IS CS-2021S-02

题目来源：Problem 2 日期：2024-07-12 题目主题：CS-Formal Languages-Language Substitution

具体题目

Let $\Sigma$ be the set $\{a,b\}$ of letters. For a word $w\in {\Sigma }^{\ast }$ and two languages $L_a,L_b\subseteq {\Sigma }^{\ast }$ over $\Sigma$, we define the language $w\{a\mapsto L_a,b\mapsto L_b\}\subseteq {\Sigma }^{\ast }$ as follows, by induction on $w$.

$$ϵ\{a\mapsto L_a,b\mapsto L_b\}=\{ϵ\}$$ $$(aw)\{a\mapsto L_a,b\mapsto L_b\}=\{w_1{w}_2\mid w_1\in L_a,w_2\in w\{a\mapsto L_a,b\mapsto L_b\}\}$$ $$(bw)\{a\mapsto L_a,b\mapsto L_b\}=\{w_1{w}_2\mid w_1\in L_b,w_2\in w\{a\mapsto L_a,b\mapsto L_b\}\}$$

Here, $ϵ$ represents the empty word. For example, if $w=aba$, $L_a=\{b^n\mid n\ge 0\}$, and $L_b=\{a^n\mid n\ge 0\}$, then $w\{a\mapsto L_a,b\mapsto L_b\}=\{b^l{a}^m{b}^n\mid l,m,n\ge 0\}$. Furthermore, for languages $L,L_a,L_b\subseteq {\Sigma }^{\ast }$, we define $L\{a\mapsto L_a,b\mapsto L_b\}$ as ${\bigcup }_{w\in L}w\{a\mapsto L_a,b\mapsto L_b\}$. For example, if $L=\{a^{n}b\mid n\ge 0\}$, $L_a=\{ab\}$, and $L_b=\{a^n\mid n\ge 0\}$, then $L\{a\mapsto L_a,b\mapsto L_b\}=\{(ab{)}^m{a}^n\mid m,n\ge 0\}$.

Answer the following questions.

1. Let $L=\{(ab{)}^m{a}^n\mid m,n\ge 0\}$, $L_a=\{bb\}$, and $L_b=\{ab,a\}$. Express $L\{a\mapsto L_a,b\mapsto L_b\}$ using a regular expression.
2. Let $L^{\prime }=\{a^m{b}^n\mid m\ge n\ge 0\}$, $L_a^{\prime }=\{a^n\mid n\ge 0\}$, and $L_b^{\prime }=\{a^m{b}^m\mid m\ge 0\}$. Express $\{w\in {\Sigma }^{\ast }\mid w\{a\mapsto L_a^{\prime },b\mapsto L_b^{\prime }\}\subseteq L^{\prime }\}$ using a regular expression.
3. Let $A_0=(Q_0,\Sigma ,{\delta }_0,q_0,F_0)$, $A_1=(Q_1,\Sigma ,{\delta }_1,q_1,F_1)$, and $A_2=(Q_2,\Sigma ,{\delta }_2,q_2,F_2)$ be deterministic finite automata, and for each $i\in \{0,1,2\}$, let $L_i$ be the language accepted by $A_i$. Here, $Q_i,{\delta }_i,q_{i,0},F_i$ are the set of states, the transition function, the initial state, and the set of final states of $A_i$ ($i\in \{0,1,2\}$), respectively. Assume that the transition functions ${\delta }_i\in Q_i\times \Sigma \to Q_i$ ($i\in \{0,1,2\}$) are total. Give a non-deterministic finite automaton that accepts $L_0\{a\mapsto L_1,b\mapsto L_2\}$, with a brief explanation. You may use $ϵ$-transitions.
4. For $A_i$ and $L_i$ ($i\in \{0,1,2\}$) in question (3), give a deterministic finite automaton that accepts $\{w\in {\Sigma }^{\ast }\mid w\{a\mapsto L_i,b\mapsto L_j\}\subseteq L_k\}$, with a brief explanation.

正确解答

1. Regular expression for $L\{a\mapsto L_a,b\mapsto L_b\}$

Given $L=\{(ab{)}^m{a}^n\mid m,n\ge 0\}$, $L_a=\{bb\}$, and $L_b=\{ab,a\}$:

$$L\{a\mapsto L_a,b\mapsto L_b\}=(bb(ab+a){)}^{\ast }(bb{)}^{\ast }$$

This expression represents the language where every $a$ in the original language is replaced by $bb$ and every $b$ is replaced by either $ab$ or $a$.

2. Regular expression for $\{w\in {\Sigma }^{\ast }\mid w\{a\mapsto L_a^{\prime },b\mapsto L_b^{\prime }\}\subseteq L^{\prime }\}$

Given $L^{\prime }=\{a^m{b}^n\mid m\ge n\ge 0\}$, $L_a^{\prime }=\{a^n\mid n\ge 0\}$, and $L_b^{\prime }=\{a^m{b}^m\mid m\ge 0\}$, for $w\in {\Sigma }^{\ast }$, suppose $w^{\prime }=w\{a\mapsto L_a^{\prime },b\mapsto L_b^{\prime }\}$.

