Question 42

1. Language $L_1 = \{a^m b^n a^n b^m \mid m, n \ge 1\}$:
   This exhibits nested (LIFO) matching:
   • Push $m$ copies of $a$ onto the stack.
   • Push $n$ copies of $b$ onto the stack.
   • Match and pop $b$'s with the subsequent $n$ copies of $a$.
   • Match and pop $a$'s with the final $m$ copies of $b$.
     A PDA can completely verify this using a single stack. Hence, $L_1$ is context-free.

2. Language $L_2 = \{a^m b^n a^m b^n \mid m, n \ge 1\}$:
   This has cross-serial dependencies (the first $a^m$ must match the second $a^m$, and the first $b^n$ must match the second $b^n$). By the Pumping Lemma for Context-Free Languages, $L_2$ cannot be accepted by any PDA. Hence, $L_2$ is not context-free.

3. Language $L_3 = \{a^m b^n \mid m = 2n + 1\}$:
   Strings are of the form $a^{2n+1} b^n$. For each $b$, there are exactly two $a$'s (plus one extra initial $a$). A PDA can push one symbol for every two $a$'s (or push all $a$'s and pop two for each $b$). Hence, $L_3$ is deterministic context-free.

Thus, $L_1$ and $L_3$ only are context-free (Option B).