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Given the following language:

$L = \{ a^ib^j \mid \lvert i - j \rvert \le 2 \} $

I am trying to prove that it is not regular. On the one hand my intuition tells me that the language is non-regular as there is no way of tracking $a^{i's}$ and $b^{j's}$.

However, when I try to prove that it is irregular using the pumping lemma I have trouble finding which word I should use to arrive at a contradiction.

Any suggestions?

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  • $\begingroup$ You can take the word $a^pb^p$, where $p$ is the constant promised by the pumping lemma. $\endgroup$
    – Yuval Filmus
    May 2, 2021 at 18:08
  • $\begingroup$ @YuvalFilmus Thanks! I just proved it with your word :) $\endgroup$
    – jaime martinez
    May 2, 2021 at 19:07

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You can use the word $a^pb^p$, where $p$ is the constant promised by the pumping lemma.

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