K-map 5-var example

$f (x_1 , \dots , x_5 ) = \sum m(1, 4, 6, 7, 9, 10, 12, 15, 17, 19, \\ 20, 23, 25, 26, 27, 28, 30, 31) + D(8, 16, 21, 22).$

Karnaugh Maps terminology

• Literal : a single variable or its complement.
• Implicant: A product term that indicates for which $$f = 1$$
All minterms are implicants.
• Prime Implicant (PI): If an implicant cannot be "combined" into fewer literals.
• Essential Prime Implicant (EPI): A PI that is the only PI to cover some 1 on the K-map.
• Cover : PI's that account for all $$f = 1$$.
• Cost : Number of gates and inputs excluding input complements.
Example: $$f = \sum m(0, 1, 2, 3, 7) = \bx_1 + x_1 x_2 x_3$$
$$\bar{x_1}$$ $$x_1$$
$$\bar{x_2}$$ $$x_2$$ $$\bar{y}$$
$$\bar{x_3}$$ 1 1 0 0
$$x_3$$ 1 1+1 1 0
• Literals are $$\bx_1, x_1, x_2, x_3$$
• Implicants are $$\bx_1$$, $$\bx_1 x_2 x_3, \dots$$, $$x_1 x_2 x_3$$
• PI's are $$\bx_1$$ and $$x_2 x_3$$
• EPI's are $$\bx_1$$ and $$x_2 x_3$$
• Cost is 9 = 6 inputs + 2 AND gate + 1 OR gate

Quine McCluskey method

$f (x 1 , \dots , x 4 ) = \sum m(0, 4, 8, 10, 11, 12, 13, 15)$

Finding PIs

$P = \{ 11*0, 101*, 110*, 1*11, 11*1, **00 \}$

Minimum Cover

$P = \{ 11*0, 101*, 110*, 1*11, 11*1, **00 \}$ $f = x_1 \bx_2 x_3 + x_1 x_2 x_4 + \bx_3 \bx_4$

Quine McCluskey method

$f (x 1 , \dots , x 4 ) = \sum m(0, 2, 5, 6, 7, 8, 9, 13) + D(1, 12, 15)$

Finding PIs

$P = \{ 00*0, 0*10, 011*, *00*, **01, 1*0*, *1*1\}$

Minimum Cover

$P = \{ 00*0, 0*10, 011*, *00*, **01, 1*0*, *1*1\}$ $f = \bx_1 x_3 \bx_4 + \bx_2 \bx_3 + x_2 x_4$

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