Consider the following context-free grammar $G$. $$S \to abaABAbba$$ $$A \to aaBBAb \mid bBabaa$$ $$B \to aBb \mid ab$$ In the above grammar, $S$ is the start symbol, $a$ and $b$ are terminal symbols, and $A$ and $B$ are non-terminal symbols. Let $L(G)$ be the language generated by the grammar $G$. For a string $s \in L(G)$, let $n_1(s)$ be the number of $a$'s in $s$ and $n_2(s)$ be the number of $b$'s in $s$. Which of the following statements is/are true?

Let $n_a(w)$ and $n_b(w)$ denote the number of $a$'s and $b$'s in a string or sentential form $w$.

1. For non-terminal $B$:

   $$B \to aBb \mid ab$$

   Any terminal string derived from $B$ has the form $a^k b^k$ where $k \ge 1$. Therefore, for any string $w_B$ derived from $B$:

   $$n_a(w_B) = n_b(w_B) \ge 1$$

2. For non-terminal $A$:

   • The production $A \to aaBBAb$ introduces $2$ terminal $a$'s, $1$ terminal $b$, two $B$'s, and one $A$. The direct terminal contribution has $\Delta(n_a - n_b) = 2 - 1 = +1$, and the ratio of terminal $a$'s to $b$'s is $\frac{2}{1} = 2$.

   • The production $A \to bBabaa$ introduces $3$ terminal $a$'s, $2$ terminal $b$'s, and one $B$. The direct terminal contribution has $\Delta(n_a - n_b) = 3 - 2 = +1$, and the ratio of terminal $a$'s to $b$'s is $\frac{3}{2} = 1.5 \le 2$.

   Since $B$ always contributes equal numbers of $a$'s and $b$'s ($n_a(w_B) = n_b(w_B)$), any string $w_A$ derived from $A$ satisfies:

   $$n_a(w_A) > n_b(w_A) \quad \text{and} \quad n_a(w_A) \le 2 n_b(w_A)$$

3. For the start symbol $S$:

   $$S \to abaABAbba$$

   The terminal prefix and suffix contribute $3$ $a$'s and $3$ $b$'s ($n_a = 3, n_b = 3$). The body contains $2$ occurrences of $A$ and $1$ occurrence of $B$.

   Therefore, for every string $s \in L(G)$:

   $$n_1(s) = 3 + 2 n_a(w_A) + n_a(w_B)$$

   $$n_2(s) = 3 + 2 n_b(w_A) + n_b(w_B)$$

   • Since $n_a(w_A) > n_b(w_A)$ and $n_a(w_B) = n_b(w_B)$:

     $$n_1(s) > n_2(s) \implies n_1(s) \ge n_2(s)$$

     Hence, (B) is true and (A) is false.

   • Since $3 \le 2(3)$, $n_a(w_B) \le 2 n_b(w_B)$, and $n_a(w_A) \le 2 n_b(w_A)$ for all derived sub-strings:

     $$n_1(s) \le 2 n_2(s)$$

     Hence, (D) is true and (C) is false.

Thus, statements (B) and (D) are true.