The Karnaugh Map

THE THE KARNAUGH KARNAUGH

MAPMAP

GROUP MEMBERS

K MAP

• MAURICE KARNAUGH INTRODUCED K-MAP IN 1953• THE KARNAUGH MAP, ALSO KNOWN AS THE K-MAP, IS A METHOD TO SIMPLIFY

BOOLEAN ALGEBRA EXPRESSIONS.• THE KARNAUGH MAP REDUCES THE NEED FOR EXTENSIVE CALCULATIONS BY

TAKING ADVANTAGE OF HUMANS' PATTERN-RECOGNITION CAPABILITY.• THE REQUIRED BOOLEAN RESULTS ARE TRANSFERRED FROM A TRUTH TABLE

ONTO A TWO-DIMENSIONAL GRID WHERE THE CELLS ARE ORDERED IN GRAY CODE, AND EACH CELL POSITION REPRESENTS ONE COMBINATION OF INPUT CONDITIONS, WHILE EACH CELL VALUE REPRESENTS THE CORRESPONDING OUTPUT VALUE. OPTIMAL GROUPS OF 1S OR 0S ARE IDENTIFIED.

• THESE TERMS CAN BE USED TO WRITE A MINIMAL BOOLEAN EXPRESSION REPRESENTING THE REQUIRED LOGIC.

• ADVANTAGES• REDUCES EXTENSIVE CALC.• REDUCES EXPRESSION WITHOUT BOOLEAN THEOREMS• USED FOR MINIMIZING CIRCUITS• LESS TIME CONSUMING• LESS SPACE CONSUMING

• DISADVANTAGES• TEDIOUS FOR MORE THAN 5 VARIABLES• SOME EXAMPLES ARE SOLVED IN FEW SECONDS BY BOOLEAN THEOREMS EASILY

THE KARNAUGH MAPTHE KARNAUGH MAP

• NUMBER OF SQUARES = NUMBER OF COMBINATIONS

• EACH SQUARE REPRESENTS A MINTERM

• 2 VARIABLES 4 SQUARES



THE 3 VARIABLE K-MAP

• THERE ARE 8 CELLS AS SHOWN:CC

ABAB 00 11

0000

0101

1111

1010

CBA CBA

CBA BCA

CAB ABC

CBA CBA

THE 4-VARIABLE K-MAP

CDCDABAB 0000 0101 1111 1010

0000

0101

1111

1010 DCBA

DCAB

DCBA

DCBA

DCBA

DCAB

DCBA

DCBA

CDBA

ABCD

BCDA

CDBA

DCBA

DABC

DBCA

DCBA

ADJACENT SQUARES

• A LARGER NUMBER OF ADJACENT SQUARES ARE COMBINED, WE OBTAIN A PRODUCT TERM WITH FEWER LITERALS.1 SQUARE = 1 MINTERM = THREE LITERALS.

2 ADJACENT SQUARES = 1 TERM = TWO LITERALS.

4 ADJACENT SQUARES = 1 TERM = ONE LITERAL.

8 ADJACENT SQUARES ENCOMPASS THE ENTIRE MAP AND PRODUCE A FUNCTION THAT IS ALWAYS EQUAL TO 1.

• IT IS OBVIOUSLY TO KNOW THE NUMBER OF ADJACENT SQUARES IS COMBINED IN A POWER OF TWO SUCH AS 1,2,4, AND 8.

8

CELL ADJACENCY

CDCDABAB

0000 0101 1111 1010

0000010111111010

K-MAP SOP MINIMIZATION

• THE K-MAP IS USED FOR SIMPLIFYING BOOLEAN EXPRESSIONS TO THEIR MINIMAL FORM.

• A MINIMIZED SOP EXPRESSION CONTAINS THE FEWEST POSSIBLE TERMS WITH FEWEST POSSIBLE VARIABLES PER TERM.

• GENERALLY, A MINIMUM SOP EXPRESSION CAN BE IMPLEMENTED WITH FEWER LOGIC GATES THAN A STANDARD EXPRESSION.

MAPPING A STANDARD SOP EXPRESSION

FOR AN SOP EXPRESSION IN STANDARD FORM: A 1 IS PLACED ON THE K-

MAP FOR EACH PRODUCT TERM IN THE EXPRESSION.

EACH 1 IS PLACED IN A CELL CORRESPONDING TO THE VALUE OF A PRODUCT TERM.

EXAMPLE: FOR THE PRODUCT TERM , A 1 GOES IN THE 101 CELL ON A 3-VARIABLE MAP.

CBA

CCABAB 00 11

0000

0101

1111

1010

CBA CBA

CBA BCA

CAB ABC

CBA CBA1

CCABAB 00 11

0000

0101

1111

1010

MAPPING A STANDARD SOP EXPRESSION (FULL EXAMPLE)

THE EXPRESSION: CBACABCBACBA

000 001 110 100

1 1

1

1

MAPPING A NONSTANDARD SOP EXPRESSION

• A BOOLEAN EXPRESSION MUST BE IN STANDARD FORM BEFORE YOU USE A K-MAP.• IF ONE IS NOT IN STANDARD FORM, IT MUST BE

CONVERTED.

• YOU MAY USE THE PROCEDURE MENTIONED EARLIER OR USE NUMERICAL EXPANSION.

