Actually I found that the set of context-sensitive Languages, $\mathbf{CSL}$ ($\mathbf{=NSPACE}(O(n))$ $= \mathbf{LBA}$ accepted languages) are not so widely discussed as $\mathbf{REG}$ (regular languages) or $\mathbf{CFL}$ (context-free languages).
And also the open problem $\mathbf{DSPACE}(O(n)) \stackrel?= \mathbf{NSPACE}(O(n))$ is not so famous as the "analogous" problem: $\mathbf{P} \stackrel?= \mathbf{NP}$.
So I have two questions:
- Is there a language in $\mathbf{CSL}$ which couldn't be proved to be in $\mathbf{DSPACE}(O(n))$?
- Moreover: Is there a language $L$ in $\mathbf{CSL}$ which is "complete" in the following sense: if we can prove that $L$ is in $\mathbf{DSPACE}(O(n))$ we get that $\mathbf{DSPACE}(O(n)) = \mathbf{CSL}$?
Notice: I have asked these questions in cs.stackexchange and got some interesting Answers but not a concrete (proven) example for such a language: https://cs.stackexchange.com/questions/79528/a-language-in-nspaceon-and-very-likely-not-in-dspaceon