GATE CS 2025 Set 1 Q34 — Regular and Context-free Languages

To determine the regularity of the two languages, we analyze them individually:

1. Analysis of $L_1$:

$$L_{1}=\left\{\alpha \beta \alpha \mid \alpha \in\{a, b\}^{+} \text{ and } \beta \in\{a, b\}^{+}\right\}$$

Let us intersect $L_1$ with the regular language $R = a b^+ a b^+$:

$$L_1 \cap R = \left\{ a b^n a b^m \mid n, m \ge 1 \text{ and } n \neq m \right\}$$

To see why $n \neq m$, if $n = m$, then any choice of prefix and suffix $\alpha$ starting with $a$ and ending with $b$ (i.e., $\alpha = a b^n$) forces $\beta$ to be empty, which is not allowed because $\beta \in \{a,b\}^+$. Since $\left\{ a b^n a b^m \mid n \neq m \right\}$ is a well-known non-regular language (provable by the pumping lemma) and regular languages are closed under intersection, $L_1$ is not regular.

2. Analysis of $L_2$:

$$L_{2}=\left\{\alpha \beta \alpha \mid \alpha \in\{a\}^{+} \text{ and } \beta \in\{a, b\}^{+}\right\}$$

Since $\alpha \in \{a\}^+$, we can write $\alpha = a^k$ for some $k \ge 1$. Any string in $L_2$ is of the form $a^k \beta a^k$ with $\beta \in \{a, b\}^+$. By choosing $k = 1$, we can express any such string as:

$$a \gamma a \quad \text{where } \gamma \in \{a, b\}^+$$

Thus, $L_2$ is simply the set of all strings over $\{a, b\}$ that start with $a$, end with $a$, and have a length of at least $3$. This can be represented by the regular expression:

$$a(a+b)^+a$$

Since it can be defined by a regular expression, $L_2$ is regular.