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?
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Sign up to join this communityLet $L \subseteq A^\star$ 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?
Your question is a bit unclear, and when we clarify it, it becomes true.
If by "deterministic context-free 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.