Computer Science/Discrete Mathematics Seminar I

We show that for all functions $t(n) \geq n$, every multitape Turing machine running in time $t$ can be simulated in space only $O(\sqrt{t \log t})$. This is a substantial improvement over Hopcroft, Paul, and Valiant's simulation of time $t$ in $O(t/\log t)$ space from about 50 years ago [FOCS 1975, JACM 1977]. Among other results, our simulation implies that bounded fan-in circuits of size $s$ can be evaluated on any input in only $\sqrt{s} \cdot \poly(\log s)$ space, and that there are explicit problems solvable in $O(n)$ space which require essentially $n^2$ time on a multitape Turing machine, thereby making a little progress on the P versus PSPACE problem.

Our simulation reduces the problem of simulating multitape Turing machines to an implicitly defined Tree Evaluation instance with nice parameters, leveraging the remarkable space-efficient algorithm for Tree Evaluation recently found by Cook and Mertz [STOC 2024].