GATE CS 2001 Q1.4 — Regular and Context-free Languages

To determine the correctness of the statements, let us analyze each language individually:

1. Statement $S_1$: $L_1 = \left\{ 0^{2n} \mid n \geq 1 \right\}$

   This language contains strings of $0$s of even length (excluding the empty string): $L_1 = \{00, 0000, 000000, \dots\}$. We can easily represent this language using the regular expression (00)+. Since a regular expression exists, $L_1$ is a regular language. Thus, $S_1$ is correct.

2. Statement $S_2$: $L_2 = \left\{0^m1^n0^{m+n} \mid m \geq 1 \text{ and } n \geq 1 \right\}$

   In this language, the number of trailing $0$s must equal the sum of the leading $0$s and the middle $1$s. This requires matching and counting across non-adjacent blocks, which cannot be done with a finite automaton (finite memory). Using the pumping lemma, we can easily show that $L_2$ is not regular (it is a context-free language). Thus, $S_2$ is incorrect.

Therefore, only statement $S_1$ is correct.