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I've been watching a lot of videos about how to prove this and I am getting mixed information about how to do so.

In the case of contradiction, I am getting two different approaches and I want to know which is correct.

Example:

L = {a^n b^n+1 c^n+1: n>=0}

One source says I can Suppose that the language is context free, choose the sub-strings for u,v,x,y while maintaining that three constraints, that being:

  1. uv^ixy^iz is in A for all i>=0
  2. |vy| > 0
  3. |vxy|<= p

and then show that the string that is pumped is not in the language, even while maintaining the constraints of a CFL. If the string pumped is not in the language, that how I can show the contradiction.

Another Source says there are certain rules that must be followed when choosing vxy, that is vxy must straddle the midpoint... if you divided the string into 5 parts and a constraint isn't met, then that alone is proving that the language is not a CFL.

Basically, which approach is right? what are the rules to dividing up u,v,x,y?

msc
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  • You're going to have a tough time proving that language is context-free, since it's not. – user2357112 Mar 28 '18 at 01:37
  • I meant proving it's not....which approach is correct? – msc Mar 28 '18 at 01:38
  • https://en.m.wikipedia.org/wiki/Pumping_lemma_for_context-free_languages – Millie Smith Mar 28 '18 at 01:52
  • that just tells me definition and proof, not how to divide up u,v,x,y which was my question at the end – msc Mar 28 '18 at 02:00
  • @msc Look at the formal definition. It holds the answer. Putting the upside down e into words, "There exists a u,v,w,x,y such that ....". You have to find it or prove that such a division of the string s does not exist. And the rules for u,v,w,x,y are defined in the formal proof. – Millie Smith Mar 28 '18 at 07:01
  • @MillieSmith so it doesn't matter how uvwxy is split up...it just has to satisfy the conditions? now rules about vxy straddling some midpoint of the string or anything? – msc Mar 28 '18 at 16:13
  • Yes, it just has to satisfy the conditions described in the formal definition. There is no rule there about vxy straddling the middle. Maybe they were referring to the fact that there's at least some repetition for sufficiently large p in a context free grammar. – Millie Smith Mar 28 '18 at 16:24

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