Are permutations of context-free languages context-free?

Question

Given a context-free language $L$, define the language $p(L)$ as containing all permutations of strings in $L$ (i.e. all strings in $L$ such that the order of symbols is not important). Is $p(L)$ context-free?

I found two papers dealing with similar, but not identical, questions:

Generating all permutations by context-free grammars in Chomsky normal form by Asveld (2003) deals with finite languages.
Permutations are not context-free: An application of the interchange lemma by Main (1982) deals with "permutation languages", i.e. sets of strings of the form $w x p(x) z$, where $p(x)$ is a any permutation of $x$. Also, the result is limited to alphabets with 16 symbols.

Have you tried applying the techniques shown in our reference question? — Raphael, Apr 15 '13 at 12:16

score 5 · Accepted Answer · answered Apr 15 '13 at 14:34

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Start with simple context-free (or even regular) languages, and see what happens. For instance determine $p(L)$ for $L = (ab)^*$ and $L=(abc)^*$.

answered Apr 15 '13 at 14:34

Hendrik Jan

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A penny just dropped in my head, thanks. – Raphael Apr 15 '13 at 14:45

score 1 · Answer 2 · answered Dec 30 '23 at 15:38

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No, in general not. For example the MIX language, consisting of equal numbers of $\{a,b,c\}$ in any order is not context-free but is the permutation of the regular language $(abc)^*$.

answered Dec 30 '23 at 15:38

Alexander Clark

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Are permutations of context-free languages context-free?

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