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I have a hard time creating a context-free grammar for the language

$$L=\{(a^n)^*(b^m)^*y\mid y\in\{a,b\}^*,\ |y|=n+m \}\,.$$

Any help breaking down the problem is appreciated, since I'm getting nowhere with this one...

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    $\begingroup$ Your definition of the language was a little unclear to me. I've tried to write it in more easily parsed notation: could you check that I've put the brackets in the right place? $\endgroup$ – David Richerby Apr 2 '17 at 14:39
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    $\begingroup$ For April fool's this language is a little late. Since any star contains $\varepsilon$ all I can make of this is $L = \{a,b\}^*$. $\endgroup$ – Hendrik Jan Apr 2 '17 at 16:40
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I'm not sure to have fully understood your notation.

As production rules you could consider:

$S \rightarrow aSa \ |aSb \ | S_1 | \epsilon$

$S_1 \rightarrow bS_1a | \ bS_1b \ | \ \epsilon$

In $S$ for each $a$ you add an $a$ or a $b$ or you can go to $S_1$ and then never come back to $S$ again.

In $S_1$ for each $b$ you add an $a$ or a $b$.

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  • $\begingroup$ Ok, so I've totally misunderstood the language that OP specified I guess. $\endgroup$ – abc Apr 2 '17 at 14:43
  • $\begingroup$ Well, it wasn't specified very clearly at all. Maybe I misunderstood it! Let's see if the asker clarifies. $\endgroup$ – David Richerby Apr 2 '17 at 15:04
  • $\begingroup$ Most probably your answer is OK. The original formulation may have used * for concatenation. $\endgroup$ – Hendrik Jan Apr 3 '17 at 16:32

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