How to show that the following language is not regular with the pumping lemma?

$$L=\left\{0^{n\lceil\log_2 n\rceil} \,\middle|\, n\in \mathbb{N}-\{0\}\right\}.$$

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    $\begingroup$ Possible duplicate of How to prove that a language is not regular? $\endgroup$ – David Richerby Nov 3 '18 at 10:06
  • $\begingroup$ This seems like homework, what did you try, and where are you stuck? $\endgroup$ – Pål GD Nov 4 '18 at 22:37
  • $\begingroup$ Did you consider accepting my answer? (This comment will be removed.) $\endgroup$ – John L. Apr 29 at 5:24

(Dense-length lemma for regular languages) Let $R$ be an regular language that has infinitely many words. Then there exists a number $d$ such that for any word in $R$, there is another word such that the difference of the lengths of both words is less than $d$.

The above lemma says, basically, the lengths of words in a regular language cannot be too sparse anywhere. It can be proved easily using the pumping lemma for regular languages.

Let us take a look at $L$. The lengths of its word are $\{n\lceil\log_2 n\rceil \,\mid n\in \Bbb{N}-\{0\}\}$. When $n$ goes larger and larger, the lengths become sparser and sparser because of the factor $\lceil\log_2n\rceil$. In fact, if $n>2^d$, $m>2^d$ and $n\not=m$, then $\left|n\lceil\log_2n\rceil-m\lceil\log_2m\rceil\right| > d$. So $L$ cannot be regular because of the above lemma.

Exercise 1. Prove the dense-length lemma for regular languages.

Exercise 2. Prove the dense-length lemma also holds for context-free languages that have infinitely many words.

Exercise 3. Does the dense-length lemma hold for context-sensitive languages?

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  • $\begingroup$ Hint for exercise 1, the pumping lemma for regular languages. $\endgroup$ – John L. Apr 29 at 5:27

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