IS CS-2022S-03

题目来源：Problem 3 日期：2024-07-01 题目主题：CS-Formal Languages-String Pattern Matching and Finite Automata

解题思路

这道题目涉及到形式语言理论中的几个重要概念：正则语言、有限自动机、上下文无关语言和下推自动机。

在本题中，我们处理字母表 ${\Sigma }_1=\{a,b\}$ 和 ${\Sigma }_2=\{t,f\}$ 上的字符串及其转换。首先，我们要定义如何将一个字符串 $w\in {\Sigma }_1^{\ast }$ 转换成另一个字符串 ${\Sigma }_2^{|w|}$，其中 $|w|$ 表示字符串的长度。这个转换基于在字符串 $w^{\prime }$ 中找到的子串 $w$ 的起始位置，并在这些位置替换成 $t$，其他位置替换成 $f$。

我们需要理解给定的函数 $f_w$ 和 $f_w^{\ast }$ 的定义，然后应用这些概念来解答问题。每个小问都建立在前面的基础上，逐步深入到更复杂的概念。第四问与第三问紧密相关，我们可以利用第三问的思路来解答第四问。

Solution

1. Compute $f_{aba}(babababa)$

Let’s analyze the string babababa and mark the positions where “aba” appears as a subword:

b a b a b a b a
  ^ ^ ^
      ^ ^ ^
          ^ ^ ^

Now, let’s replace these starting positions with ‘t’ and the rest with ‘f’:

$f_{aba}(babababa)=ftftftff$

2. Express $f_{ab}({\Sigma }_1^{\ast })$ by using a regular expression

To construct this regular expression, we need to consider all possible ways aba can appear in a string:

1. The string might start with aba: $(tf{)}^{\ast }$
2. There might be any number of a’s or b’s before ab: $(f^{\ast }tf{)}^{\ast }$
3. The string might end with any number of a’s or b’s: $f^{\ast }$

Combining these, we get:

$(f^{\ast }tf{)}^{\ast }{f}^{\ast }$

Therefore, $f_{ab}({\Sigma }_1^{\ast })=(f^{\ast }tf{)}^{\ast }{f}^{\ast }$

3. Non-deterministic finite automaton for $f_w^{\ast }(L)$

Let $A=(Q,{\Sigma }_1,\delta ,q_0,F)$ be the given DFA that accepts $L$. Let $w_n\in {\Sigma }_1$ be the $n$-th figure of word $w$. We can construct an NFA $A^{\prime }=(Q^{\prime },{\Sigma }_2,{\delta }^{\prime },q_0^{\prime },F^{\prime })$ that accepts $f_w^{\ast }(L)$ as follows:

1. $Q^{\prime }=Q\times \{0,1,\dots ,|w|\}$
2. $q_0^{\prime }=(q_0,0)$
3. $F^{\prime }=\{(q,i)\mid q\in F,0\le i\le |w|\}$
4. For the transition function ${\delta }^{\prime }$:
   • For each $(q,i)\in Q^{\prime }$ and $\lambda \in {\Sigma }_1$:
     • If $i<|w|$ and $\lambda =w_{i+1}$:
       • Add $ϵ$-transition from $(q,i)$ to $(\delta (q,\lambda ),i+1)$
     • If $i<|w|$ and $\lambda \ne w_{i+1}$:
       • Add $f$-transition from $(q,i)$ to $(\delta (q,\lambda ),0)$
     • If $i=|w|$:
       • Add $t$-transition from $(q,i)$ to $(\delta (q,\lambda ),0)$

This NFA simulates the DFA $A$ while keeping track of potential matches of $w$ during each of the transitions in $A$. When a complete match is found, it accepts ‘t’, otherwise ‘f’.

4. Prove or disprove: If $L$ is context-free, then $f_w^{\ast }(L)$ is context-free

This proposition is true. We can prove it by constructing a pushdown automaton (PDA) that accepts $f_w^{\ast }(L)$, using a similar approach to the NFA construction in Question 3.

