# Confused about decomposition in Context Free Pumping lemma

I am trying to decide whether the following language is context free:

$$L = \{ a^nb^{3n}c^n \, | \, n \geq 0 \}$$

Assume $$L$$ is context-free. Let $$p$$ be the pumping length given by the Pumping Lemma. Choose string $$s = a^pb^{3p}c^p ∈ L$$. Then, $$|s| ≥ p$$. Take an arbitrary decomposition $$s = uvxyz$$ where $$|vxy| \le p$$ and $$|vy| > 0$$.

Case 1:

i. $$v,y ∈ a^*$$

ii. $$v,y ∈ b^*$$

iii. $$v,y ∈ c^*$$

Here, uv0xy0z ∉ L, due to incorrect number of symbols.

Case 3: i. $$v ∈ aa* and y ∈ bb*$$ ii. $$v ∈ bb* and y ∈ cc*$$

Here, uv0xy0z ∉ L, since #a's ≠ #c's.

So, s cannot be pumped, which contradicts the Pumping Lemma. Hence, L is not context-free.

However, after checking my answer, it seems that I missed that I missed one case where:

$$v ∈ aa^*bb^* or \,\,y ∈ aa^*bb^*$$ $$v ∈ bb^*cc^* or \,\,y ∈ bb^*cc^*$$

Here, uv2xy2z ∉ L, due to incorrect ordering of symbols.

But this doesn't make sense to me; how can this mess up the ordering of the symbols because wouldn't it follow that for example X would be the symbols after v?

Thanks guys,

• Could someone fix my equations at the end also btw; I'm not sure why it didn't format correctly. Commented Sep 7, 2023 at 7:16
• Welcome to Computer Science! Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction.
– D.W.
Commented Sep 7, 2023 at 15:51
• Note that in math mode you need to use \{, rather than {. Also in math mode, use * rather than \*.
– D.W.
Commented Sep 7, 2023 at 15:52
• I edited the beginning of the question to help, but please edit the rest to make it readable. Commented Sep 20, 2023 at 19:17
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Commented Sep 20, 2023 at 19:17