K map

K-MAP TOOL

SOLVING EQUATIONS MADE EASY…..

SUBMITTED BY

DEEPAK R -

2SD08EE022

KUMAR PATIL -2SD08EE030

NAGAPPA D V -

2SD08EE033

UNDER THE GUIDANCE OF

PROF. H N VIJAYMURTHY

Problem statement

Development of an application software called K-Map software tool which gives the simplified

Boolean equation.

NEED TO SIMPLIFY EQUATIONS..?

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 the products are taken (AND) – e.g.: (A+B+C) (D+E+F) (G+H+I)

• 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 3-input 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

main ( )

read input

validate input

derive truth table

generate k-map

solve k-map

solve for 2 var

quad

pair

solve for 3 var

octet

quad

pair

solve for 4 var

1

octet

quad

pair

display

2var

3var

4var

Architectural design

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 check validity of input if valid form the input array

case 3: if minterm with truth table values

check validity of input

if valid form the input array

case 4: if maxterm with truth table values

check validity of input

if valid form the input array } // 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

solve3var (k mat [ ][ ])

if n is 4

solve4var (k mat [ ][ ])

} // 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 [ ][ ])

// 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 ] as 1;

else Octet;

Quad;

Pair ;

Single ;

Display ( ); } // end

Algorithm Octet (k_mat , flag)// input : no of variable (n) , k_mat

// 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

set all positions in the flag matrix as 1

}

}

} // 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

set all positions in the flag matrix as 1

}

}

} // end of Pair

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.

application

CONCLUSION

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 equations

Input is processed in various different forms of input to achieve simplicity

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

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

Any queries…??

Thank you..