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If $L$ is a Context Free language, it can happen that for some $n$, all words of length $n$ are in $L$. If we consider the set $A_L$ of such lengths represented in unary, we may guess that such set is Context Free (and hence regular), but it is not the case.

More formally; if $L \in CF$ define:

$A_L = \{ 1^n \mid |w|=n \Rightarrow w \in L \}$

There are CF languages for which $A_L \notin REG$.

The example I have in mind uses the sequence of tape configurations (alternating straight/reverse order like in the proof of the undecidability of $L = \Sigma^*$) of a valid Turing machine computation that on input $x$ (in binary), writes $1^x$ on the tape and halts.

Before spending more time in formalizing it, I wonder if there is a simpler example, or if I can find it in some books/papers (I made some searches but probably I'm using the wrong terms).

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The shortest word in $A_L$ is not bounded by a recursive function in the size of a given context-free grammar describing $L$. See here for more results in that direction: https://doi.org/10.4230/LIPIcs.STACS.2020.16

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    $\begingroup$ Thank you! I had to download it from arXiv because the one from your link misses some parts. By the way, the set $A_L$ built from the CF language $L$ they used to prove the undecidability of "Exists $\ell$ s.t. $\Sigma^\ell \subseteq L$?" (and the unboundedness w.r.t. the size of the grammar) is regular :-) $\endgroup$ – Marzio De Biasi Jul 14 '20 at 13:10

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