IS CS-2023S-01

题目来源：Problem 1 日期：2024-08-17 题目主题：CS-Formal Languages-Operations on Languages

解题思路

本题考察了对给定语言进行操作后生成新语言的过程，涉及正则表达式、上下文无关文法（CFG），以及有限状态自动机（DFA）的构造等内容。需要运用语言和自动机理论中的概念来回答问题，并根据给定语言的性质来推导出新的语言表示。

Solution

Question 1: Regular Expression for $\Gamma (L_2)$

Let $L_2=\{(ab{)}^n\mid n\ge 0\}$. The language $L_2$ consists of strings of even length formed by repeating the substring “ab”.

Solution: To find $\Gamma (L_2)$, consider what $\Gamma (L)$ does. For each string $v\in \Gamma (L_2)$, there exists a string $w$ of the same length such that $vw\in L_2$.

Given that $L_2$ consists of strings of the form $(ab{)}^n$, the possible strings $v$ must have lengths that can be paired with some $w$ such that their concatenation results in a string from $L_2$.

The key observation is that $v$ can be any prefix of strings from $L_2$. This gives us the possible regular expression:

$$\Gamma (L_2)=(a+b{)}^{\ast }$$

This is because for any string $v$ formed from letters $a$ and $b$, there is a corresponding $w$ such that $vw\in L_2$ as long as the length condition is satisfied.

Question 2: Context-Free Grammar for $\Gamma (L_3)$

Let $L_3=\{a^n{b}^n{a}^m{b}^m\mid n\ge 0,m\ge 0\}$. This language consists of strings where the numbers of $a$s and $b$s are matched in pairs.

Solution: To generate $\Gamma (L_3)$, consider the strings that can form the beginning of valid strings in $L_3$. Specifically, $\Gamma (L_3)$ includes any prefix of $L_3$ that can be completed to a string in $L_3$. Therefore, $\Gamma (L_3)$ includes strings of the form $a^p{b}^q{a}^r{b}^s$ where $0\le p\le n$, $0\le q\le n$, $0\le r\le m$, and $0\le s\le m$.

A context-free grammar $G=(V,\Sigma ,R,S)$ generating $\Gamma (L_3)$ can be defined as follows:

• Variables: $S,A,B,C,D$

• Alphabet: $\Sigma =\{a,b\}$

• Rules:

  • $S\to AB\mid ϵ$
  • $A\to aA\mid B$
  • $B\to bB\mid C$
  • $C\to aC\mid D$
  • $D\to bD\mid ϵ$

• Start Symbol: $S$

This grammar generates strings in the form of prefixes of the strings in $L_3$.

Question 3: DFA for $\Gamma (L_{\mathcal{M}})$

Let $\mathcal{M}=(Q,\Sigma ,\delta ,q_0,F)$ be a deterministic finite automaton (DFA) that accepts a language $L_{\mathcal{M}}$. We need to construct a DFA ${\mathcal{M}}^{\prime }$ that accepts $\Gamma (L_{\mathcal{M}})$.

Solution: To construct ${\mathcal{M}}^{\prime }$, consider that $\Gamma (L_{\mathcal{M}})$ consists of strings $v$ such that there exists some $w$ with $|v|=|w|$ and $vw\in L_{\mathcal{M}}$.

• States: The states of ${\mathcal{M}}^{\prime }$ will be pairs of states from $Q\times Q$, representing the DFA being in a state corresponding to the prefixes $v$ and $w$ of equal length.
• Transitions: For each $(p,q)\in Q\times Q$ and for each letter $a\in \Sigma$, if $\delta (p,a)=p^{\prime }$ and $\delta (q,a)=q^{\prime }$, then $(p,q)\stackrel{a}{\to }(p^{\prime },q^{\prime })$.
• Start State: The start state is $(q_0,q_0)$.
• Final States: A pair $(p,q)$ is a final state if there exists some $p^{\prime }\in Q$ and $q^{\prime }\in F$ such that $(p,q)$ transitions to $(p^{\prime },q^{\prime })$ by some string.

This DFA ${\mathcal{M}}^{\prime }$ accepts exactly the strings in $\Gamma (L_{\mathcal{M}})$ by simulating the original DFA on both halves of the string in parallel.

Question 4: Context-Free Language Proposition

Proposition: “For every context-free language $L\subseteq {\Sigma }^{\ast }$, $\Gamma (L)$ is a context-free language.”

Solution: The proposition is false. A counterexample can be constructed as follows:

Consider the context-free language $L=\{a^n{b}^n{c}^n\mid n\ge 1\}$, which is known to be context-free. However, $\Gamma (L)$ would include strings like $a^p{b}^q{c}^r$ where $p,q,r$ do not necessarily follow the strict condition $p=q=r$. The language $\Gamma (L)$ can be shown to be non-context-free because it requires balancing different parts of the string in a way that a context-free grammar cannot generally handle.

This counterexample demonstrates that $\Gamma (L)$ is not necessarily context-free even if $L$ is.

知识点

RegularExpressionContextFreeGrammarDeterministicFiniteAutomatonContextFreeLanguage

解题技巧和信息

1. Constructing Regular Expressions: Understanding the structure of the language is crucial to forming the correct regular expression.
2. Designing CFGs: Consider the form of prefixes when designing CFGs for $\Gamma (L)$.
3. DFA Construction: For automaton-based constructions, consider pairs of states to account for parallel processing of string halves.
4. Non-context-free Languages: Remember that some operations can lead to languages that are not context-free even if the original language is.

重点词汇

• Prefix: 前缀
• Context-free grammar (CFG): 上下文无关文法
• Deterministic finite automaton (DFA): 确定性有限自动机
• Regular expression: 正则表达式

参考资料

1. Hopcroft, J.E., Motwani, R., Ullman, J.D. (2006). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. Chapter 2, 5.
2. Sipser, M. (2012). Introduction to the Theory of Computation. Cengage Learning. Chapter 2, 4.