The students should be familiar with the following terms in Boolean Algebra before going through this module on K-MAPS..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
1. Subject : Computer Science (083)
Boolean Algebra
Topic : Minimization of Boolean Expressions
Using Karnaugh Maps (K-Maps)
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Submitted By :
Poonam Chopra
PGT Computer Science
Mount Abu Public School
Sec-5, Rohini,Delhi.
2. Learning Objectives :
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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
3. Previous Knowledge :
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The students should be familiar with the
following terms in Boolean Algebra before going
through this module on K-MAPS
x
y
x+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
4. 10/11/2015 Karnaugh Maps 4
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
7. Boolean expressions are practically implemented in the
form of GATES (Circuits).
A minimized Boolean expression means less number of
gates which means
Simplified Circuit
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MINIMIZATION OF BOOLEAN
EXPRESSION
WHY we Need to simplify (minimize) expressions?
8. 10/11/2015 Karnaugh Maps 8
MINIMIZATION OF BOOLEAN
EXPRESSION
Different methods
Karnaugh
Maps
Algebraic
Method
9. Karnaugh Maps
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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
10. 10/11/2015 Karnaugh Maps 10
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. Correspondence between the
Karnaugh Map and the Truth Table
for the general case of a two Variable Problem
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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.)
12. 10/11/2015 Karnaugh Maps 12
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
13. Karnaugh maps consist of a set of 22 squares where 2 is the
number of variables
in the Boolean expression being minimized.
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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
14. For three and four variable expressions
Maps with 23 = 8 and 24 = 16 cells are used.
Each cell represents a MINTERM or a MAXTERM
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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
15. 10/11/2015 Karnaugh Maps 15
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
16. 10/11/2015 Karnaugh Maps 16
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
17. 10/11/2015 Karnaugh Maps 17
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
19. OCTET REDUCTION ( Group of 8:)
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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
20. OCTET REDUCTION ( Group of 8:)
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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
21. OCTET REDUCTION ( Group of 8:)
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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
22. OCTET REDUCTION ( Group of 8:)
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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
23. OCTET REDUCTION ( Group of 8:)
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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
24. QUAD REDUCTION ( Group of 4)
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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
25. QUAD REDUCTION ( Group of 4)
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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
26. QUAD REDUCTION ( Group of 4)
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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
27. SINGLE CELL REDUCTION
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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
28. PAIR REDUCTION ( Group of 2)
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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 )
29. 10/11/2015 Karnaugh Maps 29
• Groups may not include any cell containing a zero
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)
30. 10/11/2015 Karnaugh Maps 30
•Groups may be horizontal or vertical, but not diagonal.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)
31. 10/11/2015 Karnaugh Maps 31
• 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)
32. 10/11/2015 Karnaugh Maps 32
•Each group should be as large as possible.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)
33. 10/11/2015 Karnaugh Maps 33
•Each cell containing a 1 must be in at least one group.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)
34. 10/11/2015 Karnaugh Maps 34
•Groups may overlap.
NEXT
Karnaugh Maps - Rules of Simplification
(SOP Expression)
35. 10/11/2015 Karnaugh Maps 35
• 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)
36. 10/11/2015 Karnaugh Maps 36
• 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)
37. 10/11/2015 Karnaugh Maps 37
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. 10/11/2015 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. 10/11/2015 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.