[Solved]: Using the pumping lemma to prove that a language is context-free

Problem Detail: 

I am new to automata theory.

Could you give me a little hand with the correct use of the pumping lemma? I understand now how to proof a language is not context-free, but how do I use the pumping lemma to proof it is?

Do I find one way to break the word into: uvwxy so that v and x can be pumped? Is that enough and can I pick it in such a way that u or w or y are empty. I understand from the lemma that as long as v or x are not empty it is good. But can only w be empty and the rest be at least one character?

I did read on this topic, but I am just trying to make sure that what I understood is correct and there aren't other not so obvious things I should consider. Please help.

Asked By : Justina

Answered By : Renato Sanhueza

I read How to prove that a language is not context-free? and it has a very detailed example of a demonstration using the pumping lemma. But you are looking for some general understanding of the method so I will tell you the steps I follow when I use pumping lemma to prove that $L$ is not CFL:

1) You suppose that $L$ is CFL. The idea is find a contradiction using the pumping lemma.

2) By pumping lemma exists a constant $N>0$ associated to $L$. Use a generic constant because if you use a fixed one like $N=1$ for example, and don't find a contradiction you have to try with $N=2,3,4,...$.

3) You have to choose a word $\sigma \in L$ such that $|\sigma|\geq N$. You have to use your creativity to choose a word that allows you to easily reach a contradiction in the next steps.

4) By the pumping lemma exists a factorization of your word $\sigma=uvwxy$ and the following always complies: $$|vwx|\leq N$$ $$|vx|\geq 1$$ $$\sigma_i=uv^iwx^iy \in L \forall i\in\mathbb{N}_0$$

5) So the next step is to prove that the factorization does not exists. Your job if to check that all possible factorizations in the form $uvwxy$ don't complies with the 3 points of the pumping lemma. This generate a contradiction in your initial assumption then $L$ can't be CFL.

6) If you find a factorization of your particular word this mean nothing! Remember that the pumping lemma says:

If $L$ is CFL $\Rightarrow$ exist a factorization$\dots$

So you have two options at this point. The first is suspect that maybe your language is CFL and try to write a context-free grammar or draw a push-down automaton that denotes $L$. In second option you can assume that the word $\sigma$ was bad pick and try with another word to find a contradiction.

This is the general mechanics of the pumping lemma. I recommend you to try to understand the theory behind it or you will probably make mistakes in your test.

You can't use the pumping lemma to show that a language $L$ is context-free.. If you want to prove that $L$ is CFL the standard way is to write a CFG or draw a PDA like I said before.

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