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 Kmap

Cover : PI's that account for all \( f = 1 \)
Example: \( f = \sum m(0, 1, 2, 3, 7) = \bx_1 + x_1 x_2 x_3\)
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 Kmap.

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
Find the cost of the \( g = \overline{(x_1\bx_2 + x_3)}(\bx_4 + x_4)\)
Cost = 2 AND + 2 OR + 1 NOT + 9 inputs = 14
Find the cost of the \( f = (x_2 + x_3)(x_3 + x_4)(\bx_1 + \bx_2 + \bx_3 + \bx_4)\)
Cost = 3 OR + 1 AND + (8+3) inputs = 15
Kmap neighborhood property
\begin{align}
f &= x_1 \bx_3 \bx_4 + x_2 \bx_3 \bx_4 + x_1 \bx_2 \bx_3 &
\end{align}
  \(\bar{x}_1\)  \(x_1 \) 
  \(\bar{x}_2\)  \(x_2 \)  \(\bar{x}_2\) 
\(\bar{x}_3\)  \(\bar{x}_4\) 
0  1  1  1 
\(x_4\) 
0  0  0  1 
\(x_3\) 
0  0  0  0 
\(\bar{x}_4\) 
0  0  0  0 
Adjacent cells of Kmaps differ only by one variable
Karnaugh Maps for 5variable
Karnaugh Maps for 6variable
KMaps for POS minimization
\[ f(x_1, x_2, x_3) = \prod M(4, 5, 6) \]
\[ f(x_1, x_2, x_3) = \prod M(4, 5, 6) \]
\[ f(x_1, x_2, x_3) = (\bx_1 + x_3)(\bx_1 + x_2) \]
Cost = 9 = 6 inputs + 2 OR gate + 1 AND gate
\[ f(x_1, x_2, x_3) = \prod M(4, 5, 6) \]
\[ \bar{f}(x_1, x_2, x_3) = \sum m(4, 5, 6) \]
 \(\bar{x}_1\)  \(x_1 \) 
 \(\bar{x}_2\)  \(x_2 \)  \(\bar{x}_2\) 
\(\bar{x}_3\) 
0  0  1  1 
\(x_3\) 
0  0  0  1 
\[ \bar{f} = \color{red}{x_1 \bx_3} + \color{blue}{x_1 \bx_2 } \]
\[ f = \color{red}{(\bx_1 + x_3)}\color{blue}{(\bx_1 + x_2)} \]
Minimization process

Draw Kmap for \( f \) and \( \bar{f} \)

Find cost for each. Choose the minimum cost implementation.
Incompletely specified functions
\[f(x_1, \dots, x_4) = \sum m(2, 4, 5, 6, 10) + D(12, 13, 14, 15).\]
\[f(x_1, \dots, x_4) = \sum m(2, 4, 5, 6, 10) + D(12, 13, 14, 15).\]
  \(\bar{x}_1\)  \(x_1 \) 
  \(\bar{x}_2\)  \(x_2 \)  \(\bar{x}_2\) 
\(\bar{x}_3\)  \(\bar{x}_4\) 
0  1  d  0 
\(x_4\) 
0  1  d  0 
\(x_3\) 
0  0  d  0 
\(\bar{x}_4\) 
1  1  d  1 
\[f = \color{blue}{x_2 \bx_3} + \color{red}{x_3 \bx_4} \]
Cost = 6 inputs + 2 AND + 1 OR = 7
  \(\bar{x}_1\)  \(x_1 \) 
  \(\bar{x}_2\)  \(x_2 \)  \(\bar{x}_2\) 
\(\bar{x}_3\)  \(\bar{x}_4\) 
0  1  d  0 
\(x_4\) 
0  1  d  0 
\(x_3\) 
0  0  d  0 
\(\bar{x}_4\) 
1  1  d  1 
\[\bar{f} = \color{blue}{\bx_2 \bx_3} + \color{red}{\bx_3 x_4} \]
\[f = \color{blue}{(x_2 + x_3)} \color{red}{(x_3 + \bx_4)} \]
Cost = 6 inputs + 2 OR + 1 AND = 7