A student wrote two context-free grammars **G1** and **G2** for generating a single C-like array declaration. The dimension of the array is at least one. For example, int a[10][3]; The grammars use $D$ as the start symbol, and use six terminal symbols **int ; id [ ] num**. Grammar **G1** $D \rightarrow \mathbf{int} \; L;$ $L \rightarrow \mathbf{id} \; [E$ $E \rightarrow \mathbf{num}]$ $E \rightarrow \mathbf{num}] \; [E$ Grammar **G2** $D \rightarrow \mathbf{int} \; L;$ $L \rightarrow \mathbf{id} \; E$ $E \rightarrow E \; [\mathbf{num}]$ $E \rightarrow [\mathbf{num}]$ Which of the grammars correctly generate the declaration mentioned above?

Let's analyze both grammars to see if they generate declarations of the form $\mathbf{int} \; \mathbf{id} \; [\mathbf{num}] \dots [\mathbf{num}];$ for any dimension $\ge 1$:

1. Grammar G1:

   • Derivation starts with: $$D \Rightarrow \mathbf{int} \; L; \Rightarrow \mathbf{int} \; \mathbf{id} \; [E;$$

   • If dimension is 1: $$E \Rightarrow \mathbf{num}]$$ Yields: $\mathbf{int} \; \mathbf{id} \; [\mathbf{num}];$

   • If dimension is 2: $$E \Rightarrow \mathbf{num}] \; [E \Rightarrow \mathbf{num}] \; [\mathbf{num}]$$ Yields: $\mathbf{int} \; \mathbf{id} \; [\mathbf{num}][\mathbf{num}];$

   • In general, $E$ generates one or more repetitions of $\mathbf{num}]$ separated by $[$, correctly forming $\mathbf{id}[\mathbf{num}][\mathbf{num}]\dots[\mathbf{num}]$. Thus, G1 generates the required declarations.

2. Grammar G2:

   • Derivation starts with: $$D \Rightarrow \mathbf{int} \; L; \Rightarrow \mathbf{int} \; \mathbf{id} \; E;$$

   • $E$ is a left-recursive non-terminal producing one or more $[\mathbf{num}]$ pairs: $$E \Rightarrow^+ [\mathbf{num}][\mathbf{num}]\dots[\mathbf{num}]$$

   • Combining with $\mathbf{int} \; \mathbf{id}$, it yields $\mathbf{int} \; \mathbf{id}[\mathbf{num}][\mathbf{num}]\dots[\mathbf{num}];$.

   • Thus, G2 also correctly generates the required declarations.

Therefore, Both G1 and G2 correctly generate the declaration, making (A) the correct option.