What is Karnaugh Maps and how to simplify the complex boolean expression

What is the Karnaugh Maps?

Karnaugh Maps offer a graphical method of reducing a digital circuit to its minimum number of gates. The map is a simple table containing 1s and 0s that can express a truth table or complex Boolean expression describing the operation of a digital circuit. The map is then used to work out the minimum number of gates needed, by graphical means rather than by algebra. Karnaugh maps can be used on small circuits having two or three inputs as an alternative to Boolean algebra, and on more complex circuits having up to 6 inputs, it can provide quicker and simpler minimization than Boolean algebra.

The Karnaugh map uses the following rules for the simplification of expressions by grouping together adjacent cells containing ones

• Groups may not include any cell containing a zero
  

• Groups may be horizontal or vertical, but not diagonal.
  

• Groups must contain 1, 2, 4, 8, or in general 2ⁿ cells.
  That is if n = 1, a group will contain two 1's since 2¹ = 2.
  If n = 2, a group will contain four 1's since 2² = 4.
  

• Each group should be as large as possible.
  

• Each cell containing a one must be in at least one group.
  

• Groups may overlap.
  

• Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.
  

• There should be as few groups as possible, as long as this does not contradict any of the previous rules.
  

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Summary:

1. No zeros allowed.
2. No diagonals.
3. Only power of 2 number of cells in each group.
4. Groups should be as large as possible.
5. Everyone must be in at least one group.
6. Overlapping allowed.
7. Wrap around allowed.
8. The fewest number of groups possible.