Learning Kmap

Problem Areas in Boolean Algebra

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

10/11/2015Karnaugh Maps1

Submitted By :Poonam ChopraPGT Computer ScienceMount Abu Public SchoolSec-5, Rohini,Delhi.

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The operator keywordOverloading Unary operatorsOverloading Binary operatorsConstructors as conversion routinesConverting between basic and user-defined types

LEAD IN: Overloaded Unary Operators

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 :

10/11/2015Karnaugh Maps3The students should be familiar with the following terms in Boolean Algebra before going through this module on K-MAPSx

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

10/11/2015Karnaugh Maps4 MinimizationOfBoolean 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

Karnaugh MapsINDEX


References

For K-Map Minimizer Downloadhttp://karnaugh.shuriksoft.comhttp://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 EXPRESSIONWHY we Need to simplify (minimize) expressions?

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Unary operators require no arguments because they automatically refer to the object that calls them.

For ageClass, the ++ and -- operators would intuitively increment and decrement the age data member.

LEAD IN: Return Values


MINIMIZATION OF BOOLEAN EXPRESSIONDifferent methods

Karnaugh MapsAlgebraic Method

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Unary operators require no arguments because they automatically refer to the object that calls them.

For ageClass, the ++ and -- operators would intuitively increment and decrement the age data member.

LEAD IN: Return Values

Karnaugh Maps10/11/2015Karnaugh Maps9WHAT 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.

Dont 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 0s and 1s within a map helps in visualizing, leading directly to Simplified Boolean Expression

Karnaugh Maps(Contd.)

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Correspondence between the Karnaugh Map and the Truth Table for the general case of a two Variable Problem


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A \ B

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The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem.

Truth Table

Karnaugh Map

A \ B

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Truth Table2 Variable K-Map Karnaugh Maps(Contd.)

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10/11/2015Karnaugh Maps12Drawing 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

Karnaugh maps consist of a set of 22 squares where 2 is the number of variables in the Boolean expression being minimized.



Truth Table

Karnaugh Map

Truth Table2 Variable K-MapKarnaugh Maps(Contd.)

A \ B

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Minterm

AB

AB

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



Truth Table

Karnaugh Map

4 Variable K-Map 24 = 16 CellsKarnaugh Maps(Contd.)

BCA

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A B \ C D

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3 Variable K-Map 23 = 8 Cells

10/11/2015Karnaugh Maps15Minimization 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

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121315148911 101111

NEXT

10/11/2015Karnaugh Maps16Minimization 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

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121315148911 10

NEXT

10/11/2015Karnaugh Maps17Minimization Steps (SOP Expression)

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

1100110101110110 0

wxyz00 01 11 1000

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3245761

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131514

8911

10Ensure 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)=(1,3,4,5,7,10,11,12,14,15)10/11/2015Karnaugh Maps18PAIR (m4,m5)REDUNDANTGROUP

1100110101110110

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wxyz00 01 11 1000

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3245761

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131514

8911

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QUAD (m1,m3,m5,m7)

QUAD(m10,m11,m14,m15)

QUAD(m3,m7,m11,m15)REDUNDANT Group

PAIR (m4,m12) Minimized Expression : xyz + wy + wz

OCTET REDUCTION ( Group of 8:)10/11/2015Karnaugh Maps190011001100110011

W XYZ0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z0 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


W XYZ0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z0 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)1001100110011001

W XYZ0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z0 0W.X

0 1W.X

1 1W.X

1 0W.X0132457612131514891110

NEXT


W XYZ0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z0 0W.X

0 1W.X

1 1W.X

1 0W.X

0132457612131514891110OCTET(m4,m5,m6,m7,m12, m13, m14,m15)

NEXT


MAP ROLLING

OCTET(m0,m1,m2,m3M8,m9,m10,m11)1111000000001111

W XYZ0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z0 0W.X

0 1W.X

1 1W.X

1 0W.X0132457612131514891110

QUAD REDUCTION ( Group of 4)10/11/2015Karnaugh Maps241100111101110110 0

WXYZ

3245761

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131514

8911

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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. Z0 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

QUAD REDUCTION ( Group of 4)10/11/2015Karnaugh Maps25

MAP ROLLING

QUAD (m1,m3,m9,m11)QUAD(m4,m6,m12,m14)1110111111110110 0

WXYZ

3245761

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131514

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0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z0 0W.X

0 1W.X

1 1W.X

1 0W.X

NEXT

QUAD REDUCTION ( Group of 4)10/11/2015Karnaugh Maps26QUAD(m0,m2,m8,m10)

1001000000001001 0

WXYZ3245761

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131514

8911

100 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z0 0W.X

0 1W.X

1 1W.X

1 0W.X

CORNER ROLLING

SINGLE CELL REDUCTION 10/11/2015Karnaugh Maps271100110100000010

wxyz00 01 11 1000

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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)10/11/2015Karnaugh Maps28YZMAP ROLLINGPAIR(m0,m2)0000000001101001 0

WX3245761

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131514

8911

10PAIR(m5,m7)0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z0 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

NEXTKarnaugh Maps - Rules of Simplification (SOP Expression)


Groups may be horizontal or vertical, but not diagonal.



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.



Each group should be as large as possible.



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



Groups may overlap.



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.



There should be as few groups as possible, as long as this does not contradict any of the previous rules.

NEXTKarnaugh Maps - Rules of Simplification

(SOP Expression)


No 0s allowed in the groups. No diagonal grouping allowed. Groups should be as large as possible. Only power of 2 number of cells in each group. Every 1 must be in at least one group. Overlapping allowed. Wrap around allowed. Fewest number of groups are considered. Redundant groups ignored

Karnaugh Maps - Rules of Simplification

(SOP Expression)


Minimalization logic function with 3-10inputs. Draw karnaugh map Draw shema Covert 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-MapsName: 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.

Learning Kmap

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