# Is $UL\neq PSPACE$ known?

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

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

$$\mathbf{UL}$$ 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.
• How about $\oplus L$ vs $PSPACE$? – 1.. Feb 26 at 1:22
• $\oplus \mathbf{L} \subseteq \mathbf{DSPACE}(\log^2 n)$ as well. One way to prove it is a variation of Savitch's algorithm. – William Hoza Feb 26 at 6:33