(0,1,8,9)
b’c’ X X
X
X
X
(0,2,8,10) b’d’ X
X
X
X X
(2,6,10,14) cd’
X
X
(1,5)
a’c’d
X
X
(5,7)
a’bd
X X
(6,7)
a’bc
We choose first prime implicants that cover the
most minterms.
Should this strategy always work??
CMPUT 329 - Computer
Organization and Architecture II
86
Prime Implicants
Prime Implicant Chart
(Example)
minterms
0 1 2 5 6 7 8 9 10 14
X X
(0,1,8,9)
b’c’ X X
X
X
X
(0,2,8,10) b’d’ X
X
X
X X
(2,6,10,14) cd’
X
X
(1,5)
a’c’d
X
X
(5,7)
a’bd
X X
(6,7)
a’bc
Therefore our minimum solution is:
f(a,b,c,d) = b’c’ + cd’ + a’bd
CMPUT 329 - Computer
Organization and Architecture II
87
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
Which ones are the essential prime implicants
in this chart?
There is no essential prime implicants, how we proceed?
CMPUT 329 - Computer
Organization and Architecture II
88
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
Also, all implicants cover the same number of
minterms. We will have to proceed by trial and error.
F(a,b,c) = a’b’
CMPUT 329 - Computer
Organization and Architecture II
89
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
Also, all implicants cover the same number of
minterms. We will have to proceed by trial and error.
F(a,b,c) = a’b’ + bc’
CMPUT 329 - Computer
Organization and Architecture II
90
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
Thus, we get the minimization:
F(a,b,c) = a’b’ + bc’ + ac
CMPUT 329 - Computer
Organization and Architecture II
91
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
Lets try another set of prime implicants.
CMPUT 329 - Computer
Organization and Architecture II
92
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
Lets try another set of prime implicants.
F(a,b,c) = a’c
CMPUT 329 - Computer
Organization and Architecture II
93
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
Lets try another set of prime implicants.
F(a,b,c) = a’c + b’c’
CMPUT 329 - Computer
Organization and Architecture II
94
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
Lets try another set of prime implicants.
F(a,b,c) = a’c + b’c’+ ab
CMPUT 329 - Computer
Organization and Architecture II
95
Cyclic Prime Implicant
Chart
0
1
2
5
6
7
000 
001 
010 
101 
110 
111 
0,1
0,2
1,5
2,6
5,7
6,7
000-0
-01
-10
1-1
11-
Prime Implicants
F(a,b,c) =  m(0, 1, 2, 5, 6, 7)
minterms
0 1 2 5 6 7
(0,1) a’b’ X X
X
(0,2) a’c X
X
X
(1,5) b’c
X
X
(2,6) bc’
X
X
(5,7) ac
X X
(6,7) ab
This time we obtain:
F(a,b,c) = a’c + b’c’+ ab
CMPUT 329 - Computer
Organization and Architecture II
96
Cyclic Prime Implicant
Chart
F(a,b,c) = a’b’ + bc’ + ac
F(a,b,c) = a’c + b’c’+ ab
Which minimal form is better?
Depends on what terms we must form for other
functions that we must also implement.
Often we are interested in examining all minimal
forms for a given function.
Thus we need an algorithm to do so.
CMPUT 329 - Computer
Organization and Architecture II
97
Petrick’s Method
S. R. Petrick. A direct determination of the irredundant
forms of a boolean function from the set of prime
implicants. Technical Report AFCRC-TR-56-110, Air
Force Cambridge Research Center, Cambridge,
MA, April, 1956.
Goal: Given a prime implicant chart, determine
all minimum sum-of-products solutions.
CMPUT 329 - Computer
Organization and Architecture II
98
Prime Implicants
Petrick’s Method
An Example
minterms
0 1 2 5 6 7
P1 (0,1) a’b’ X X
X
P2 (0,2) a’c X
X
X
P3 (1,5) b’c
X
X
P4 (2,6) bc’
X
X
P5 (5,7) ac
X X
P6 (6,7) ab
Step 1: Label all the rows in the chart.
Step 2: Form a logic function P with the logic
variables P1, P2, P3 that is true when
all the minterms in the chart are covered.
CMPUT 329 - Computer
Organization and Architecture II
99
Prime Implicants
Petrick’s Method
An Example
minterms
0 1 2 5 6 7
P1 (0,1) a’b’ X X
X
P2 (0,2) a’c X
X
X
P3 (1,5) b’c
X
X
P4 (2,6) bc’
X
X
P5 (5,7) ac
X X
P6 (6,7) ab
The first column has an X in rows P1 and P2.
Therefore we must include one of these rows
in order to cover minterm 0. Thus the following
term must be in P:
(P1 + P2)
CMPUT 329 - Computer
Organization and Architecture II
100
Prime Implicants
Petrick’s Method
An Example
minterms
0 1 2 5 6 7
P1 (0,1) a’b’ X X
X
P2 (0,2) a’c X
X
X
P3 (1,5) b’c
X
X
P4 (2,6) bc’
X
X
P5 (5,7) ac
X X
P6 (6,7) ab
Following this technique, we obtain:
P = (P1 + P2) (P1 + P3) (P2 + P4) (P3 + P5) (P4 + P6) (P5 + P6)
P = (P1 + P2) (P1 + P3) (P4 + P2) (P5 + P3) (P4 + P6) (P5 + P6)
P = (P1 + P2) (P1 + P3) (P4 + P2) (P4 + P6) (P5 + P3) (P5 + P6)
P = (P1 + P2 P3) (P4 + P2 P6) (P5 + P3 P6)
CMPUT 329 - Computer
Organization and Architecture II
101
Petrick’s Method
An Example
P = (P1 + P2) (P1 + P3) (P2 + P4) (P3 + P5) (P4 + P6) (P5 + P6)
P = (P1 + P2) (P1 + P3) (P4 + P2) (P5 + P3) (P4 + P6) (P5 + P6)
P = (P1 + P2) (P1 + P3) (P4 + P2) (P4 + P6) (P5 + P3) (P5 + P6)
P = (P1 + P2 P3) (P4 + P2 P6) (P5 + P3 P6)
P = (P1 P4 + P1 P2 P6 + P2 P3 P4 + P2 P3 P6) (P5 + P3 P6)
P = P1 P4 P5 + P1 P2 P5 P6 + P2 P3 P4 P5 + P2 P3 P5 P6
+ P1 P3 P4 P6 + P1 P2 P3 P6 + P2 P3 P4 P6 + P2 P3 P6
P = P1 P4 P5 + P1 P2 P5 P6 + P2 P3 P4 P5 + P1 P4 P3 P6 + P2 P3 P6
CMPUT 329 - Computer
Organization and Architecture II
102
Petrick’s Method
An Example
P = P1 P4 P5 + P1 P2 P5 P6 + P2 P3 P4 P5 + P1 P4 P3 P6 + P2 P3 P6
This expression says that to cover all the minterms
we must include the terms in line P1 and line P4 and
line P5, or we must include line P1, and line P2, and
line P5, and line P6, or …
Considering that all the terms P1, P2, … have the same
cost, how many minimal forms the function has?
The two minimal forms are
F = a’b’ + bc’ + ac
P1 P4 P5
and
P2 P3 P6:
F = a’c’ + b’c + ab
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Organization and Architecture II
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