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I want to write a cfg for the following language:

$ L = {((ab)^n)^m }$ $m,n >= 0$

this language produces (abababababab) where:
$n=2, m=3 \\ or \\ n=3, m=2$

I have no idea what to do with it!

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No need for a CFG: this language is regular!

$$L = (ab)^*$$

Any number of $ab$s is in the language: just set $n$ to 1, and $m$ to the number of repetitions. So all you need is a Kleene star.

If you really want a CFG, you can do it like this:

$$S \rightarrow SX \mid \varepsilon$$ $$X \rightarrow ab$$

This is the "standard" way of translating a Kleene star into a CFG.

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  • $\begingroup$ Knowing this makes the grammar easier but the question about how to write a CFG for this regular (and, therefore, context-free) language remains. $\endgroup$ – David Richerby Mar 21 at 14:55
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    $\begingroup$ @DavidRicherby Added a translation $\endgroup$ – Draconis Mar 21 at 14:59

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