Let $ACFG$ be the language of all encodings $(C,x)$ where $C$ is a context-free grammar that generates a language containing $x$, i.e. $ACFG$ is the acceptance problem for context-free grammars.
It is easy to show that $ACFG$ is decidable. Parsing algorithms like CKY are proof of this fact.
Are $ACFG$ also regular? Is it also context-free? I would guess nay to both, but how would one formally show these facts?
Let's prove a more general result:
Let $\phi\colon \{0,1\}^* \to 2^{\{0,1\}^*}$ be an arbitrary function mapping binary strings to languages over the binary alphabet. Suppose that the range of $\phi$ is infinite. Then the language $L_\phi = \{(x,y) : y \in \phi(x)\}$ is not regular.
(The language is over the alphabet consisting of $0,1$ as well as parentheses and comma.)
For the proof, we use Myhill—Nerode theory. Let $(x_n)_{n \in \mathbb{N}}$ be encodings of languages such that $\phi(x_i) \neq \phi(x_j)$ for $i \neq j$. Then the words $(x_i,$ and $(x_j,$ are inequivalent modulo $L_\phi$: taking any word $y \in \phi(x_i) \Delta \phi(x_j)$, the word $y)$ separates the words $(x_i,$ and $(x_j,$.
Interestingly, $L_\phi$ could be context-free. Let $\phi(0^n) = \{1^n\}$ and $\phi(w) = \emptyset$ if $w$ contains $1$. Then $L_\phi = \{(0^n,1^n) : n \in \mathbb{N}\}$, which is context-free.
