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I need to prove that the following language is not regular

$\{c^mb^na^n \mid n>0,m\geq0\}$

But I am not sure how to do that for this particular one. I vaguely understand pumping lemma, but every example I see is in the form

$\{a^nb^n \mid n>0\}$

Or something similiar. This does not help me with my problem, because I dont fully understand pumping lemma and am not sure how to do it for two variables that have no dependency on each other. I know to take a length P from L, and I also need a string s>=P. Which then becomes P=xyz, but it is this step that confuses me.

tyuip
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1 Answers1

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There are many ways to solve this. Here are some:

  1. If $L = \{c^m b^n a^n : n>0, m\geq 0\}$ were regular then so would $L \cap b^*a^*$ be.

  2. The words $\{ b^n : n > 0 \}$ are pairwise inequivalent modulo $L$, so $L$ is irregular according to Myhill-Nerode.

  3. The pumping lemma works if you ignore the $c$ part.

Yuval Filmus
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