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I am currently learning the pumping lemma, and encountered the following question, which I am unable to solve:

Prove that $L = \{ 0^n \mid \text{$n$ is power of 2}\}$ is not regular.

I considered $w = 0^{2^n}$, where $n$ is the pumping lemma constant. Then I divided $w$ into $xyz$, where $y \ne \epsilon$ and $|xy| \leq n$. Hence, $|y|$ will be between 1 and $n$. So, $|xy^kz|$ satisfies $L$ if $|y| = 2$ for all values of $k$ and it is within the bound. So, how is $L$ irregular?

The question is to prove it is irregular but here it is coming as regular.

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    $\begingroup$ The pumping lemma requires $xy^kz$ to be in the language for all values of $k$, not just for some value. $\endgroup$ – rici Aug 21 '19 at 1:47
  • $\begingroup$ @rici so, if |y| = 2, all values of k will be valid $\endgroup$ – Shantanu Shinde Aug 21 '19 at 2:14
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    $\begingroup$ All multiples of 2 are not powers of 2. $\endgroup$ – rici Aug 21 '19 at 4:07
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    $\begingroup$ Though it's not the case here, some irregular languages satisfy the pumping lemma. $\endgroup$ – Yuval Filmus Aug 21 '19 at 7:45
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If you choose $k = 2$ then, writing $|y| = m$, we get $$ xy^2z = 0^{2^n+m}. $$ Since $1 \leq m \leq n$, we have $$ 2^n < 2^n+m \leq 2^n+n. $$ Cantor's theorem shows that $n < 2^n$, and in particular, $$ 2^n < 2^n+m < 2^{n+1}. $$ Therefore $xy^2z \notin L$.


Let me take this opportunity to mention another misconception. There are some non-regular languages which satisfy the pumping lemma. Therefore the fact that you couldn't prove that a certain language is non-regular using the pumping lemma does not imply that the language is regular.

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