GATE CS 2022 Q37 — Context-free Grammars and Push-down Automata

To determine the nature of the languages $L_1$ and $L_2$, we analyze them individually:

Analysis of $L_1 = \{ a^n w a^n \mid w \in \{a,b\}^* \}$:

• If $n \ge 0$, choosing $n = 0$ gives $L_1 = \{a,b\}^*$, which is regular.
• If $n \ge 1$, any string in $L_1$ must start with $a$ and end with $a$. By choosing $n = 1$, we can write any such string as $a w a$ for some $w \in \{a,b\}^*$. Thus, $L_1 = a(a+b)^*a$, which is regular.

In either standard interpretation of $n$, $L_1$ is a regular language.

Analysis of $L_2 = \{ w x w^R \mid w, x \in \{a,b\}^*, |w|, |x| > 0 \}$:

• Since $|w| \ge 1$, the simplest choice for $w$ is a single character, i.e., $w = a$ or $w = b$.
• If $w = a$, then $w^R = a$, giving strings of the form $a x a$ where $|x| \ge 1$.
• If $w = b$, then $w^R = b$, giving strings of the form $b x b$ where $|x| \ge 1$.

Any string of length $\ge 3$ that starts and ends with the same character belongs to $L_2$. Thus, we can express $L_2$ as the regular expression: $$L_2 = a(a+b)^+a + b(a+b)^+b$$ Hence, $L_2$ is also a regular language.

Since every regular language is also a context-free language (CFL):

• A is TRUE: Both $L_1$ and $L_2$ are regular.
• B is TRUE: Both $L_1$ and $L_2$ are context-free.
• C is TRUE: $L_1$ is regular and $L_2$ is context-free (since $L_2$ is regular, it is also context-free).
• D is FALSE: They are regular.