Language generated by $S \to aAb|Sb$, $A \to aAb|ab$

Question

Let $G = (\{A,S\}, \{a,b\}, S, P\}$ be the grammar with the following productions: \begin{align} & S \to aAb | Sb \\ & A \to aAb | ab \end{align}

1. What is the language $L(G)$ generated by the grammar?
2. Prove your answer to 1. by induction.

Here is my attempt. I think that the language generated by $G$ is $$L= \{a^nb^{n+k} \mid n\geq2,k\geq0. \} $$ I need to prove that $L = L(G)$ by induction on length, but I don't know how to do it properly because I have two parameters ($n,k$). There also need to be two directions to the proof.

I found this post, but it is of no help to me.

What is the base case? I tried looking about distinguishing whether the length is even or odd, and I think I got the pattern between $(n,k)$ correctly, but I really don't have a clue how to start the proof.

Answer 1

Start by showing that $L(A)$ (the language generated by the grammar if the starting symbol is $A$) is $L_A = \{ a^n b^n \mid n \geq 1 \}$. You prove this by double inclusion:

• In order to prove that $L_A \subseteq L(A)$, prove by induction on $n$ that $A \Rightarrow^* a^nb^n$.
• In order to prove that $L(A) \subseteq L_A$, prove by induction on the length of the derivation that if $A \Rightarrow^* \alpha$ then either $\alpha = a^n b^n$ for some $n \ge 1$, or $\alpha = a^n A b^n$ for some $n \geq 0$. (In fact we won't need this direction, but it's good practice.)

Now we can show that $L = L(G)$, again by double inclusion:

• In order to prove that $L \subseteq L(A)$, prove by induction on $k$ that $S \Rightarrow^* aAb^{k+1}$.
• In order to prove that $L(A) \subseteq L$, prove by induction on the length of the derivation that if $S \Rightarrow^* \alpha$ then either $\alpha = S b^k$ for some $k \geq 0$, or $\alpha = a^{n+1}Ab^{n+k+1}$ for some $n,k \geq 0$, or $\alpha = a^{n+2} b^{n+k+2}$ for some $n,k \geq 0$.