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?



Are context-free languages with context-free complements necessarily deterministic context-free?
It seems that the answer to your question is no. See [here](https://cstheory.stackexchange.com/questions/4263/the-class-cfl-cap-co-cfl). 


