Let $L \subseteq A^\star$ be a formal language over $A$ generated by a contextfree grammar, and $L' = A^\star  L$ be the relative complement in $A^\star$.
If $L$ and $L'$ are both contextfree, are they necessarily deterministic contextfree?
Let $L \subseteq A^\star$ be a formal language over $A$ generated by a contextfree grammar, and $L' = A^\star  L$ be the relative complement in $A^\star$. If $L$ and $L'$ are both contextfree, are they necessarily deterministic contextfree? 


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


Your question is a bit unclear, and when we clarify it, it becomes true. If by "deterministic contextfree grammar" you mean, as usual, an LR(k) grammar for some k, then Knuth proved in his seminal paper ("On the translation of languages from left to right", 1965) that the languages defined are the same as those defined by deterministic PDAs. These are the DFCLs, and the DFCLs are closed under complement. So both your L and L' are DFCLs and hence CFLs, and your last premise is redundant. Your question really comes down to: are the DFCL's closed under complement  and they are. 

