Minimization Methods
4-Variable Karnaugh Map (Review)
f(a,b,c,d)
ab
c
cd
00
01
11
a
10
00
01
11
10
m0
m1
m3
m2
m4
m5
m7
m6
m12
m13
m8
m9
m15
m11
b
m14
m10
d
Digital Logic Fundamentals
6
© ptb/dkb (January 19, 2024)
5 and 6 Variable Karnaugh Maps
bc de
a=0
00
01
11
10
8
00
01
11
10
0
1
3
2
5
13
9
00
10
24
28
00
16
01
11
10
6
12
8
ab=01
10
01
21
27
01
11
10
0
1
3
2
5
13
9
00
11
23
10
10
18
28
ab=11
00
01
16
21
10
26
cd ef
00
01
56
53
58
cd ef
00
01
10
Digital Logic Fundamentals
7
40
44
33
37
45
41
43
10
50
54
62
59
01 36
11
55
63
32
18
22
51
57
00
11
49
61
10
30
26
60
ab=10
23
31
48
11
19
27
00
11
17
29
6
14
10
01 52
30
15
25
24
22
7
11
01 20
19
31
00
4
cd ef
14
17
29
25
00
11
01 20
11
7
15
11
bc de
a=1
ab=00
4
12
cd ef
11
35
39
47
10
34
38
46
42
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Karnaugh Map Based Minimization
• The purpose of using K-maps is to aid in the simplification of logic
expressions.
• Use of 1’s in K-maps gives minterm SOP form directly. Use of 0’s in K-maps
gives maxterm POS form directly.
• Minterms (maxterms) that can be combined differ in exactly one bit
position.
• Minimization using K-maps is done by grouping “adjacent” minterms
(maxterms).
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16
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• Conditions for grouping:
• Minterm (maxterm) groups are restricted to have size that is
a power of 2, e.g., 1, 2, 4, 8, …
• All K-map cells in a group must have a 1 as their K-map entry
for grouping minterms (or 0 as their K-map entry for
maxterm groups).
• All minterms (maxterms) must be adjacent; they must differ
in exactly i bits where, 2i is the size of the group.
• To minimize a function, find the set of groups which
include all the required minterms; the function is the
sum of these groups (or if maxterms are used then the
function is the product of the groups).
• The larger the groups, the better. Why?
Digital Logic Fundamentals
17
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Example: Grouping Minterms
f(w,x,y,z) yz
wx
00
01
11
1
1
1
1
00
10
0001 = m1
0011 = m3
1000 = m8
1001 = m9
1011 = m11
1100 = m12
01
Groupings:
11
1
10
1
{m1,m3}
{m9,m11}
{m1,m9}
{m8,m9}
{m3,m11}
{m8,m12}
Simplified Expression:
Digital Logic Fundamentals
Minterms:
{m1,m3,m9,m11}
f(w,x,y,z) = {m1,m3,m9,m11} + {m8,m12}
 x z  wyz
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Example: Grouping Maxterms
f(w,x,y,z) yz
wx
00
01
11
10
00
0
1
1
0
01
0
0
0
0
11
1
0
0
0
10
1
1
1
0
Maxterms:
Groupings (note: all groupings are not listed):
0000 = M0
0010 = M2
0100 = M4
0101 = M5
0110 = M6
0111 = M7
1010 = M10
1101 = M13
1110 = M14
1111 = M15
{M0,M2,M4,M6}, {M4,M5,M6,M7}, {M5,M7,M13,M15},
{M2,M6,M10,M14}, {M6,M7,M14,M15}
Simplified Expression: f(w,x,y,z) = {M0,M2,M4,M6}{M5,M7,M13,M15}
{M2,M6,M10,M14}
 w  z x  z y  z 
Digital Logic Fundamentals
21
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Logic Minimization: Example 1
x
yz
00
0
1
1
1
01
11
10
1
1
1
Sum of Products:
x
yz
00
01
11
10
0
1
0
1
1
1
1
0
0
1
Product of Sums:
F(x,y,z) = y  z  x  z 
F(x,y,z) = z  x y
# INV
= 2
# AND2 = 1
# OR2 = 2
# INV = 2
# AND2 = 1
# OR2 = 1
 For example 1 SOP is SMALLER
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Digital Logic Fundamentals
22
Logic Minimization: Example 2
wx
yz
00
01
11
10
00
0
1
0
0
01
0
1
1
0
11
1
1
1
1
10
1
1
0
0
POS:
SOP:
F w , x , y , z  w x  w y  y z  x z
F w , x , y , z  w  z x  y 
# INV
= 1
# AND2 = 1
# OR2 = 2
# INV
= 1
# AND2 = 4
# OR2 = 3
 For example 2 POS is SMALLER
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Digital Logic Fundamentals
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5-Variable Minimization
Minimize: F  a, b, c, d , e     2,5,7,8,10,13,15,17,19, 21, 23, 24, 29,31
bc de
a=0
1
1
1
bc de
a=1
00
00
01
1
01
Digital Logic Fundamentals
10
1
1
11
11
1
1
11
10
01
00
01
10
00
1
1
11
F  ce  abe  bcd e  acd e
10
1
1
1
1
24
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Don’t Care Input Combinations
• Sometimes functions are incompletely specified; the function is not
defined for some minterms.