Proof Sketch: Let $M=(Q,{\Sigma }_1,\Gamma ,\delta ,q_0,Z_0,F)$ be a PDA that accepts $L$. Let $w_n\in {\Sigma }_1$ be the $n$-th symbol of word $w$. We can construct a PDA $M^{\prime }=(Q^{\prime },{\Sigma }_2,\Gamma ,{\delta }^{\prime },q_0^{\prime },Z_0,F^{\prime })$ that accepts $f_w^{\ast }(L)$ as follows:

1. $Q^{\prime }=Q\times \{0,1,\dots ,|w|\}$
2. $q_0^{\prime }=(q_0,0)$
3. $F^{\prime }=\{(q,i)\mid q\in F,0\le i\le |w|\}$
4. For the transition function ${\delta }^{\prime }$:
   • For each $((q,i),\lambda ,\gamma )\in Q^{\prime }\times ({\Sigma }_2\cup \{ϵ\})\times \Gamma$:
     • If $i<|w|$ and $\lambda =w_{i+1}$:
       • For each $(p,\alpha )\in \delta (q,\lambda ,\gamma )$, add $((p,i+1),\alpha )$ to ${\delta }^{\prime }((q,i),ϵ,\gamma )$
     • If $i<|w|$ and $\lambda \ne w_{i+1}$:
       • For each $(p,\alpha )\in \delta (q,\lambda ,\gamma )$, add $((p,0),\alpha )$ to ${\delta }^{\prime }((q,i),f,\gamma )$
     • If $i=|w|$ and $a=t$:
       • For each $\lambda \in {\Sigma }_1$ and $(p,\alpha )\in \delta (q,\lambda ,\gamma )$, add $((p,0),\alpha )$ to ${\delta }^{\prime }((q,i),t,\gamma )$

Explanation:

• Similar to the NFA construction in Question 3, the state $(q,i)$ represents that we are in state $q$ of the original PDA and have matched $i$ symbols of $w$.
• The $ϵ$-transitions simulate the original PDA’s transitions for matching the next symbol of $w$.
• The $f$-transition occurs for positions where $w$ is not matched, resetting the match counter to 0.
• The $t$-transition occurs when we complete a match of $w$, also resetting the match counter to 0.

This PDA $M^{\prime }$ simulates the computations of $M$ while keeping track of occurrences of $w$ and outputting the corresponding string in ${\Sigma }_2^{\ast }$. Therefore, $f_w^{\ast }(L)$ is context-free.

Note: The key difference between this construction and the one in Question 3 is that we’re now working with a PDA instead of a DFA, which allows us to handle the stack operations necessary for context-free languages. However, the core idea of tracking partial matches of $w$ remains the same.

知识点

DFANFAPDA正则表达式上下文无关语言下推自动机

难点思路

1. 理解函数 $f_w$ 和 $f_w^{\ast }$ 的定义及其在字符串和语言上的作用。
2. 构造接受 $f_w^{\ast }(L)$ 的非确定性有限自动机，需要巧妙地结合原 DFA 和模式匹配。
3. 将 NFA 构造的思路扩展到 PDA，以证明 $f_w^{\ast }(L)$ 的上下文无关性。

解题技巧和信息

1. 在处理形式语言问题时，尝试从简单的例子开始，然后推广到一般情况。
2. 在构造自动机或文法时，考虑如何将问题的不同方面（如这里的 PDA 转换和模式匹配）结合起来。
3. 对于复杂的语言操作，考虑如何使用现有的形式语言工具（如 NFA、PDA）来模拟这些操作。
4. 注意识别问题之间的联系，如第三问和第四问之间的关系，可以帮助简化解题过程。
5. 在扩展 NFA 到 PDA 时，要注意保持状态转换的基本结构，同时增加对栈操作的处理。

重点词汇

• deterministic finite automaton (DFA) 确定性有限自动机
• non-deterministic finite automaton (NFA) 非确定性有限自动机
• pushdown automaton (PDA) 下推自动机
• context-free grammar (CFG) 上下文无关文法
• regular expression 正则表达式
• $ϵ$-transition $ϵ$-转换

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), Chapter 3 (Regular Expressions and Languages), Chapter 5 (Context-Free Grammars and Languages), Chapter 6 (Pushdown Automata)
2. Formal Languages and Automata Theory by C. K. Nagpal. Chapters on Regular Languages, Context-Free Languages, and Pushdown Automata