Analysis of $L_1$ : CONTEXT-FREE
- $L_1=\left\{w \in\{a, b\}^* \mid w=u v\right.$ for some $u, v \in\{a, b\}^{+}$where $\left.n_a(u)=n_b(v)\right\}$
- This language is context-free. It can be recognized by a non-deterministic pushdown automaton (NPDA).
- How it works: An NPDA can non-deterministically "guess" where the string $u$ ends and $v$ begins.
1. While reading the part it assumes is $u$, it pushes a symbol onto the stack for every a it sees.
2. When it guesses the split point, it starts reading the part it assumes is $v$.
3. While reading $v$, it pops a symbol from the stack for every b it sees.
4. If the stack is empty at the end of the string, that particular path of computation accepts. Because a valid accepting path exists, the language is context-free.
Analysis of $L_2$ : CONTEXT-FREE
- $L_2=L_A \cap L_B$, where $L_A=\left\{x \# y \mid x, y \in\{0,1\}^{+}, x=y^R\right\}$ and $L_B$ is from the regex $0^* 1^* \# 1^* 0^*$.
- This language is context-free based on a crucial closure property.
1. $L_A$ is the language of marked palindromes, a classic example of a context-free language.
2. $\quad L_B$ is described by a regular expression, which by definition makes it a regular language.
3. Key Rule: The family of context-free languages is closed under intersection with regular languages.
- Since we are intersecting a CFL ( $L_A$ ) with a regular language ( $L_B$ ), the result ( $L_2$ ) must be context-free.
Analysis of L3:
The language is defined as the intersection of two other languages: $L_3=L_C \cap L_D$, where:
- $L_C=\left\{a^i b^i c^j \mid i, j \geq 1\right\}$
- $L_D=\left\{a^i b^j c^j \mid i, j \geq 1\right\}$
Analyze the Intersection ($L_C \cap L_D$ ):
The intersection of $L_C$ and $L_D$ is therefore the language $\left\{a^n b^n c^n \mid n \geq 1\right\}$.
The language $\left\{a^n b^n c^n \mid n \geq 1\right\}$ is the canonical example of a language that is not context-free. A PDA has only one stack, which allows it to compare two counts (like matching $a^n$ to $b^n$ ), but it does not have enough memory to verify a third independent count.
