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How can I show that $\{a^ib^jc^k|i=0 \lor j=k\}$ is not regular?

I tried applying the pumping lemma but it does seem to have a pumping length of 1?

Alternatively there is the Myhill–Nerode theorem. Are $\{a^ib^j|i=0 \lor j\in\mathbb{N}\}$ infinite many equivalence classes?

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  • $\begingroup$ it involves the language a^nb^n which is not regular and FM can't count unlimited number. $\endgroup$ – Mr. Sigma. Dec 3 '18 at 8:43
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Consider the intersection of your language with $ab^*c^*$.

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  • $\begingroup$ Then we would get the nonregular language $b^nc^n$ but which theorem can I use on the intersection? $\endgroup$ – Mark Regev Nov 17 '18 at 11:51
  • $\begingroup$ Closure under intersection. $\endgroup$ – Yuval Filmus Nov 17 '18 at 15:17
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Myhill-Nerode theorem: For every j, j' the strings $ab^j$ and $ab^{j'}$, $j ≠ j'$, can be distinguished - because one can be followed by $c^j$ and the other can't.

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You could try with Pumping Theory (P.T.) of Regular Languages, that is to assume the language is regular and pick ${n}$ in P.T. as ${n=j=k}$. Then follow the three rules in P.T. for a contradiction.

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