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Subject : Computer Science (083)
Unit : Boolean Algebra
Topic : Minimization of Boolean
Expressions Using
Karnaugh Maps (K-Maps)

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Learning Objectives :
After successfully completing this module students should
be able to:
Understand the Need to simplify (minimize)
expressions
List Different Methods for Minimization
Karnaugh Maps
Algebraic method
Use Karnaugh Map method to minimize the Boolean
expression

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Previous Knowledge :
The students should be familiar with the
following terms in Boolean Algebra before
going through this module on K-MAPS
xx
yy
x+yx+y
Boolean variable, Constants and Operators
Postulates of Boolean Algebra
Theorems of Boolean Algebra
Logic Gates- AND, OR, NOT, NAND, NOR
Boolean Expressions and related terms
MINTERM (Product Term)
MAXTERM (Sum Term)
Canonical Form of Expressions

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Minimization
Of
Boolean Expressions
Who Developed it
NEED For Minimization
Different Methods
What is K-Map
Drawing a K-Map
Minimization Steps
Important Links
Recap. K-Map Rules
(SOP Exp.)
K-Map Quiz
EXIT
INDEX

5. 01/30/15 Karnaugh Maps 5
References
For K-Map Minimizer Download
http://karnaugh.shuriksoft.com
Thomas C. Bartee, DIGITAL
COMPUTER FUNDAMENTALS,
McGraw Hill International.
Computer Science (Class XII)
By Sumita Arora
http://www.ee.surrey.ac.uk/Projects/Labview/
minimisation/karrules.html

6. 01/30/15 Karnaugh Maps 6
The End

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Boolean expressions are practically implemented
in the form of GATES (Circuits).
A minimized Boolean expression means less
number of gates which means
Simplified Circuit
MINIMIZATION OF BOOLEAN
EXPRESSION
WHY we Need to simplify (minimize) expressions?

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MINIMIZATION OF BOOLEAN
EXPRESSION
Different methods
Karnaugh
Maps
Algebraic
Method

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Karnaugh Maps
WHAT is Karnaugh Map (K-Map)?
A special version of a truth table
Karnaugh Map (K-Map) is a GRAPHICAL display
of fundamental terms in a Truth Table.
Don’t require the use of Boolean Algebra
theorems and equation
Works with 2,3,4 (even more) input variables
(gets more and more difficult with more
variables)
NEXT

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K-maps provide an alternate way of simplifying
logic circuits.
One can transfer logic values from a Truth Table
into a K-Map.
The arrangement of 0’s and 1’s within a map
helps in visualizing, leading directly to
Simplified Boolean Expression
Karnaugh Maps……… (Contd.)
NEXT

11. 01/30/15 Karnaugh Maps 11
Correspondence between the
Karnaugh Map and the Truth Table
for the general case of a two Variable Problem
A B
0 0
0 1
1 0
1 1
F
a
b
c
d
A B 0 1
0 a b
1 c d
Truth Table
2 Variable K-Map
Karnaugh Maps……… (Contd.)

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Drawing a Karnaugh Map (K-Map)
K-map is a rectangle made up of certain number
of SQUARES
For a given Boolean function there are 2N
squares where N is the number of variables
(inputs)
In a K-Map for a Boolean Function with 2
Variables f(a,b) there will be 22
=4 squares
Each square is different from its neighbour by
ONE Literal
Each SQUARE represents a MAXTERM or
MINTERM
NEXT

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Karnaugh maps consist of a set of 22
squares
where 2 is the number of variables
in the Boolean expression being minimized.
Truth Table 2 Variable K-Map
Karnaugh Maps……… (Contd.)
A B 0 1
0 0 1
1 1 11
A B F
0 0 0
0 1 1
1 0 1
1 1 1
Minterm
A’B’
A’B
A B’
A B
Maxterm
A + B
A + B’
A’ + B
A’ + B’
NEXT

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 For three and four variable expressions
Maps with 23
= 8 and 24
= 16 cells are used.
Each cell represents a MINTERM or a
MAXTERM
4 Variable K-Map 24
= 16 Cells
Karnaugh Maps……… (Contd.)
BC
A
00 01 11 10
0
1
A B C D 00 01 11 10
00
01
11
10
3 Variable K-Map
23
= 8 Cells

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Minimization Steps (SOP Expression with 4 var.)
The process has following steps:
Draw the K-Map for
given function as
shown
Enter the function
values into the K-Map
by placing 1's and 0's
into the appropriate
Cells
A B C D 00 01 11 10
00 0
0
0
1
0
3
0
2
01 0 0 0 0
11 1 1 0 0
10 1 1 0 0
0
5
0
4
0
7
0
6
0
0
12 13 15 14
8 9 11 10
1
1 1
1
NEXT

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Minimization Steps (SOP Expression)
Form groups of adjacent
1's. Make groups as large
as possible.
Group size must be a power
of two. i.e. Group of
• 8 (OCTET),
• 4 (QUAD),
• 2 (PAIR) or
• 1 (Single)
A B C D 00 01 11 10
00 0
0
0
1
0
3
0
2
01 0 0 0 0
11 1 1 0 0
10 1 1 0 0
0
5
0
4
0
7
0
6
0
0
12 13 15 14
8 9 11 10
NEXT

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Minimization Steps (SOP Expression)
Select the
least number
of groups
that cover all
the 1's.
1100
1101
0111
0110
0
wx
yz
00 01 11 10
00
01
11
10
3 2
4 5
7 6
1
12 13 15 14
8 9 11 10
Ensure that every 1 is in a group.
1's can be
part of more
than one
group.
Eliminate
Redundant
Groups
NEXT