Since $w^{\prime }\subseteq L^{\prime }$, we can express any element of $w^{\prime }$, $w_i^{\prime }$ as $a^x{a}^y{b}^y$ where $x,y\ge 0$. This implies that $w^{\prime }$ contains $a^x$ followed by $a^y{b}^y$ for some $x,y\ge 0$.

Since all of the $b$ in $w_i^{\prime }$ can only come from $b$ in $w$, we can reverse the substitution process to get $w=a^x{b}^y$, where $y$ can only be $0$ or $1$.

Therefore, the regular expression for $\{w\in {\Sigma }^{\ast }\mid w\{a\mapsto L_a^{\prime },b\mapsto L_b^{\prime }\}\subseteq L^{\prime }\}$ is:

$$a^{\ast }(ϵ+b)$$

3. NFA for $L_0\{a\mapsto L_1,b\mapsto L_2\}$

We will construct an NFA that accepts $L_0\{a\mapsto L_1,b\mapsto L_2\}$ using $ϵ$-transitions. The NFA will have the same structure as $A_0$, but the transitions will be replaced based on the input letter with the transitions from $A_1$ and $A_2$, and $ϵ$-transitions will be used to connect the states.

For example, supposing the original transitions for the input letter $a$ in $A_0$ are $q_{0,0}\stackrel{a}{\to }{q}_{0,1}$, we will replace these transitions with the corresponding transitions from $A_1$:

$$q_{0,0}\stackrel{ϵ}{\to }{q}_{1,0}\to \dots \to F_{1,i}\stackrel{ϵ}{\to }{q}_{0,1}$$

Similarly, for the input letter $b$, we will replace the transitions with the corresponding transitions from $A_2$.

The final states of the NFA will be those states where the original final states of $A_0$ are reached after the substitution process.

Explanation: Since the language $L_0\{a\mapsto L_1,b\mapsto L_2\}$ is obtained by substituting the strings in $L_1$ and $L_2$ for $a$ and $b$ in the strings of $L_0$, the NFA needs to simulate this substitution process by transitioning to the corresponding states in $A_1$ and $A_2$ based on the input letter.

4. DFA for $\{w\in {\Sigma }^{\ast }\mid w\{a\mapsto L_i,b\mapsto L_j\}\subseteq L_k\}$

To construct a DFA for $\{w\in {\Sigma }^{\ast }\mid w\{a\mapsto L_i,b\mapsto L_j\}\subseteq L_k\}$:

Explanation:

• We need to track the states of $A_i$, $A_j$, and $A_k$ simultaneously.
• The DFA will have states $(q_i,q_j,q_k)$, where $q_i\in Q_i$, $q_j\in Q_j$, and $q_k\in Q_k$.
• The initial state is $(q_{i0},q_{j0},q_{k0})$.
• The transition function will be defined as:
  • $(q_i,q_j,q_k)\stackrel{a}{\to }({\delta }_i(q_i,a),q_j,{\delta }_k(q_k,w_{L_i}))$ for all $w_{L_i}\in L_i$.
  • $(q_i,q_j,q_k)\stackrel{b}{\to }(q_i,{\delta }_j(q_j,b),{\delta }_k(q_k,w_{L_j}))$ for all $w_{L_j}\in L_j$.
• The final states are those where the third component is a final state in $F_k$.

This DFA ensures that as we read $w$, we keep track of the corresponding states in $A_i$, $A_j$, and $A_k$ to ensure the substitution process results in strings that belong to $L_k$.

知识点

语言替换正则表达式NFADFA

难点解题思路

1. 语言替换的正则表达式表达形式，需要理解替换过程以及结果语言的模式。
2. 识别满足特定替换条件的字符串集合，需要考虑原语言和替换后的语言的关系。
3. 构造非确定性有限自动机 (NFA) 以处理语言替换，需要使用 $ϵ$-transitions 连接不同自动机的状态。
4. 构造确定性有限自动机 (DFA) 来接受满足条件的字符串集合，需要同时跟踪多个自动机的状态。

解题技巧和信息

• 在处理语言替换问题时，理解每一步替换过程对最终结果的影响非常重要。
• 构造 NFA 和 DFA 时，注意状态之间的转换关系以及如何利用 $ϵ$-transitions 连接不同语言的自动机。

重点词汇

• $ϵ$-transitions: $ϵ$-转换
• Regular Expression: 正则表达式
• Non-deterministic Finite Automaton (NFA): 非确定性有限自动机
• Deterministic Finite Automaton (DFA): 确定性有限自动机

参考资料

1. Michael Sipser, “Introduction to the Theory of Computation”, Chapter 2
2. John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman, “Introduction to Automata Theory, Languages, and Computation”, Chapter 3