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I need help solving this question:

Is $L = \{ w \in \{a,b,c\}^* \mid n_a(w) = n_b(w) = 2n_c(w)\}$ a context-free language? That is the number of $a$'s equal the number of $b$'s equal twice the number of $c$'s in the string $w$.

First, I think that the language is context-free, but i'm having trouble finding a context free grammar to prove it.

So far I have:

\begin{align} S &\to SASBS \mid SBSAS \mid λ \\ A &\to a \\ B &\to b \\ \end{align}

I'm not sure how to get twice the number of $c$'s, could someone please show me/correct me?

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1 Answer 1

Hint: The language is (unfortunately) not context free. Recall the Pumping Lemma for context free languages, and suppose that $p$ is the pumping length of $L$. Now consider the string $w = \mathtt{a}^{2p} \mathtt{b}^{2p} \mathtt{c}^p$. (Note that no substring of $w$ of length $\leq p$ can contain $\mathtt{a}$s, $\mathtt{b}$s and $\mathtt{c}$s.)

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