First, we state here a theorem that is well-known in computability theory:

$L=\{xx\mid x\in\Sigma^*\}\notin CFL$ for every fixed $|\Sigma|\geq2$

And, the standard proof is using pumping lemma. At first sight, we rarely face the language $L$ exactly as defined above in a broader theoretical research. Instead, what might be more frequent is $$L'=\{\text{some-simple-manipulation}(x)x\mid x\in\Sigma^*\}$$ Perhaps, those simple manipulation are necessary for unambiguous encoding. For example, symbol doubling, prefix, postfix, etc. Those are quite frequent and unavoidable.

Now, the question is: Can the above theorem still hold under these simple manipulations?

Symbol doubling (as commented below) is a good starter, if one feels the question too broad.

Or even worse, there may be some simple manipulation that push $L$ down to CFL again.

  • $\begingroup$ Unless you specify the simple manipulations, the question will be hard to answer. Does reverse count, for example? For other manipulations, I would guess that standard closure properties would show that the modified languages are still not context-free. $\endgroup$ – Yuval Filmus Oct 18 '18 at 7:19
  • $\begingroup$ Take symbol doubling as a starter, $001011$ indicates $011$ (the second in each pair terminates the current string) $\endgroup$ – Thinh D. Nguyen Oct 18 '18 at 7:21
  • $\begingroup$ Note that PDA is only one-direction less from being Turing-complete. $\endgroup$ – Thinh D. Nguyen Oct 18 '18 at 7:22
  • $\begingroup$ I don't understand what symbol doubling is. Could you please explain more? $\endgroup$ – xskxzr Oct 18 '18 at 9:31
  • $\begingroup$ @xskxzr I guess it means duplicating each symbol of the alphabet, e.g. $x=abcde$ to $x'=aabbccddee$. $\endgroup$ – chi Oct 18 '18 at 12:51

Note $\left\{x^{\mathrm{R}}x\mid x\in\Sigma^*\right\}$ is context free, where $x^{\mathrm{R}}$ means the reverse of $x$.

Or even worse, there may be some simple manipulation that push $L$ down to CFL again.

So if you consider reversing as a simple manipulation, the answer is indeed yes.

  • $\begingroup$ I just realize that symbol doubling like in my question can still be pumped to contradiction. $\endgroup$ – Thinh D. Nguyen Oct 18 '18 at 14:31

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