KARNAUGH MAP

A B C D   f 
0 0 0 0   0
0 0 0 1   0
0 0 1 0   0
0 0 1 1   0
0 1 0 0   1
0 1 0 1   0
0 1 1 0   0 
0 1 1 1   0
1 0 0 0   1
1 0 0 1   1
1 0 1 0   1
1 0 1 1   1 
1 1 0 0   1
1 1 0 1   0
1 1 1 0   1
1 1 1 1   1.

The binary digits in the map represent the function's output for any given combination of inputs. We write 0 in the upper leftmost corner of the map because f = 0 when A = 0, B = 0, C = 1, D = 0. Similarly we mark the bottom right corner as 1 because A = 1, B = 0, C = 0, D = 0 gives f = 1.

After the Karnaugh map has been constructed our next task is to find the minimal terms to use in the final expression. These terms are found by encircling the 1's in the map. The encirclings can only encompass 2ⁿ fields, where n is an integer ≥ 0 (1, 2, 4, 8...). They should be as large as possible. The optimal encirclings in this map are marked by the green, red and blue lines.

For each of these encirclings we find those variables that have the same state in each of the fields in the encircling. For the first encircling (the red one) we find that:

• The variable A maintains the same state (1) in the whole encircling, therefore it should be included in the term for the red encircling.
• Variable B does not maintain the same state (it shifts from 1 to 0), and should therefore be excluded.
• Similarly D changes but C does not.

For the green encircling we see that A and B maintains the same state, C and D changes. But B is 0 and has to be negated before it can be included.

The second term becomes AB'.

By working the blue encircling the same way we find the term BC'D' and our final expression for the function is ready: AC + AB' + BC'D'.

The inverse of a function is solved in the same way by encircling the 0's instead.

It is worth mentioning is that the number of product terms for an encircling P is:

P = 2log(n/x)

References

• Maurice Karnaugh, The Map Method for Synthesis of Combinational Logic Circuits, Trans. AIEE. pt I, 72(9):593-599, November 1953.