It is unknown whether $P\subseteq CSL$ or $P\not\subseteq CSL$, where$P$ is the set of all languages decidable in polynomial time on a deterministic Turing machine, and$CSL$ is the class of context-sensitive languages, known to be equivalent to $NSPACE(O(n))$, the languages decided by linear-bounded automata.For many open questions, there is a tendency towards one answer (a la most experts believe that $P\neq NP$). Is there something like this for this question?In particular, would either answer have unexpected consequences? I can only see expected (but unproven) consequences:If $P\subseteq CSL$, then $P\subseteq NSPACE(O(n))\subsetneq NSPACE(O(n^2))$ (space hierarchy theorem), hence $P\subsetneq PSpace$.If $P\not\subseteq CSL$, then there is a language $l\in P\setminus NSPACE(O(n))$ and therefore $l\in P\setminus NL$, hence $NL\subsetneq P$.(Acknowledgement: The second consequence of these two was pointed out by Yuval Filmus at https://cs.stackexchange.com/questions/69614/)
What is the conjectured relationship between P (PTime) and Type 1 (context-sensitive) languages?
cc.complexity theory;fl.formal languages;polynomial time
If $\mathrm{P\subseteq CSL}$, then $\mathrm{P\subseteq DSPACE}(n^2)$. By a padding argument, this implies$$\mathrm{DTIME}(t(n))\subseteq\mathrm{DSPACE}\bigl(t(n)^\epsilon\bigr)$$for every superpolynomial well-behaved function $t(n)$ and every $\epsilon>0$. I believe such a strong advantage of space over time is not expected to be true. The best currently known result in this direction is$$\mathrm{DTIME}(t(n))\subseteq\mathrm{DSPACE}(t(n)/\log t(n)),$$due to Hopcroft, Paul, and Valiant.