KMAP

04/09/23 Karnaugh Maps 1

Subject : Computer Science (083)

Unit : Boolean AlgebraTopic : Minimization of Boolean

Expressions Using Karnaugh Maps (K-Maps)


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


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


MinimizationOf

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


References

For K-Map Minimizer Downloadhttp://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


The End


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?


MINIMIZATION OF BOOLEAN EXPRESSION

Different methods

Karnaugh Maps

Algebraic Method


Karnaugh MapsWHAT 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


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


Correspondence between the Karnaugh Map and the Truth Table

for the general case of a two Variable Problem

A B0 00 11 01 1

Fabcd

A \ B 0 1

0 a b

1 c d

Truth Table

2 Variable K-Map



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 MINTERMNEXT


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


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


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


BCA

00 01 11 10

0

1

A B \ C D 00 01 11 1000

01

11

103 Variable K-Map

23 = 8 Cells


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


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


Minimization Steps (SOP Expression)

Select the least number of groups that cover all the 1's.

1100

1101

0111

0110

0

wx

yz00 01 11 10

00

01

11

10

3 2

4 57 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


Example: Reduce f(wxyz)=Σ(1,3,4,5,7,10,11,12,14,15)

PAIR (m4,m5)REDUNDANTGROUP1100

1101

0111

0110

0

wx

yz00 01 11 10

00

01

11

10

3 2

4 57 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


OCTET REDUCTION ( Group of 8:)

0011

0011

0011

0011W X

YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z

0 0W.X

0 1W.X

1 1W.X

1 0W.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



0110

0110

0110

0110W X

YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z

0 0W.X

0 1W.X

1 1W.X

1 0W.X

OCTET

(m1,m3,m5,m7,m9, m11, m13,m15)

NEXT



MAP ROLLING

OCTET(m0,m2,m4,m6,

m8, m10, m12,m14)

1001

1001

1001

1001W X

YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z

0 0W.X

0 1W.X

1 1W.X

1 0W.X

0 1 3 2

4 57

6

12 13 1514

8 9 11 10

NEXT



0000

1111

1111

0000W X

YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z

0 0W.X

0 1W.X

1 1W.X

1 0W.X

0 1 3 2

4 57

6

12 13 1514

8 9 11 10

OCTET

(m4,m5,m6,m7,m12, m13, m14,m15)

NEXT



MAP ROLLING

OCTET(m0,m1,m2,m3

M8,m9,m10,m11)

1111

0000

0000

1111W X

YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z

0 0W.X

0 1W.X

1 1W.X

1 0W.X

0 1 3 2

4 57

6

12 13 1514

8 9 11 10


QUAD REDUCTION ( Group of 4)

1100

1111

0111

0110

0

WX

YZ

3 2

4 57 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 0W.X

0 1W.X

1 1W.X

1 0W.X

•The term gets reduced by 2 literals i.e. 2 variables change within the group of 4( QUAD )

NEXT



MAP ROLLING

QUAD (m1,m3,m9,m11)

QUAD(m4,m6,m12,m14)1110

1111

1111

0110

0

WX

YZ

3 2

4 57 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 0W.X

0 1W.X

1 1W.X

1 0W.X

NEXT



QUAD(m0,m2,m8,m10)

1001

0000

0000

1001

0

WX

YZ

3 2

4 57 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 0W.X

0 1W.X

1 1W.X

1 0W.X

CORNER ROLLING


SINGLE CELL REDUCTION

1100

1101

0000

0010wx

yz00 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


PAIR REDUCTION ( Group of 2)

YZ

MAP ROLLINGPAIR

(m0,m2)

0000

0000

0110

1001

0

WX

3 2

45 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 0W.X

0 1W.X

1 1W.X

1 0W.X

•The term gets reduced by 1 literals i.e. 1 variables change within the group of 2( PAIR )


• Groups may not include any cell containing a zero

NEXT

Karnaugh Maps - Rules of Simplification (SOP Expression)


•Groups may be horizontal or vertical, but not diagonal.

NEXT



• 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



•Each group should be as large as possible.

NEXT



•Each cell containing a 1 must be in at least one group.

NEXT



•Groups may overlap.

NEXT



• 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



• 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)


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)


• 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


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.

KMAP

Education

Transcript of KMAP