Question 44

1. Analysis of $L_2$:
   • Every string in $L_2$ starts with $a$, ends with $a$, and has length at least $3$.
   • Conversely, if any string $w \in \{a, b\}^*$ starts with $a$, ends with $a$, and has $|w| \ge 3$, we can write $w = a \beta a$ by choosing $\alpha = a \in \{a\}^+$ and $\beta \in \{a, b\}^+$.
   • Therefore, $L_2 = a(a + b)^+a$, which is a regular language.
2. Analysis of $L_1$:
   • In $L_1$, $\alpha$ can be any string in $\{a, b\}^+$. Consider strings of the form $w = a^p b a a^p b$, where $p$ is the pumping length: the only way to decompose $w = \alpha \beta \alpha$ is $\alpha = a^p b$ and $\beta = a$, which requires matching the arbitrary count $p$ of $a$'s at both ends. By the pumping lemma, $L_1$ is not a regular language.

Hence $L_1$ is not regular, whereas $L_2$ is regular.