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$L\neq DSPACE[\omega(\log n)]$ is known.

Is $UL\neq DSPACE[\omega(\log n)]$ and $UL\neq PSPACE$ known?

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$\mathbf{UL}$ is contained in $\mathbf{NL}$, which is contained in $\mathbf{DSPACE}(\log^2 n)$ by Savitch's theorem, which is strictly contained in $\mathbf{PSPACE}$ by the space hierarchy theorem, so $\mathbf{UL}$ is strictly contained in $\mathbf{PSPACE}$. It would be surprising if e.g. $\mathbf{UL} = \mathbf{DSPACE}(\log^2 n)$, but I don't think it's known to be false.

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  • $\begingroup$ How about $\oplus L$ vs $PSPACE$? $\endgroup$
    – Turbo
    Commented Feb 26, 2021 at 1:22
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    $\begingroup$ $\oplus \mathbf{L} \subseteq \mathbf{DSPACE}(\log^2 n)$ as well. One way to prove it is a variation of Savitch's algorithm. $\endgroup$ Commented Feb 26, 2021 at 6:33
  • $\begingroup$ Posted variant cstheory.stackexchange.com/questions/48491/…. $\endgroup$
    – Turbo
    Commented Feb 26, 2021 at 6:40

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