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I am looking for a language L with the following properties:

  1. L should not be context-free.

  2. L's complement should not be context-free. (Everything you see in textbooks as prime examples of non-context-free languages seem to fail this second requirement.)

  3. L shouldn't be too hard, For example, I know that undecidable languages fit the first two requirements, but what I want is a simpler language which can be recognized by a slightly "improved" automaton model, e.g. a probabilistic pushdown automaton.

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Here is another example:

$ \mathtt{L} = \{ x\#y \mid x \in \mathtt{EQ},y \in \overline{\mathtt{EQ}} \} $,
where $ \mathtt{EQ}=\{ a^nb^nc^n \mid n \geq 0 \} $ and $ \overline{\mathtt{EQ}} $ is the complement of $\mathtt{EQ}$.

It is a well known fact that $\mathtt{EQ}$ is not in $ \mathsf{CFL} $.

Assume that $ \mathtt{L} $ is recognized by a PDA $ \small \mathcal{P_1}$. We construct a new PDA $ \small \mathcal{P}'$. On input $w$, $ \small \mathcal{P}'$ simulates $ \small \mathcal{P}_1$ on the string $w\#a$. Since $ \small \mathcal{P}'$ clearly recognizes $ \mathtt{EQ} $, we conclude that $ \mathtt{L} \notin \mathsf{CFL}$.

Similarly, assume that the complement of $ \mathtt{L} $ is recognized by a PDA $ \small \mathcal{P}_2$. We build another PDA $ \small \mathcal{P}''$. On input $w$, $\small \mathcal{P}''$ simulates $\small \mathcal{P}_2$ on the string $\#w$. $ \small \mathcal{P}''$ also recognizes $ \mathtt{EQ} $, so $ \mathtt{L} $ can not be in $\mathsf{coCFL}$ either.

$ \mathtt{EQ}$ can be recognized by a (one-way) probabilistic one-counter automaton (P1CA) with any desired error bound (Freivalds, 1979). So, it is not hard to show that $\mathtt{L}$ can also be recognized by a P1CA with any desired error bound.

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  • $\begingroup$ Even better than Dominik's answer, since it also describes a PPDA recognizing the language! (Dominik's is a tally language, and I have no idea how to build a PPDA that's superior to a PDA regarding a tally language.) $\endgroup$ – Cem Say Nov 25 '11 at 22:40
  • $\begingroup$ @CemSay: PPDAs cannot recognize any tally nonregular language with bounded error, too Kaneps et al. $\endgroup$ – Abuzer Yakaryilmaz Nov 25 '11 at 23:26
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How about $L:=\{a^{n^2}\mid n\in\mathbb{N}\}$? It is easy to see that $L$ and its complement are not regular, and hence (as we are dealing with a unary alphabet) not context-free.

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  • $\begingroup$ That's it, thanks. This is what my question asked for, so I'm accepting it, but I would appreciate any other examples very much. $\endgroup$ – Cem Say Nov 24 '11 at 9:41
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$QSAT$ or even $SAT$ are examples, unless $P = PSPACE$ or $P = NP$ respectively. $SAT$ is an example, as it is $NP$-complete and $CFL \subseteq P$.

$QSAT$ (true quantified boolean formulas) is $PSPACE$-complete, and is a CSL, recognizable by a LBA.

For unconditional examples you can take an arbitrary $EXP$-complete problem, such as generalized Chess or Go.

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  • $\begingroup$ Yes, thanks, but any even simpler ones, preferably those in the class P, please? $\endgroup$ – Cem Say Nov 24 '11 at 9:35

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