patternMinor
A language in NSPACE(O(n)) and very likely not in DSPACE(O(n))
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Problem
Actually I found that the set of context-sensitive Languages, $\mathbf{CSL}$ ($\mathbf{=NSPACE(O(n)) = LBA}$ accepted languages) are not so widely discussed as $\mathbf{REG}$ (regular languages) or $\mathbf{CFL}$ (context-free languages).
And also the open problem $\mathbf{DSPACE(O(n))} =^{?} \mathbf{NSPACE(O(n))}$ is not so famous as the "analogous" problem: "$\mathbf{P} =^{?} \mathbf{NP}$".
Well, is there really such an analogy:?
And also the open problem $\mathbf{DSPACE(O(n))} =^{?} \mathbf{NSPACE(O(n))}$ is not so famous as the "analogous" problem: "$\mathbf{P} =^{?} \mathbf{NP}$".
Well, is there really such an analogy:?
- Is there a language in $\mathbf{CSL}$ which couldn't be proved to be in $\mathbf{DSPACE(O(n))}$ (like $\mathbf{NP}$ complete languages)?
- 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 $\mathbf{DSPACE(O(n)) = NSPACE(O(n))}$?
- (Maybe just a matter of opinion) Are both problems on the same level of difficulty?
Solution
The more well-known version of these questions is the $\mathsf{L} \stackrel?= \mathsf{NL}$ question. If $\mathsf{L} = \mathsf{NL}$ then a (slightly tricky) padding argument shows that $\mathsf{DSPACE}(n) = \mathsf{NSPACE}(n)$, and so $\mathsf{DSPACE}(n) \neq \mathsf{NSPACE}(n)$ implies the well-known conjecture $\mathsf{L} \neq \mathsf{NL}$.
The conjecture $\mathsf{L} \neq \mathsf{NL}$ is considered (by some) to be more approachable than the conjecture $\mathsf{P} \neq \mathsf{NP}$. I'm not sure many people have an opinion on the conjecture $\mathsf{DSPACE}(n) \neq \mathsf{NSPACE}(n)$.
The bigger picture here is whether Savitch's theorem, which states that $\mathsf{NSPACE}(t(n)) \subseteq \mathsf{DSPACE}(t(n)^2)$ for reasonable $t(n) \geq \log n$, is tight. While $\mathsf{NPSPACE} = \mathsf{PSPACE}$, I think that most people believe that $\mathsf{NSPACE}(n^k) \neq \mathsf{DSPACE}(n^k)$. On the other hand, I'm not sure that people believe that $t(n)^2$ is the optimal blowup; perhaps a smaller exponent also works, at least in some cases. See for example a recent arXiv paper, The parameterized space complexity of model-checking bounded variable first-order logic, by Yijia Chen, Michael Elberfeld, and Moritz Müller.
The conjecture $\mathsf{L} \neq \mathsf{NL}$ is considered (by some) to be more approachable than the conjecture $\mathsf{P} \neq \mathsf{NP}$. I'm not sure many people have an opinion on the conjecture $\mathsf{DSPACE}(n) \neq \mathsf{NSPACE}(n)$.
The bigger picture here is whether Savitch's theorem, which states that $\mathsf{NSPACE}(t(n)) \subseteq \mathsf{DSPACE}(t(n)^2)$ for reasonable $t(n) \geq \log n$, is tight. While $\mathsf{NPSPACE} = \mathsf{PSPACE}$, I think that most people believe that $\mathsf{NSPACE}(n^k) \neq \mathsf{DSPACE}(n^k)$. On the other hand, I'm not sure that people believe that $t(n)^2$ is the optimal blowup; perhaps a smaller exponent also works, at least in some cases. See for example a recent arXiv paper, The parameterized space complexity of model-checking bounded variable first-order logic, by Yijia Chen, Michael Elberfeld, and Moritz Müller.
Context
StackExchange Computer Science Q#79528, answer score: 8
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