# Is this language is Context-free language or not?

Is anybody can help me please to determine is this language is Context-free language or not?

L={wvw | w,v∈{a,b,c}+}

for example:

part of the language: acbac, abcab, bbcbb

not part of the language: abab, aa, abcc

I tried to prove using pumping lemma for context-free languages like that:

I take this word from language w=aacbbaa. I split the word to 5 parts uvxyz and if I can pump u and y and the word stays inside language the language is context-free:

u-aa, v-cc, x-b, y=bb, z=aa
after I pump v and y the word is still in language->
u-aa, v-cccc, x-b, y=bbbb, z=aa ->
aaccccbbbbbaa


But the answer inside answers section is that the language is not context-free.

What wrong with my prof?

• Same language here: "Is {xyx | |x|≥1} context-free?" cs.stackexchange.com/q/11629/4287, where one of the answers suggests the pumping lemma can be used to show the language is not context-free. Mar 8, 2019 at 22:48

## 1 Answer

You can't use the pumping lemma to prove that a language is context-free (see, e.g., here). The pumping lemma is only useful to prove that the language is not context-free. See our reference questions for details on how to prove that a language is or isn't context-free: How to prove that a language is context-free?, How to prove that a language is not context-free?.

So, one thing wrong with your proof is where you say "if... the word stays inside language the language is context-free" -- that's not correct.

• D.W. , thanks for post.If pumping lemma not works for any part(y and u) in word I can say that langauge is not context free? Mar 8, 2019 at 19:30
• @Michael, sorry, this site isn't designed for back-and-forth interactive assistance. I encourage you to study our reference material, and if you need more interactive help, perhaps you can find a friend, tutor, or teaching assistant to help you.
– D.W.
Mar 8, 2019 at 19:31
• D.W. only one question is this language is context free or not? Mar 8, 2019 at 20:30