Technische Informatik
Minimierung mit KV-Diagrammen
Thorsten Thormählen
17. November 2022
Teil 5, Kapitel 2
Thorsten Thormählen
17. November 2022
Teil 5, Kapitel 2
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Typ | Schriftart | Beispiele |
---|---|---|
Variablen (Skalare) | kursiv | $a, b, x, y$ |
Funktionen | aufrecht | $\mathrm{f}, \mathrm{g}(x), \mathrm{max}(x)$ |
Vektoren | fett, Elemente zeilenweise | $\mathbf{a}, \mathbf{b}= \begin{pmatrix}x\\y\end{pmatrix} = (x, y)^\top,$ $\mathbf{B}=(x, y, z)^\top$ |
Matrizen | Schreibmaschine | $\mathtt{A}, \mathtt{B}= \begin{bmatrix}a & b\\c & d\end{bmatrix}$ |
Mengen | kalligrafisch | $\mathcal{A}, B=\{a, b\}, b \in \mathcal{B}$ |
Zahlenbereiche, Koordinatenräume | doppelt gestrichen | $\mathbb{N}, \mathbb{Z}, \mathbb{R}^2, \mathbb{R}^3$ |
$b$ | $a$ | $y$ | DNF |
---|---|---|---|
0 | 0 | 1 | $(\overline{a} \land \overline{b})$ |
0 | 1 | 0 | |
1 | 0 | 1 | $\lor (\overline{a} \land b)$ |
1 | 1 | 0 |
x2 | x1 | x0 | y | DNF |
---|---|---|---|---|
0 | 0 | 0 | 0 | |
0 | 0 | 1 | 0 | |
0 | 1 | 0 | 1 | (¬x2∧x1∧¬x0) |
0 | 1 | 1 | 0 | |
1 | 0 | 0 | 0 | |
1 | 0 | 1 | 0 | |
1 | 1 | 0 | 1 | ∨(x2∧x1∧¬x0) |
1 | 1 | 1 | 0 |
$x_1$ | $x_0$ | $y$ | DNF |
---|---|---|---|
0 | 0 | 1 | $(\overline{x_1} \land \overline{x_0})$ |
0 | 1 | 0 | |
1 | 0 | 1 | $\lor (x_1 \land \overline{x_0})$ |
1 | 1 | 0 |
$a$ | $b$ | $c_{\text{in}}$ | $s$ | $c_{\text{out}}$ |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 1 | 0 |
0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 1 | 1 | 1 |
x1 | x0 | y | KV-Zelle |
---|---|---|---|
0 | 0 | 1 | 0 |
0 | 1 | 0 | 1 |
1 | 0 | 1 | 2 |
1 | 1 | 0 | 3 |
x2 | x1 | x0 | y | KV-Zelle |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 1 |
0 | 1 | 0 | 1 | 2 |
0 | 1 | 1 | 0 | 3 |
1 | 0 | 0 | 1 | 4 |
1 | 0 | 1 | 1 | 5 |
1 | 1 | 0 | 1 | 6 |
1 | 1 | 1 | 1 | 7 |
x3 | x2 | x1 | x0 | y | KV-Zelle |
---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 2 |
0 | 0 | 1 | 1 | 1 | 3 |
0 | 1 | 0 | 0 | 0 | 4 |
0 | 1 | 0 | 1 | 1 | 5 |
0 | 1 | 1 | 0 | 1 | 6 |
0 | 1 | 1 | 1 | 0 | 7 |
1 | 0 | 0 | 0 | 0 | 8 |
1 | 0 | 0 | 1 | 1 | 9 |
1 | 0 | 1 | 0 | 1 | 10 |
1 | 0 | 1 | 1 | 0 | 11 |
1 | 1 | 0 | 0 | 1 | 12 |
1 | 1 | 0 | 1 | 0 | 13 |
1 | 1 | 1 | 0 | 1 | 14 |
1 | 1 | 1 | 1 | 0 | 15 |
$y = \overline{x}_2 \overline{x}_1 x_0 \lor x_2 \overline{x}_1 x_0 = \overline{x}_1 x_0$
$x_0$
$x_1$
$x_2$
$y = x_2 \overline{x}_1 x_0 \lor x_2 \overline{x}_1 \overline{x}_0 = x_2 \overline{x}_1$ |
$x_0$
$x_1$
$x_2$
$y = x_2 x_1 x_0 \lor x_2 \overline{x}_1 x_0 = x_2 x_0$ |
$x_0$
$x_1$
$x_2$
$y = x_2 \overline{x}_1 \overline{x}_0 \lor x_2 x_1 \overline{x}_0 = x_2 \overline{x}_0$ |
$x_0$
$x_1$
$x_2$
$y = x_2$ |
|
$x_0$
$x_1$
$x_2$
$y = \overline{x}_1 x_0 \lor x_2 \overline{x}_1$ |
|
$x_0$
$x_1$
$x_2$
$y = x_1 x_0 \lor x_2$ |
|
$x_0$
$x_1$
$x_2$
$x_3$
$x_0$
$x_1$
$x_2$
$x_3$
|
Die Blöcke können auch über die Ränder hinweg gebildet werden.
