TITLE: Can more polynomial time compensate for less polynomial memory?
QUESTION [12 upvotes]: I'm wondering what is known about the relation between time and memory for polynomial-time algorithms (which are necessarily also polynomial-space). In particular, I would like to learn what is known about less time vs. less memory, in the following sense:
Is it known whether (say) any language $L$ recognized by a polynomial-time algorithm $A$ which is polynomial-space $O(n^s)$ ($s>2$ say), can also be recognized by a polynomial-time algorithm $B$ which is polynomial-space $O(n^{s/2})$? Presumably any $B$ with less memory would need significantly more time, but can we restrict this extra time to polynomial-time? 
So, recapping: is there for poly-time algorithms, some general way to compensate for less memory, in polynomial time? Answers and references greatly appreciated.   
update: Dan Brumleve and Timothy Chow have given valuable insights and references in the comments and in this chat. It seems that time-space trade-offs like the one in this question are still a bridge too far.

REPLY [3 votes]: I suspect that what you're hoping for—i.e., that for all $a$ and $b$ there exists $c$ such that  $\mathrm{DTISP}(n^a, n^b) \subseteq \mathrm{DTISP}(n^c, n^{b/2})$—is false, but for sure nobody has proved this.  In fact I suspect it may not even be true that $\mathrm{DTISP}(n^a,n^b) \subseteq\mathrm{DSPACE}(n^{b/2})$ (i.e., I suspect you cannot compensate for the loss of space with any amount of time).  Certainly, if we consider all problems solvable in space $O(n^b)$ (with no restrictions on time), then 
the space hierarchy theorem tells us that we will be able to solve certain problems that are simply unsolvable in space $O(n^{b/2})$, no matter how much time we give ourselves.  But settling your question will probably require proving a timespace hierarchy theorem, and very little is known unconditionally about such things (see this question on CS Theory Stackexchange for some information).  As another illustration of our ignorance about time-space tradeoffs, note that it is not known whether a problem that can be solved in polynomial time, and that can also be solved (possibly by a different algorithm) in polylogarithmic space, is in SC (i.e., is solvable is polynomial time and polylogarithmic space simultaneously).