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I have a question. Assume $L = \{ a^m b^m \mid m ≥ 1 \}$ is not a regular language. Prove that $I = \{ a^{5n} b^{3m} c^n d^m \mid m,n ≥ 0 \}$ is not a regular language.

I can prove it with pumping lemma but the question is to solve it by using the assumption.

Thanks.

Jonas Kölker
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user10578515
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1 Answers1

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Here's a hint: we know that $L = \{x^m y^m \mid m ≥ 1 \}$ is a non-regular language.

Is there some part of $I = \{ a^{5n} b^{3m} c^n d^m \mid m,n ≥ 0\}$ which looks similar to $x^m y^m$?

Could you somehow extract that part?

If you could, could you use the fact that $L$ is not regular to conclude that the extracted part is not regular?

Given this, can you then conclude that $I$ is not regular?

Jonas Kölker
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