[Math] Are context-free languages with context-free complements necessarily deterministic context-free

Question

Let $L \subseteq A^\star$ be a formal language over $A$ generated by a context-free grammar, and $L' = A^\star – L$ be the relative complement in $A^\star$.

If $L$ and $L'$ are both context-free, are they necessarily deterministic context-free?

Answer 1

It seems that the answer to your question is no. See here.