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Example: Reduce f(wxyz)=Σ(1,3,4,5,7,10,11,12,14,15)
PAIR (m4,m5)
REDUNDANTGROUP
1100
1101
0111
0110
0
wx
yz
00 01 11 10
00
01
11
10
3 2
4 5
7 6
1
12 13 15 14
8 9 11 10
QUAD (m1,m3,m5,m7)
QUAD
(m10,m11,m14,m15)
QUAD
(m3,m7,m11,m15)
REDUNDANT
Group
PAIR (m4,m12)
Minimized Expression : xy’z’ + wy + w’z

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OCTET REDUCTION ( Group of 8:)
0011
0011
0011
0011
W X
YZ 0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
OCTET
(m0,m1,m4,m5,m8,
m9, m12,m13)
•The term gets reduced by 3 literals i.e. 3 variables
change within the group of 8 ( Octets )
NEXT

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OCTET REDUCTION ( Group of 8:)
0110
0110
0110
0110
W X
YZ 0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
OCTET
(m1,m3,m5,m7,m9,
m11, m13,m15)
NEXT

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OCTET REDUCTION ( Group of 8:)
MAP ROLLING
OCTET
(m0,m2,m4,m6,
m8, m10, m12,m14)
1001
1001
1001
1001
W X
YZ 0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
0 1 3 2
4 5
7
6
12 13 15
14
8 9 11 10
NEXT

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OCTET REDUCTION ( Group of 8:)
0000
1111
1111
0000
W X
YZ 0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
0 1 3 2
4 5
7
6
12 13 15
14
8 9 11 10
OCTET
(m4,m5,m6,m7,m12,
m13, m14,m15)
NEXT

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OCTET REDUCTION ( Group of 8:)
MAP ROLLING
OCTET
(m0,m1,m2,m3
M8,m9,m10,m11)
1111
0000
0000
1111
W X
YZ 0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
0 1 3 2
4 5
7
6
12 13 15
14
8 9 11 10

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QUAD REDUCTION ( Group of 4)
1100
1111
0111
0110
0
WX
YZ
3 2
4 5
7 6
1
12 13 15 14
8 9 11 10
QUAD
(m1,m3,m5,m7)
QUAD
(m10,m11,m14,m15)
QUAD
(m4,m5,m12,m13)
0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
•The term gets reduced by 2 literals i.e. 2 variables
change within the group of 4( QUAD )
NEXT

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QUAD REDUCTION ( Group of 4)
MAP ROLLING
QUAD
(m1,m3,m9,m11)
QUAD
(m4,m6,m12,m14)
1110
1111
1111
0110
0
WX
YZ
3 2
4 5
7 6
1
12 13 15 14
8 9 11 10
0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
NEXT

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QUAD REDUCTION ( Group of 4)
QUAD
(m0,m2,m8,m10)
1001
0000
0000
1001
0
WX
YZ
3 2
4 5
7 6
1
12 13 15 14
8 9 11 10
0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
CORNER ROLLING

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SINGLE CELL REDUCTION
1100
1101
0000
0010
wx
yz
00 01 11 10
00
01
11
10
SINGLE CELL (m1)
SINGLE CELL (m12)
QUAD
(m10,m11,m14,m15)
•The term is not reduced in a single cell

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PAIR REDUCTION ( Group of 2)
YZ
MAP ROLLING
PAIR
(m0,m2)
0000
0000
0110
1001
0
WX
3 2
4
5 7 6
1
12 13 15 14
8 9 11 10
PAIR
(m5,m7)
0 0 0 1 1 1 1 0
Y.Z Y.Z Y. Z Y. Z
0 0
W.X
0 1
W.X
1 1
W.X
1 0
W.X
•The term gets reduced by 1 literals i.e. 1 variables
change within the group of 2( PAIR )

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• Groups may not include any cell containing a zero
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)

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•Groups may be horizontal or vertical, but not diagonal.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)

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• Groups must contain 1, 2, 4, 8, or in general 2n
cells.
• That is if n = 1, a group will contain two 1's since 21
= 2.
• If n = 2, a group will contain four 1's since 22
= 4.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)

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•Each group should be as large as possible.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)

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•Each cell containing a 1 must be in at least one group.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)

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•Groups may overlap.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)

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• Groups may wrap around the table.
• The leftmost cell in a row may be grouped with
the rightmost cell and
• The top cell in a column may be grouped with the
bottom cell.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)

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• There should be as few groups as possible,
as long as this does not contradict any of
the previous rules.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)

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1. No 0’s allowed in the groups.
2. No diagonal grouping allowed.
3. Groups should be as large as possible.
4. Only power of 2 number of cells in each group.
5. Every 1 must be in at least one group.
6. Overlapping allowed.
7. Wrap around allowed.
8. Fewest number of groups are considered.
9. Redundant groups ignored
Karnaugh Maps - Rules of Simplification
(SOP Expression)

38. 01/30/15 Karnaugh Maps 38
• Minimalization logic function with 3-10inputs.
• Draw karnaugh map
• Draw shema
• Cońvert to NOR and NANDS.
Karnaugh map minimalization software is
freeware.
Karnaugh Minimizer is a tool for developers
of small digital devices and radio amateurs,
also for those who is familiar with Boolean
algebra, mostly for electrical engineering
students.
Important Links…
K-Min

39. 01/30/15 Karnaugh Maps 39
Who Developed K-Maps…
• Name: Maurice Karnaugh, a telecommunications
engineer at Bell Labs. While designing digital logic
based telephone switching circuits he developed a
method for Boolean expression minimization.
• Year : 1950 same year that Charles M. Schulz
published his first Peanuts comic.