This document describes a K-Map software tool that simplifies Boolean equations. The tool reads in a Boolean expression with up to 4 variables in sum-of-products or product-of-sums form, generates a Karnaugh map, and uses it to minimize the expression. Algorithms are provided for solving 2, 3, and 4 variable maps. The tool could aid in designing sequential circuits and simplifying expressions frequently in other applications. Its use of different input forms and deductive reasoning achieves simplified output.

1. K-MAP
TOOL
SOLVING EQUATIONS
MADE EASY…..

2. SUBMITTED BY
DEEPAK R
-
2SD08EE022
KUMAR PATIL
-
2SD08EE030
NAGAPPA D V
2SD08EE033
-

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

4. NEED TO SIMPLIFY EQUATIONS..?

5. 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

6. 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)
•

7. 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.

8. 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

9. 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

11. 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

12. 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

13. 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

14. 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

15. 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();

16. 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 ( );

17. 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

18. 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

19. 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

20. 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

21. 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.

22. 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

23. 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.

24. 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

25. Any queries…??

26. Thank
you..