# Is this a context free language? I need to make PDA but I don't think it is doable

I got a question: Design a pushdown automata that can recognize strings in L= {$$a^n b^{2n} c^{3n} | n ≥ 0$$} . I tried to think and design it, but I couldn't find it. The best that I can think of is L= {$$a^n b^{m} c^{m+n} | m,n ≥ 0$$} . I only can make sure that c appears as much as a+b appear. So, I'm wondering, is this language context free? Could we make a PDA for it? Thank you.

• Even $a^nb^nc^n, n \geq 0$ is not context-free (proof). You can create a proof that $a^nb^{2n}c^{3n}$ is not CF similarly. Jul 8, 2022 at 16:34

For $$L= \{a^n b^{2n} c^{3n} | n ≥ 0 \}$$ you will be failed to make PDA with one-stack. Because here is two comparisons, one $$a$$ should be checked against with $$2b$$ and $$3c,$$ but when you complete one $$a$$ checked against with $$2b$$ then stack will be empty, and there will be pending of $$c$$'s computation. This two comparison intuitively proved that your language is CSL accepted by LBA.
Consider this homomorphism: $$f(a)=a\ ,f(b)=bb,\ f(c)=ccc$$
What is $$f^{-1}(L)?$$