Let the operation $$Perm(L) = \{ w | \exists u \in L \text{ such that } u \text{ is a permutation of } w \}$$

Prove that both regular languages and CFLs aren't closed under $Perm(L)$.

I've tried to use several well-known languages (like $\{0^n1^n\}$) and applying $Perm(L)$ and afterward manipulate them or using the pumping lemma in order to get a contradiction, but nothing worked out.


1 Answer 1



  1. For regular languages, consider $Perm((01)^*) \cap 0^* 1^*$.

  2. For context-free languages, consider $Perm(0^n 1^n 2^m 3^m) \cap 0^* 2^* 1^* 3^*$.

  • 1
    $\begingroup$ And also $Perm((012)^*) \cap 0^*1^*2^*$, which jumps from regular ouside context-free. $\endgroup$ May 19, 2015 at 13:59

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