outputs really don’t matter when these input combinations occur, or
these input combinations never occur in normal operation.
• These input combinations are known as don’t care conditions.
• Don’t cares are entered as X’s in the K-map (sometimes D or d are also used).
• During simplification, X’s can be treated as 1’s or 0’s.
Digital Logic Fundamentals
25
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Don’t Care: Example
G a , b , c , d    0,4,8,12 ,13   d 5 ,11 
G
cd
00
ab
01
00
1
01
1
X
11
1
1
10
1
11
10
X
G a , b , c , d   b c  c d
Digital Logic Fundamentals
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Example: Combinational Logic Design
• Design a circuit that accepts binary numbers between
1 and 5 and generates an output equal to the number
of 1’s in the input. Use only 2-input logic gates.
• Determine the number of inputs and outputs
• No. of inputs = 3
• No. of outputs = 2
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Digital Logic Fundamentals
27
2. K-maps and logic minimization
1. Create a truth table
Inputs
Outputs
b2
b1
b0
z1
z0
0
0
0
X
X
0
0
1
0
1
0
1
0
0
1
z0 b1 b0
00
b2
01
0
X
1
1
1
0
1
1
1
0
1
0
0
0
1
0
1
0
1
1
0
1
1
0
X
X
1
1
1
X
X
Digital Logic Fundamentals
10
1
X
X
z 0  b0  b2 b1
z1 b1 b0
00
b2
© ptb/dkb (January 19, 2024)
11
1
or
01
X
11
10
1
1
X
X
z1  b2 b0  b1 b0
z1  b0 b2  b1 
28
3. Gate level implementation
z1  b0 b2  b1 
z 0  b0  b2 b1
How fast can you run this circuit (or what is the throughput) ?
(i.e., how often can you supply new inputs to this circuit)
© ptb/dkb (January 19, 2024)
Digital Logic Fundamentals
29
Example:
• Design a circuit that will increment a BCD digit by 1 and produce an output BCD digit.
X cd
W cd
ab
00
a
b
c
d
W X
Y Z
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
1
0
0
0
1
1
0
0
1
1
0
1
0
0
11 X
0
1
0
0
0
1
0
1
10 1
0
1
0
1
0
1
1
0
0
1
1
0
0
1
1
1
0
1
1
1
1
0
0
0
1
0
0
0
1
0
0
1
1
0
0
1
0
0
0
0
1
0
1
0
X
X
X X
X
X
X X
X
X
X X
…
1
1
1
1
01
11
10
00
01
1
01
X
1
X
X
11 X
X
X
10
W  ad  bcd
10
1
X
X
X
X
X  bc  bd  bcd
Y cd
ab
01 1
X
11
1
00
00
Z cd
00
01
11
10
ab
00
01
11
10
00
1
1
00 1
1
01
1
1
01 1
1
11 X
X
X
X
11 X
X
X
10 1
10
Y  a c d  cd
© ptb/dkb (January 19, 2024)
ab
Digital Logic Fundamentals
X
X
X
X
X
Z d
30
Do it yourself
•
Obtain minimum SOP and POS expressions for the following functions.
For selected problems, implement circuits using NAND and NOR gates.
