We know that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{P}$ and that $\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{L}^2 \subseteq $ $\mathsf{polyL}$, where $\mathsf{L}^2 = \mathsf{DSPACE}(\log^2 n)$. We also know that $\mathsf{polyL} \neq \mathsf{P}$ because the latter has complete problems under logarithmic space many-one reductions while the former does not (due to the space hierarchy theorem). In order to understand the relationship between $\mathsf{polyL}$ and $\mathsf{P}$, it may help to first understand the relationship between $\mathsf{L}^2$ and $\mathsf{P}$.What are the consequences of $\mathsf{L}^2 \subseteq \mathsf{P}$, or even $\mathsf{L}^{k} \subseteq \mathsf{P}$?
What are the consequences of $\mathsf{L}^2 \subseteq \mathsf{P}$?
cc.complexity theory;complexity classes;conditional results;structural complexity
The following is an obvious consequence: $\mathsf{L}^2 \subseteq \mathsf{P}$ would imply $\mathsf{L} \subsetneq \mathsf{P}$ and therefore $\mathsf{L} \neq \mathsf{P}$.By the space hierarchy theorem $\mathsf{L} \subsetneq \mathsf{L}^2$. If $\mathsf{L}^2 \subseteq \mathsf{P}$ then $\mathsf{L} \subsetneq \mathsf{L}^2 \subseteq \mathsf{P}$. 