Karnaugh 图法 (Karnaugh Map Method)

概述 (Overview)

Karnaugh 图法（简称 K 图法）是一种用于简化布尔代数表达式的图形工具。它通过将逻辑表达式以图表形式排列，便于识别和消除冗余的变量，从而简化逻辑电路的设计。

The Karnaugh Map (K-Map) is a graphical tool used to simplify Boolean algebra expressions. It arranges logic expressions in a tabular form, making it easier to identify and eliminate redundant variables, thereby simplifying the design of logic circuits.

操作步骤 (Steps)

绘制 K 图 (Draw the K-Map):
- 根据变量的数量确定 K 图的尺寸。例如，3 个变量的 K 图是 8 个单元格，4 个变量的 K 图是 16 个单元格。
- 列出变量的所有可能组合，并在图表中相应位置填写这些组合。
Draw the K-Map:
- Determine the size of the K-Map based on the number of variables. For example, a K-Map for 3 variables has 8 cells, and for 4 variables, it has 16 cells.
- List all possible combinations of the variables and fill them in the corresponding positions on the map.
填入真值表 (Fill in the Truth Table Values):
- 将真值表中的输出值填写到 K 图的相应单元格中。
Fill in the Truth Table Values:
- Write the output values from the truth table into the corresponding cells of the K-Map.
识别并圈出最小项 (Identify and Circle the Min-terms):
- 在 K 图中找到所有输出为 1 的单元格，并将它们圈出。相邻的 1 可以形成组，每组中的 1 应尽量多，但每组的大小必须是 2 的幂（1, 2, 4, 8 等）。
Identify and Circle the Min-terms:
- Locate all cells in the K-Map with an output of 1 and circle them. Adjacent 1s can form groups, where each group should be as large as possible but must be a power of 2 (1, 2, 4, 8, etc.).
写出简化后的表达式 (Write the Simplified Expression):
- 根据圈出的最小项，写出简化后的布尔表达式。每个圈代表一个简化项，该项由圈内单元格共有的变量决定。
Write the Simplified Expression:
- Based on the circled min-terms, write the simplified Boolean expression. Each circle represents a simplified term, determined by the variables common to all cells in the circle.

示例 (Example)

假设我们有一个真值表如下：

Assume we have the following truth table:

A	B	C	Output
0	0	0	0
0	0	1	1
0	1	0	1
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

1. 绘制 K 图 (Draw the K-Map)

对于 3 个变量 A, B, C，我们的 K 图如下：

For 3 variables A, B, C, our K-Map is as follows:

   AB
    00  01  11  10
C
0   
1

2. 填入真值表 (Fill in the Truth Table Values)

将真值表中的输出值填入 K 图：

Fill in the truth table values into the K-Map:

   AB
    00  01  11  10
C
0   0   1   1   0
1   1   1   1   1

3. 识别并圈出最小项 (Identify and Circle the Minterms)

在 K 图中圈出所有输出为 1 的组：

Circle all groups of 1s in the K-Map:

   AB
    00  01  11  10
C
0   0   (1) (1)  0
1   (1) (1) (1) (1)

4. 写出简化后的表达式 (Write the Simplified Expression)

通过识别出所有组，写出简化后的布尔表达式：

By identifying all groups, write the simplified Boolean expression:

圈 1： $C = 0$ 行圈出 $B = 1$ 的两列，即 $B C^{'}$
圈 2：整个 $C = 1$ 行，即 $C$

F = B C^{'} + C

此表达式为已简化的布尔表达式，表示了所有输出为 1 的情况。

This expression is the simplified Boolean expression representing all the cases where the output is 1.

常见问题 (Common Issues)

未正确圈出所有最小项 (Not Correctly Circling All Minterms):
- 确保每个输出为 1 的单元格都被圈出，且尽量形成最大的组。
Ensure all cells with an output of 1 are circled and form the largest possible groups.
忽略变量的对称性 (Ignoring Variable Symmetry):
- K 图的排列方式有助于识别变量的对称性，从而更容易简化表达式。
The arrangement of the K-Map helps to identify the symmetry of variables, making it easier to simplify expressions.
未使用 2 的幂进行分组 (Not Using Power of 2 for Grouping):
- 组的大小必须是 2 的幂（例如 1, 2, 4, 8）。
The size of the groups must be a power of 2 (e.g., 1, 2, 4, 8).

通过遵循上述步骤和注意事项，Karnaugh 图法可以有效地简化复杂的逻辑表达式。

By following the above steps and considerations, the Karnaugh Map method can effectively simplify complex logic expressions.

Zephyr's Notes on ISCS & CBMS, UTokyo

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