F (W , X , Y , Z )   (0, 2,5, 7,8,10,13)  d (1,9,11)
F ( A, B, C )   (1, 2,3,5, 6) D(4)
F (W , X , Y , Z )   (0,3, 4,5, 6, 7,11,12,13,14,15)
F (W , X , Y , Z )   (2,3, 6, 7,8,9,12,13)
F ( A, B, C )   (4, 6)
F ( A, B, C )   (2,3)
F ( A, B, C )   (1, 2,3,5, 7)
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Digital Logic Fundamentals
31
Do it yourself
• Using Boolean axioms and/or theorems manipulate the OR function
F = X + Y so that it can be implemented with only AND elements and
INVERTERS. Use parentheses to clearly indicate the order of
intended operations. Draw circuit and verify that circuit satisfies the
OR function (easy).
Digital Logic Fundamentals
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Some Definitions
• Implicant : A product term that evaluates a function to a ‘1’ is called an
implicant of the function
• Prime implicant is an implicant that cannot be combined into another
implicant having fewer literals.
• Cover : A collection of implicants that accounts for all valuations for which
my function equals ‘1’
• Cost : Cost of the implementing a cover can be defined as the sum of the
number of gates and number of inputs to the gates.
Digital Logic Fundamentals
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Quine-McCluskey Minimization Method
• It is a tabular method involving identification of prime implicants and
minimal cover (minimum set of prime implicants)
• Prime implicants are determined using an iterative procedure in which
variables are eliminated by applying adjacency
• Minimal cover is determined using a prime implicant chart from which the
essential prime implicants and a minimal set of remaining prime implicants so
that all minterms are covered
• Suitable for use in logic minimization software
Digital Logic Fundamentals
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Example 1
f  a, b, c, d     4,5,6,8,9,10,13  d  0,7,15
•
Minimize
1.
List the minterms along with don’t cares and arrange them in groups according to
the number of 1’s in the minterms
(d) m0 0000
m4 0100
m5 0101
group 0 0000
group 1
1000
m6 0110
0101
(d) m7 0111
0110
m8 1000
group 2
m10 1010
m13 1101
(d) m15 1111
1001
1010
m9 1001
© ptb/dkb (January 19, 2024)
0100
group 3
0111
1101
group 4 1111
Digital Logic Fundamentals
After grouping we only need
to compare minterms from
adjacent groups
Don’t cares are included
in this step.
35
Example 1 (contd…)
2. Compare and combine entries in adjacent groups in column 1 and fill up entries in
column 2:
 Compare groups 0 and 1: mark those that can be combined by a check mark (√ );
If an entry cannot be combined, mark with a star. These are the prime implicants.
Column 1
0
0000 √
4
0100 √
8
1000
5
0101
6
0110
9
1001
Column 2
(0,4)
0-00
10 1010
7
0111
13 1101
15
© ptb/dkb (January 19, 2024)
1111
Digital Logic Fundamentals
36
Example 1 (contd…)
2. Compare entries in adjacent groups in column 1 and fill up entries in column 2:
 compare groups 0 and 1
Column 1
Column 2
0
0000 √
(0,4)
0-00
4
0100 √
(0,8)
-000
8
1000 √
5
0101
6
0110
9
1001
10 1010
7
0111
13 1101
15
© ptb/dkb (January 19, 2024)
1111
Digital Logic Fundamentals
37
Example 1 (contd…)
2. Compare entries in adjacent groups in column 1 and fill up entries in column 2:
 compare groups 1 and 2
Column 1
Column 2
0
0000 √
(0,4)
0-00
4
0100 √
(0,8)
-000
8
1000 √
(4,5)
010-
5
0101 √
6
0110
9
1001
10 1010
7
0111
13 1101
15
© ptb/dkb (January 19, 2024)
1111
Digital Logic Fundamentals
38
Example 1 (contd…)
2. Compare entries in adjacent groups in column 1 and fill up entries in column 2:
 compare groups 1 and 2
Column 1
© ptb/dkb (January 19, 2024)
Column 2
0
0000
√
(0,4)
0-00
4
0100
√
(0,8)
-000
8
1000
√
(4,5)
010-
5
0101
√
(4,6)
01-0
6
0110
√
(8,9)
100-
9
1001
√
(8,10)
10-0
10
1010
√
7
0111
13
1101
15
1111
Digital Logic Fundamentals
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Example 1 (contd…)
2. Compare entries in adjacent groups in column 1 and fill up entries in column 2:
 Compare remaining groups
and complete column 2
 Divide column 2 into groups
© ptb/dkb (January 19, 2024)
Column 1
Column 2
0
0000
√
(0,4)
0-00
4
0100
√
(0,8)
-000
8
1000
√
(4,5)
010-
5
0101
√
(4,6)
01-0
6
0110
√
(8,9)
100-
9
1001
√
(8,10)
10-0
10
1010
√
(5,7)
01-1
7
0111
√
(5,13)
-101
13
1101
√
(6,7)
011-
15
1111
√
(9,13)
1-01
(7,15)
-111
(13,15)
1-11
Digital Logic Fundamentals
40
Example 1 (contd…)
