IS CS-2021W-03

题目来源：Problem 3 日期：2024-07-24 题目主题：CS-Formal Languages-Operations on Languages

解题思路

这道题目主要涉及正则语言、有限自动机和上下文无关语言的概念。我们需要理解新定义的操作 $◃$，并将其应用于不同类型的语言。问题逐步深入，从具体的有限语言集合运算，到正则表达式的操作，再到有限自动机的构造，最后探讨了这个操作在更广泛的语言类别上的性质。

Solution

Question 1

Let $L_3=\{aa,b,bb\}$ and $L_4=\{a,b,ab,bb,aaa,bbab\}$. We need to find the set $L_3◃L_4$.

$$L_3◃L_4=\{w\in {\Sigma }^{\ast }\mid \exists v\in L_3\text{ such that }vw\in L_4\}$$

We check each element $v\in L_3$:

1. For $v=aa$:
   • $v\cdot w=aa\cdot w\in L_4$
   • Possible $w$ are $ϵ,a$
2. For $v=b$:
   • $v\cdot w=b\cdot w\in L_4$
   • Possible $w$ are $ϵ,b,ab,bab$
3. For $v=bb$:
   • $v\cdot w=bb\cdot w\in L_4$
   • Possible $w$ are $ϵ,ab$

Collecting all possible $w$:

$$L_3◃L_4=\{ϵ,a,b,ab,bab\}$$

Question 2

Let $L_5=(a^{\ast }b{)}^{\ast }$ and $L_6=(abba{)}^{\ast }$. We need to express $L_5◃L_6$ using a regular expression.

$L_5◃L_6=\{w\in {\Sigma }^{\ast }\mid \exists v\in L_5\text{ such that }vw\in L_6\}$

Let’s analyze this step by step:

1. Words in $L_5$ are of the form $(a^{n_1}b{a}^{n_2}b\dots a^{n_k}b)$ where $k\ge 0$ and $n_i\ge 0$ for all $i$.
2. Words in $L_6$ are repetitions of $abba$.
3. The possible prefixes from $L_5$ that could start a word in $L_6$ are:

• $ϵ$ (empty string)
• $ab$
• $abb$

Now, let’s consider what $w$ would be in each case:

• If $v=ϵ$, then $w$ can be any word in $L_6$, i.e., $(abba{)}^{\ast }$
• If $v=ab$, then $w$ must start with $ba$ and then continue with $(abba{)}^{\ast }$, i.e., $ba(abba{)}^{\ast }$
• If $v=abb$, then $w$ must start with $a$ and then continue with $(abba{)}^{\ast }$, i.e., $a(abba{)}^{\ast }$

Therefore, $L_5◃L_6$ can be expressed as the union of these possibilities:

$L_5◃L_6=(abba{)}^{\ast }\cup ba(abba{)}^{\ast }\cup a(abba{)}^{\ast }$

This can be written more compactly as:

$L_5◃L_6=(ϵ+ba+a)(abba{)}^{\ast }$

or

$L_5◃L_6=(ba\cup a)(abba{)}^{\ast }$

This regular expression captures all possible suffixes that, when concatenated with a word from $L_5$, result in a word from $L_6$.

Question 3

To construct a non-deterministic finite automaton (NFA) that accepts $\mathbf{L}({\mathbf{A}}_1)◃\mathbf{L}({\mathbf{A}}_2)$, we can follow these steps:

a) Start with the structure of ${\mathbf{A}}_1$.

b) Add $ϵ$-transitions from each state of ${\mathbf{A}}_1$ to the initial state of ${\mathbf{A}}_2$.

c) Make all final states of ${\mathbf{A}}_1$ non-final.

d) Keep the final states of ${\mathbf{A}}_2$ as final.

Formally, we can define the NFA $\mathbf{A}=(Q,\Sigma ,\delta ,q_{1,0},F)$ where:

• $Q=Q_1\cup Q_2$
• $\Sigma$ is the same as before
• $\delta (q,a)={\delta }_1(q,a)$ for $q\in Q_1,a\in \Sigma$
• $\delta (q,a)={\delta }_2(q,a)$ for $q\in Q_2,a\in \Sigma$
• $\delta (q,ϵ)=\{q|q\in Q_2\}$ for $q\in Q_1$
• $q_{1,0}$ is the initial state of ${\mathbf{A}}_1$
• $F=F_2$

This NFA simulates ${\mathbf{A}}_1$ until it non-deterministically decides to switch to ${\mathbf{A}}_2$ using an $ϵ$-transition, then continues in ${\mathbf{A}}_2$ until it reaches a final state of ${\mathbf{A}}_2$.

Question 4

The statement “For every context-free language $L$ and regular language $R$, $L◃R$ is a regular language” is false.

Counterexample:

Let $L=\{a^m{b}^n{c}^n\mid m,n\ge 0\}$ (a context-free language that is not regular)

Let $R=a^{\ast }{b}^{\ast }{c}^{\ast }$ (a regular language)

Then $L◃R=\{b^n{c}^n\mid n\ge 0\}$, which is not a regular language.

This counterexample shows that the statement is false in general.

知识点

正则语言上下文无关语言NFA正则表达式

难点思路

问题 3 和 4 可能对考生来说比较困难。

对于问题 3，关键是理解如何通过 NFA 的非确定性来模拟 $◃$ 操作。我们使用 $ϵ$-转移来允许在 ${\mathbf{A}}_1$ 的任何状态 ” 猜测 ” 是否应该开始匹配 ${\mathbf{A}}_2$ 的部分。

对于问题 4，难点在于构造合适的反例。需要选择一个非正则的上下文无关语言，并找到一个合适的正则语言，使得它们的 $◃$ 操作结果仍然是非正则的。

解题技巧和信息

1. 对于涉及新定义操作的题目，先仔细理解定义，然后尝试用具体的例子来理解操作的含义。
2. 在处理正则表达式时，考虑语言中单词的结构和可能的前缀。
3. 构造 NFA 时，利用非确定性和 $ϵ$-转移来模拟复杂的操作。
4. 在证明语言类别相关的陈述时，寻找反例通常比证明正确性更容易。尝试使用经典的非正则上下文无关语言作为起点。

重点词汇

• finite automaton 有限自动机
• non-deterministic finite automaton (NFA) 非确定性有限自动机
• regular expression 正则表达式
• context-free language 上下文无关语言
• $ϵ$-transition $ϵ$-转移
• counterexample 反例

参考资料

1. Introduction to Automata Theory, Languages, and Computation by John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman. Chapter 2 (Finite Automata) and Chapter 4 (Context-Free Grammars).
2. Sipser, M. (2012). Introduction to the Theory of Computation. Cengage Learning. Chapter 1 (Regular Languages) and Chapter 2 (Context-Free Languages).