Does $\forall k\in \mathbb N \space NP\not\subseteq TISP(poly(n),n^k)$ has any separation of classes or consequences?

My main question is can use this to show that $P \neq NP$ or some thing useful about $P \neq NP$ .

  • 1
    $\begingroup$ Why do we need the $\forall$ quantifier here? If $NP$ is not a subset of $TISP(poly(n),n)$ then your statement should also be true for all $k>1$? $\endgroup$ – C Komus Sep 3 '18 at 7:35
  • $\begingroup$ yes, they are equals. I thought they are different. $\endgroup$ – Mohsen Ghorbani Sep 3 '18 at 7:52
  • $\begingroup$ I think they are equals to $NP \not \subset TISP(poly(n),n^{o(1)})$ too. $\endgroup$ – Mohsen Ghorbani Sep 3 '18 at 7:58

This would imply that L⊊NP since L⊆TISP(poly(n),n^k) k∈N

  • 3
    $\begingroup$ It also can show $SC \neq NP$ in similar way. $\endgroup$ – Mohsen Ghorbani Sep 2 '18 at 19:50

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