3. Compare entries in adjacent groups in column 2 and fill up entries in column 3.
Repeat until nothing can be combined.
Column 1
© ptb/dkb (January 19, 2024)
Column 2
Column 3
0
0000
√
(0,4)
0-00
*
(4,5,6,7)
01--
*
4
0100
√
(0,8)
-000
*
(5,7,13,15)
-1-1
*
8
1000
√
(4,5)
010-
√
5
0101
√
(4,6)
01-0
√
6
0110
√
(8,9)
100-
*
9
1001
√
(8,10)
10-0
*
10
1010
√
(5,7)
01-1
√
7
0111
√
(5,13)
-101
√
13
1101
√
(6,7)
011-
√
15
1111
√
(9,13)
1-01
*
(7,15)
-111
√
(13,15)
11-1
√
Digital Logic Fundamentals
41
Example 1 (contd…)
• Prime Implicant Chart: It is the second part of the Quine-McCluskey
algorithm.
• It is used to select a minimum set of prime implicants.
• Essential Prime implicants are listed along the vertical axis of the table (i.e., rows of the table).
• The minterms are listed along the horizontal axis (i.e., columns of the table).
• Entry (i, j) of the table is marked if the minterm at the head of column j is contained in (covered
by) prime implicant in row i.
• 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.
Digital Logic Fundamentals
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© ptb/dkb (January 19, 2024)
Example 1 (contd…)
minterms
prime implicants
m4
m5
m6
m8
(0,4)
0-00
(0,8)
-000
X
(8,9)
100-
X
(8,10)
10-0
X
(9,13)
1-01
(4,5,6,7)
01--
(5,7,13,15)
-1-1
m9
m10
m13
X
X
X
X
X
X
X
X
X
X
Do not include don’t cares
in this table. They need
not be covered by the final
expression
• Look for minterms that are covered by a single prime implicant. The prime implicant which
contains that minterm will be in the minimal expression. Mark as “covered” all minterms
contained in such prime implicants
• Choose the minimum number of additional (from non-essential) prime implicants required to
cover the remaining minterms.
F (a, b, c, d )  a b  a b d  a c d
© ptb/dkb (January 19, 2024)
Digital Logic Fundamentals
44
Minimize: F  a, b, c, d , e     2,5,7,8,10,13,15,17,19, 21, 23, 24, 29,31
Example 2
2
8
5
10
17
24
7
13
19
21
15
23
29
31
© ptb/dkb (January 19, 2024)
00010
01000
00101
01010
10001
11000
00111
01101
10011
10101
01111
10111
11101
11111
√
√
√
√
√
√
√
√
√
√
√
√
√
√
0-010
010-0
-1000
001-1
0-101
-0101
100-1
0-111
-0111
011-1
-1101
10-11
101-1
1-101
-1111
1-111
111-1
*
*
*
√
√
√
√
√
√
√
√
√
√
√
√
√
√
Digital Logic Fundamentals
0-1-1
-01-1
0-1-1
--101
-01-1
--101
10--1
--111
-11-1
-11-1
1-1-1
1-1-1
*
√
--1-1 *
--1-1
√
*
√
√
*
45
Example 2:
m2
0-010
Prime Implicant Chart
m5
m7
m8
X
m19
X
-1000
X
m23
m24
m29
m31
X
X
X
X
X
X
10--1
X
1-1-1
--1-1
m21
X
010-0
0-1-1
m10 m13 m15 m17
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
F  a, b, c, d , e   a c d e  b c d e  a b e  c e
© ptb/dkb (January 19, 2024)
Digital Logic Fundamentals
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