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I need help with the following:
Let $k\in \mathbb{N}$, define:
$L^k=DSPACE(O(log^k(n)))$
$NL^k=NSPACE(O(log^k(n)))$
and:
$PolyL=\bigcup_{k=1}^{\infty}L^k$
$PolyNL=\bigcup_{k=1}^{\infty}NL^k$

I need to prove, disprove or to determine if it is an open question:
1. For every $k\in \mathbb{N}$ it holds that $NL^k\subseteq L^{2k}$
2. There exist $k\in \mathbb{N}$ s.t. $L^k=L^{k+1}$

The first one is easy, since it is almost immediately the result of Savitch's theorem.
The second one I'm not sure... I'd love to get some lead there.

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  • $\begingroup$ The second one "is almost immediately" resolved by a different theorem. $\;$ $\endgroup$
    – user12859
    May 24 '14 at 9:31
  • $\begingroup$ The only two theorems we've learned so far are Savitch and Immerman ($NL=CoNL$), and I don't see how Immerman has anything to do here... $\endgroup$
    – so.very.tired
    May 24 '14 at 10:51
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Use the space hierarchy theorem, which is a finite form of the undecidability of the halting problem.

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