I just need to know the truth or falsity of a simple statement.
Let $L_1$ be a context-free language over an alphabet which contains some number of characters $\Sigma$, as well as a single, special metacharacter “|”. Suppose that every string in $L_1$ contains exactly one metacharacter. Let the left half and right half of a string in $L_1$ denote the part of the string before the “|” and the part after, respectively. Let $L_2$ be the language over $\Sigma$ containing all the right halves of the strings in $L_1$.
Then is membership in $L_2$ decidable? What about if $L_1$ is restricted to be nice in some way? I'm fairly sure there are counterexamples in any case, but I can't find them.