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title: 2024-2025学年下学期期末
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## 2025春季学期计算理论基础期末考试试卷（回忆版）

> 注：这次考试的形式是五道题自行选四个做, 每题25分, 多做没有加分；此外, 原卷面是英文, 请大家注意

1. 给出接受以下语言的DFA:  
    所有以 $1$ 开头, 能被 $5$ 整除的二进制串, 如 $101, 1010, 1111$, 左侧（自动机读入的第一个符号）为最高位, 右侧为最低位

2. 请给出 DFA 化简中 State Elimination Technique 的复杂度的 tight bound, 要有明确的推理过程

3. A marble is dropped at A and B. Levers x1, x2 and x3 cause the marble tofall either to the left or to the right. Whenever a marble encounters a lever, it causes the lever to reverse after the marble passes, so the next marble will take the opposite branch. The game wins when there is a marble falls through C. The levers are all initialized to the left. Model this toy by a DFA.

    ![marble](./assets/marble.png)

4. 证明语言 $L = \{a^nb^nc^md^m (n \geq m \geq 1) \cup a^nb^mc^md^n (n \geq m \geq 1)\}$ 不是 CFL

5. 构造如下的图灵机, 其功能为：给定三个非负整数 $n, m, k$, 判断 $n + m$ 是否等于 $k$.

    输入的形式为：
    $$\underbrace{11……1}_{n\text{个}}0\underbrace{11……1}_{m\text{个}}0\underbrace{11……1}_{k\text{个}}$$
