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For proving language $\{a^nb^mc^nd^m \mid n,m > 0\}$ is not context free. Do I have to use $z = a^pb^pc^pd^p$ as pumping lemma string where $p$ is pumping length?

Or do I have to use a string that would reflect than $n$ and $m$ does not have to be the same length?

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  • $\begingroup$ Note our reference question which describes the Pumping lemma and other methods in detail. Hint: If you read $a$ and $b$ as $($, and $c$ and $d$ as $)$, you have crossing balanced parentheses. That's general a no-go pattern of CFL. $\endgroup$ – Raphael Nov 20 '14 at 17:39
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Pumping lemma lets you construct new strings belonging to the context-free language.

To prove that the language is not context-free you can take some string $s$ from L, use pumping lemma to construct a new string $s1$ which must belong to $L$ (because you used pumping lemma to construct it!). Then you show that $s1$ does not belong to L => contradiction.

Choice of original $s$ is yours. So yes, you can use $z = a^pb^pc^pd^p$ in your proof.

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