|
$x_0$
$x_1$
$x_2$
$x_3$
|
|
$x_0$
$x_1$
$x_2$
$x_3$
|
|
$c_{\text{in}}$
$b$
$a$
$c_{\text{in}}$
$b$
$a$
|
$\mathrm{f}(x_2,x_1,x_0) = \bigvee\limits_{i \in \{0,4,5,7\}} m_i$ $x_0$
$x_1$
$x_2$
$\mathrm{f}(x_2,x_1,x_0) = \overline{x}_1\overline{x}_0 \lor x_2 x_0$ |
$\mathrm{f}(x_2,x_1,x_0) = \bigvee\limits_{i \in \{0,4,5,6,7\}} m_i$ $x_0$
$x_1$
$x_2$
$\mathrm{f}(x_2,x_1,x_0) = \overline{x}_1\overline{x}_0 \lor x_2$ |
$\mathrm{f}(x_2,x_1,x_0) = \bigvee\limits_{i \in \{0,1,4,5,6,7\}} m_i$ $x_0$
$x_1$
$x_2$
$\mathrm{f}(x_2,x_1,x_0) = \overline{x}_1 \lor x_2$ |
$\mathrm{f}(x_2,x_1,x_0) = \bigvee\limits_{i \in \{0,1,3,4,5,6,7\}} m_i$ $x_0$
$x_1$
$x_2$
$\mathrm{f}(x_2,x_1,x_0) = \overline{x}_1 \lor x_0 \lor x_2$ |
|
$x_0$
$x_1$
$x_2$
$x_3$
|
|
$x_0$
$x_1$
$x_2$
$x_3$
|
x3 | x2 | x1 | x0 | y | KV-Zelle |
---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 1 | 2 |
0 | 0 | 1 | 1 | 1 | 3 |
0 | 1 | 0 | 0 | 0 | 4 |
0 | 1 | 0 | 1 | X | 5 |
0 | 1 | 1 | 0 | 1 | 6 |
0 | 1 | 1 | 1 | X | 7 |
1 | 0 | 0 | 0 | 0 | 8 |
1 | 0 | 0 | 1 | 0 | 9 |
1 | 0 | 1 | 0 | 1 | 10 |
1 | 0 | 1 | 1 | 1 | 11 |
1 | 1 | 0 | 0 | 0 | 12 |
1 | 1 | 0 | 1 | 0 | 13 |
1 | 1 | 1 | 0 | 0 | 14 |
1 | 1 | 1 | 1 | 0 | 15 |
$y = \overline{x}_3 x_1 \lor \overline{x}_2 x_ 1$
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Am Online-Quiz teilnehmen durch Besuch der Webseite:
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Variablen:
Don’t-Cares erlauben:
Ergebnis verstecken:
$a$ | $b$ | $c$ | $d$ | $l$ | $e$ | $g$ | KV-Zelle |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 1 | 0 | 0 | 2 |
0 | 0 | 1 | 1 | 1 | 0 | 0 | 3 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 4 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 5 |
0 | 1 | 1 | 0 | 1 | 0 | 0 | 6 |
0 | 1 | 1 | 1 | 1 | 0 | 0 | 7 |
1 | 0 | 0 | 0 | 0 | 0 | 1 | 8 |
1 | 0 | 0 | 1 | 0 | 0 | 1 | 9 |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 10 |
1 | 0 | 1 | 1 | 1 | 0 | 0 | 11 |
1 | 1 | 0 | 0 | 0 | 0 | 1 | 12 |
1 | 1 | 0 | 1 | 0 | 0 | 1 | 13 |
1 | 1 | 1 | 0 | 0 | 0 | 1 | 14 |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 15 |
$a$ | $b$ | $c$ | $d$ | $l$ | $e$ | $g$ | KV-Zelle |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 1 | 0 | 0 | 2 |
0 | 0 | 1 | 1 | 1 | 0 | 0 | 3 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 4 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 5 |
0 | 1 | 1 | 0 | 1 | 0 | 0 | 6 |
0 | 1 | 1 | 1 | 1 | 0 | 0 | 7 |
1 | 0 | 0 | 0 | 0 | 0 | 1 | 8 |
1 | 0 | 0 | 1 | 0 | 0 | 1 | 9 |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 10 |
1 | 0 | 1 | 1 | 1 | 0 | 0 | 11 |
1 | 1 | 0 | 0 | 0 | 0 | 1 | 12 |
1 | 1 | 0 | 1 | 0 | 0 | 1 | 13 |
1 | 1 | 1 | 0 | 0 | 0 | 1 | 14 |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 15 |
$a$ | $b$ | $c$ | $d$ | $l$ | $e$ | $g$ | KV-Zelle |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 1 | 0 | 0 | 2 |
0 | 0 | 1 | 1 | 1 | 0 | 0 | 3 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 4 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 5 |
0 | 1 | 1 | 0 | 1 | 0 | 0 | 6 |
0 | 1 | 1 | 1 | 1 | 0 | 0 | 7 |
1 | 0 | 0 | 0 | 0 | 0 | 1 | 8 |
1 | 0 | 0 | 1 | 0 | 0 | 1 | 9 |
1 | 0 | 1 | 0 | 0 | 1 | 0 | 10 |
1 | 0 | 1 | 1 | 1 | 0 | 0 | 11 |
1 | 1 | 0 | 0 | 0 | 0 | 1 | 12 |
1 | 1 | 0 | 1 | 0 | 0 | 1 | 13 |
1 | 1 | 1 | 0 | 0 | 0 | 1 | 14 |
1 | 1 | 1 | 1 | 0 | 1 | 0 | 15 |
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