K map

K-MAP
TOOL
SOLVING EQUATIONS
MADE EASY…..

SUBMITTED BY
DEEPAK R

-

2SD08EE022
KUMAR PATIL

-

2SD08EE030
NAGAPPA D V
2SD08EE033

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Problem statement
Development of an application
software called K-Map software tool
which gives the simplified Boolean
equation.

prerequisite
• BOOLEAN ALGEBRA-

Boolean algebra is formal

way to express digital

logic equations and to represent a logical design in an
alpha-numeric way. It is a language of 0’s and 1’s.
• A Boolean function is an expression formed with
binary variables which makes use of logic gates based

on Boolean algebra
e.g.F1=xyz’ where F1=1 only if x=1,y=1,z=0

Sum-of-products form
(SOP)
•

– first

the product(AND)
terms are formed then
these are summed(OR)
– e.g.: ABC + DEF + GHI

Product-of-sum form
(POS)
•

– first the sum (OR) terms are formed then
are taken (AND)
– e.g.: (A+B+C) (D+E+F) (G+H+I)

the products

It is possible to convert between these two forms
using Boolean algebra (DeMorgan’s Laws)
•

Minimization by Karnaugh Map
The Karnaugh map is a theoretical method for
the simplification of any Boolean expressions
regardless of its number of variables.
It provides simple straightforward approach for
minimizing Boolean functions.

It either regarded as pictorial form of truth
table or as an extension of the Venn diagram.

Example: Let’s do this in relation to the 3input example
S A B

Y

0

0 0

0

0

0 1

0

0

1 0

1

0

1 1

1

1

0 0

0

1

0 1

1

1

1 0

0

1

1 1

1

result: Y = S.B + S’.A

Architectural design
main ( )
read
input
validate
input

derive
truth
table

generate
k-map

solve kmap

solve for
2 var

solve for
3 var

display
solve for
4 var

2var

quad

octet

1

3var

pair

quad

octet

4var

pair

quad
pair

Algorithm

int read input ()

// Input : No of variable n , Input
// Output : Formation Of Input Truth Table Array
{
step1:

read the no of variables
if it is 2, 3 or 4
go to step 2

step2:

read the type of input

case 1:
if minterm with literals
check validity of input
if valid form the input array
case 2:
if maxterm with literals

case 3:

if minterm with truth table values

case 4:

if maxterm with truth table values

} // end of Read Input

Algorithm

int

formKmap

(int n)

// input : no of variable (n) , Integer array input[2^n ]
//output : two dimensional matrix say
//Forms the k map depending on the type of input
{

k=0;
for (i=0;i < no. of rows ;i++)
for (j=0;j < no. of columns; j++)
{

k mat [ i ][ j ] = min[k];
k++;

}
if n is 3
Swap the columns 3 & 4 in the k mat
if n is 4
Swap the columns 3 & 4 and rows 3 and 4 in the k mat
} // end of form K-map

SolveKmap (k mat [ ][ ],n )
// input : no of variable (n) , k mat
// calls other functions depending on value of n
{

if n is 2
solve2var (k mat [ ][ ])
if n is 3

if n is 4
} // end

Algorithm

solve2var(k mat [ ][ ] )

// input : no of variable (n) , k_mat
// solves the given k_mat with n=2 and calls display function

{
check for all ones in k mat
if

True

push 1 to the linked list

else if
the group formed is in pairs then send the
group value to the list
else if

}

only one variable in the matrix is one then send
its group value to the list.

Display();

Algorithm

solve3var (k_mat [ ][ ])

// input : no of variable (n) , k mat
// solves the given k mat with n=2 and calls display function
{
check for all k mat [ i ] [ j ] =1
if true

o list.Push_front (1);
set all flag [ i ] [ j ] =1;
else

Quad;

Pair;
Single;
}

Display ( );

Algorithm

solve4var (k_mat [ ][ ])

// solves the given k_mat with n=2 and calls display function
{
check for all k mat [ i ] [ j ] =1
if true
o_list.Push_front (1);

set all flag [ i ] [ j ] as 1;
else

Octet;
Quad;

Pair ;
Single ;
Display ( );

} // end

Algorithm

Octet (k_mat , flag)

// find the respective quad and push its group value to the O_list
{
for (i=0;i < 4 ;i++)
for (j=0;j < 4; j++)
{

if ( k_mat [ i ][ j ] = 1 && flag [ i ] [ j ] =0)
{
search for a octet
if exist
//push the respective group value
outeq ( i, j);

set all positions in the flag matrix as 1
}
}
} // end of octet

Algorithm

Quad (k_mat , flag)

// input : K_mat , flag
// find the respective quad and push its group value to the O_list
{
for (i=0;i < 4 ;i++)
for (j=0;j < 4; j++)
{

if ( k_mat [ i ][ j ] = 1 && flag [ i ] [ j ] =0)
{
search for a Quad
if exist
push the respective group value

}
}
} // end of Quad

Algorithm

Pair (k_mat , flag)

// input : K_mat , flag
// find the respective Pair and push its group value to the O_list
{
for (i=0;i < 4;i++)
for (j=0;j < 4; j++)
{

if ( k_mat[ i][ j ] = 1 && flag [ i ] [ j ] =0)
{
find a adjacent one
push the respective group value

}
}
} // end of Pair

application
Tool can be a part of sequential circuit designing software.
Aids to practical method of designing sequential circuits with
multi input.
It can be included in systems which involve in process of
simplifying expressions frequently.
The tool can be incorporated in educational fields as in by
the text book designers, staff and students themselves to
verify their answers.

conclusion
 K-Tool can be used for reducing Boolean expressions.
 User specified Boolean expression is converted
to is simpler form through various stages.
 User is able to get output for 2,3,4 variable
CONCLUSION
equations
 Input is processed in various different forms of
input to achieve simplicity

 Its a tool for deductive reasoning in designing
logic circuits and machines

digital

FUTURE SCOPE
 Tool can be extended to handle more number of
variables.
 Tool can be modified to handle DON’T CARE
conditions.
 Can be enhanced to obtain possible alternate
solutions.
 The gate implementation of the input and output
equations can be shown for comparison.

REFERENCES
Books:
Digital Fundamentals -Thomas L. Floyd
Digital Principles and Applications -Albert Paul Malvino and
Donald P. Leach

Digital Logic and Computer Design -M. Morris Mano

Website:

http://en.wikipedia.org/wiki/Karnaugh_map

K map

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