MAPPING A NONSTANDARD SOP EXPRESSION

NUMERICAL EXPANSION OF A NONSTANDARD PRODUCT TERMASSUME THAT ONE OF THE PRODUCT TERMS IN A

CERTAIN 3-VARIABLE SOP EXPRESSION IS . IT CAN BE EXPANDED NUMERICALLY TO STANDARD

FORM AS FOLLOWS: STEP 1: WRITE THE BINARY VALUE OF THE TWO VARIABLES

AND ATTACH A 0 FOR THE MISSING VARIABLE : 100. STEP 2: WRITE THE BINARY VALUE OF THE TWO VARIABLES

AND ATTACH A 1 FOR THE MISSING VARIABLE : 101. THE TWO RESULTING BINARY NUMBERS ARE THE

VALUES OF THE STANDARD SOP TERMS AND .

C

C

BA

CBA CBA

K-MAP SIMPLIFICATION OF SOP EXPRESSIONS

• AFTER AN SOP EXPRESSION HAS BEEN MAPPED, WE CAN DO THE PROCESS OF MINIMIZATION:

• GROUPING THE 1S• DETERMINING THE MINIMUM SOP EXPRESSION FROM THE

MAP

GROUPING THE 1S

• YOU CAN GROUP 1S ON THE K-MAP ACCORDING TO

THE FOLLOWING RULES BY ENCLOSING THOSE

ADJACENT CELLS CONTAINING 1S.

• THE GOAL IS TO MAXIMIZE THE SIZE OF THE GROUPS

AND TO MINIMIZE THE NUMBER OF GROUPS.

GROUPING THE 1S (EXAMPLE)

CCABAB 00 11

0000 11

0101 11

1111 11 11

1010

CCABAB 00 11

0000 11 11

0101 11

1111 11

1010 11 11

GROUPING THE 1S (EXAMPLE)CDCD

ABAB 0000 0101 1111 1010

0000 11 11

0101 11 11 11 11

1111

1010 11 11

CDCDABAB 0000 0101 1111 1010

0000 11 11

0101 11 11 11

1111 11 11 11

1010 11 11 11

DETERMINING THE MINIMUM SOP EXPRESSION FROM THE

MAP• THE FOLLOWING RULES ARE APPLIED TO FIND THE MINIMUM

PRODUCT TERMS AND THE MINIMUM SOP EXPRESSION:1. GROUP THE CELLS THAT HAVE 1S. EACH GROUP OF

CELL CONTAINING 1S CREATES ONE PRODUCT TERM COMPOSED OF ALL VARIABLES THAT OCCUR IN ONLY ONE FORM (EITHER COMPLEMENTED OR COMPLEMENTED) WITHIN THE GROUP. VARIABLES THAT OCCUR BOTH COMPLEMENTED AND UNCOMPLEMENTED WITHIN THE GROUP ARE ELIMINATED CALLED CONTRADICTORY VARIABLES.


MAP2. DETERMINE THE MINIMUM PRODUCT TERM FOR EACH GROUP.• FOR A 3-VARIABLE MAP:

1. A 1-CELL GROUP YIELDS A 3-VARIABLE PRODUCT TERM2. A 2-CELL GROUP YIELDS A 2-VARIABLE PRODUCT TERM3. A 4-CELL GROUP YIELDS A 1-VARIABLE PRODUCT TERM4. AN 8-CELL GROUP YIELDS A VALUE OF 1 FOR THE EXPRESSION.

• FOR A 4-VARIABLE MAP:1. A 1-CELL GROUP YIELDS A 4-VARIABLE PRODUCT TERM2. A 2-CELL GROUP YIELDS A 3-VARIABLE PRODUCT TERM3. A 4-CELL GROUP YIELDS A 2-VARIABLE PRODUCT TERM4. AN 8-CELL GROUP YIELDS A A 1-VARIABLE PRODUCT TERM5. A 16-CELL GROUP YIELDS A VALUE OF 1 FOR THE EXPRESSION.


MAP3. WHEN ALL THE MINIMUM PRODUCT TERMS ARE DERIVED

FROM THE K-MAP, THEY ARE SUMMED TO FORM THE MINIMUM SOP EXPRESSION.

DETERMINING THE MINIMUM SOP EXPRESSION FROM THE MAP (EXAMPLE)

CDCDABAB 0000 0101 1111 1010

0000 11 110101 11 11 11 111111 11 11 11 111010 11

BCA

DCA

DCACAB

DETERMINING THE MINIMUM SOP EXPRESSION FROM THE MAP (EXERCISES)

CBABCAB ACCAB

CCABAB 00 11

0000 11

0101 11

1111 11 11

1010

CCABAB 00 11

0000 11 11

0101 11

1111 11

1010 11 11

DETERMINING THE MINIMUM SOP EXPRESSION FROM THE MAP

(EXERCISES)

DBACABA CBCBAD

CDCDABAB 0000 0101 1111 1010

0000 11 11

0101 11 11 11 11

1111

1010 11 11

CDCDABAB 0000 0101 1111 1010

0000 11 11

0101 11 11 11

1111 11 11 11

1010 11 11 11

MAPPING DIRECTLY FROM A TRUTH TABLEI/PI/P O/PO/P

AA BB CC XX00 00 00 1100 00 11 0000 11 00 0000 11 11 0011 00 00 1111 00 11 0011 11 00 1111 11 11 11

CCABAB 00 11

0000

0101

1111

1010

1

11

1

The Karnaugh Map

Engineering

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