Languages that we cannot (dis)prove to be Context-Free

I'm looking for languages which are "probably not Context-Free" but we are not able to (dis)prove it using known standard techniques.

Is there a recent survey on the subject or an open problem section from a recent conference ?

Probably there are not many languages which are not known to be CF, so if you know one you can also post it as an answer.

The examples I found are:

Note: as showed by Aryeh in his answer you can build a whole class of such languages if you "link" a language to an unknown conjecture about the (non)finiteness or (non)emptiness of some sets (e.g. $$L_{Goldbach} = \{ 1^{2n} \mid 2n$$ cannot be expressed as a sum of two primes$$\}$$). I'm not quite interested in such examples.

• For your second example, I wrote a paper from my answer which is under review (and the first feedback was positive): arxiv.org/abs/1901.03913 Apr 7, 2019 at 16:22
• There are many variants of the first example that are not known to be context-free, I don't know if you want to include them as separate examples; see Chapter 10 of the linked book (Kászonyi-Katsura Theory). Apr 7, 2019 at 16:26
• @domotorp: I just gave it a look (I'm still reading chapter 2) ... they seem to me more technical attempts to attack the main problem. Apr 7, 2019 at 17:27

Another good one is the complement of the set $$S$$ of contiguous subwords (aka "factors") of the Thue-Morse sequence $${\bf t} = 0110100110010110 \cdots$$. To give some context, Jean Berstel proved that the complement of the set $$T$$ of prefixes of the Thue-Morse word is context-free (and actually something more general than that). But the corresponding result for subwords is still open.

• Great, thanks! If you saw it stated somewhere (perhaps in one of your many papers on the Thue-Morse sequence? ;-) you can add the reference (even if stated in the iterated morphism form). Apr 5, 2019 at 19:00

How about the language $$L_{TP}$$ of twin primes? I.e., all pairs of natural numbers $$(p,p')$$ (represented, say, in unary), such that $$p,p'$$ are both prime and $$p'=p+2$$? If twin primes conjecture is true, then $$L_{TP}$$ is not context-free; otherwise, it's finite.

Edit: Let me give a quick proof sketch that the twin primes conjecture implies that $$L_{TP}$$ is not context-free. Associate to any language $$L$$ its length sequence $$0\le a_1\le a_2\le\ldots$$, where the integer $$\ell$$ appears in the sequence iff there is a word of length $$\ell$$ in $$L$$. It is a consequence of the pumping lemma(s) that for $$L$$ that are regular or CFL, the length sequence satisfies the bounded differences property: there is an $$R>0$$ such that $$a_{n+1}-a_n\le R$$ for all $$n$$. It is an easy and well-known fact in number theory that the primes do not have bounded differences. Finally, any infinite subsequence of a sequence violating the bounded differences property itself must violate it.

• Nice, thanks! But I'm not quite interested in languages that are linked to unknown conjectures about the (non)finiteness of some sets. BTW if those conjectures are true the resulting language is also regular :-) Apr 5, 2019 at 13:54
• If there are infinitely many twin primes, how do you see that $L_{TP}$ is regular? Apr 5, 2019 at 14:04
• If there are infinitely many twin primes, how do you show that $L_{TP}$ is not context-free? Apr 5, 2019 at 14:48
• Oh, sorry, I didn’t notice you represent the numbers in unary. Then it is clear. (I believe that proving this for binary representation would require a considerable progress on the twin primes conjecture.) Apr 5, 2019 at 15:28
• On the contrary, Emil, the "standard" proof that the primes in binary are not context-free easily suffices to prove that every infinite set of primes is not context-free. So if there are infinitely many twin primes, the result is immediate. Apr 5, 2019 at 17:21