[Solved]: Pumping lemma problem - Choosing the right string to pump

Problem Detail: 

I have a problem finding the right string to pump for the following language:

$$L_1 = \{a^{p+q}b^rc^sd^{q+r}e^s \mid p, q, r, s \ge 0\}$$

Which string should I choose to pump? The problem is that I don't know how to handle the fact that I have $p+q$ and $q+r$?

Can I choose: $$Z = a^{2n}b^nc^nd^{2n}e^s$$

Thanks in advance.

Asked By : mrjasmin

Answered By : A.Schulz

If $k$ is the pump length from the regular pumping lemma choose $w=c^ke^k$. Clearly, $w\in L_1$, but if you pump you have to change the number of $c$'s while leaving the $e$'s unaltered. Hence after pumping the word is no longer a member of $L_1$.

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/7751