Computation Complexity polyL ⊆ QP, polyL ? P

Question

Following Computation Complexity POLYLOGSPACE,

Given that polyL / POLYLOGSPACE is the complexity class

• $⋃^∞_{=1}(()^)$, k is integer.

QP / Quasi-polynomial is the complexity class ,

• $⋃^∞_{c=1}TIME(2^{()^c})$, c is integer.

(a) How to prove that, polyL $⊆$ QP?

Reference: wiki on polyL without proof

(b) P is the Polynomial complexity class.

polyL ? P (any inclusion or equality between these two classes?)

i.e., if polyL ⊊ P, if P ⊊ polyL, or if neither is contained in the other?

Thank you!

Answer 1

(a)

From the fourth inclusion rule, Inclusion rules

$(())⊆(2^{(())})$

Here, plug in S(n) = (log n)^k, where k is an integer.

polyL ⊆ QP.

(b) polyL ≠ P, see Note to part b or PolyL-wiki.

But, it is unknown if polyL ⊊ P, if P ⊊ polyL, or if neither is contained in the other.