Karnaugh Maps

Objectives

1. Learn how to use Karnaugh maps to reduce small boolean expressions.

Notes

• This is from Carpineli chapter 2.3
• Karnaugh maps were introduced in the 1950s as a way to simplify boolean expressions.
• They are a visual trick that allows you to perform this task.
• They are useful for expressions with 4 or fewer variables.
• They can be extended higher, but there are other techniques for more variables.
• You need to start with a truth table.
• From this construct the karnaugh or k-map
  • This is a rectangular grid
    • 2x2, 2x4, 4x4
    • After this, we start adding another grid 2X 4x 4, ...
  • Across the top put the values of up to two variables in gray code.
  • Across the side put the values of up to two variables in gray code.
  • Fill in the corresponding values from the truth table.
• Next look for the largest collections of adjacent 1s
  • With periodic boundary conditions, or the top and bottom and the left and right are connected.
  • "Rectangle" these.
  • The dimensions of the rectangles need to be powers of 2: 1x1, 1x2, 2x2, 1x4, 2x4 ...
  • Try to find maximum sized "circles"
• Finally read the expressions by the constant variables in the areas circled.
  • Read the maximum sized boxes first
  • Make sure that all ones are accounted for.
  • The gray code will assure that any dimension has at least on variable that does not change
  • The product of these form a term in the expression.
  • The smallest will be a minterm.
• Let's try a two input expression

  • a	b	f(a,b)
    0	0	1
    0	1	0
    1	0	1
    1	1	1

  • We buld the k-map

  • a/b	0	1
    0	1	0
    1	1	1

  • We have two grouping here

    • a/b	0	1
      0	1	0
      1	1	1

    • Note here, b is constant: 0 (a is either 0 or 1), so this would produce the term b

    • a/b	0	1
      0	1	0
      1	1	1

    • Note here, a is constant: 1 (b is either 0 or 1), so this would produce the term a.
  • This yields an expression f(a,b) = a + b
  • Note, this is just M₁, which we could have read directly.
• K-maps are more useful with three input functions
  • We can always read any two input functions almost directly.
  • Let's try a three input function

    • a	b	c	f(a,b,c)
      0	0	0	1
      0	0	1	0
      0	1	0	1
      0	1	1	0
      1	0	0	0
      1	0	1	1
      1	1	0	0
      1	1	1	1

    • The K-Map

      • a/bc	00	01	11	10
        0	1	0	0	1
        1	0	1	1	0

      • Note the use of gray code in the column labels.
      • In this case, we have two terms

      • a/bc	00	01	11	10
        0	1	0	0	1
        1	0	1	1	0

    • The yellow cells yield a·c
    • The blue cells yield ab
    • f(a,b,c) = ab + a·c
  • A second example:
    • F(a,b,c) = m₀ + m₂ + m₃ + m ₄ + m₆

    • a	b	c	f(a,b,c)
      0	0	0	1
      0	0	1	0
      0	1	0	1
      0	1	1	1
      1	0	0	1
      1	0	1	0
      1	1	0	1
      1	1	1	0

    • a/bc	00	01	11	10
      0	1	0	1	1
      1	1	0	0	1

      • There are two "rectangles" in this map

        • a/bc	00	01	11	10
          0	1	0	1	1
          1	1	0	0	1

        • The yellow cells represent c
        • The red text represents a b
      • So the final expression is a b + c