Simulating Time With Square-Root Space
R. Ryan Williams
TL;DR
This work establishes a surprising time-space separation for multitape Turing machines by showing ${\sf TIME}[t(n)] \subseteq {\sf SPACE}[\sqrt{t(n) \log t(n)}]$ for all $t(n) \ge n$, improving the classical ${\sf TIME}[t] \subseteq {\sf SPACE}[t/\log t]$ bound. The central method reduces time-bounded computation to instances of the Tree Evaluation problem, then solves these instances space-efficiently via the Cook–Mertz algorithm, aided by a block-respecting TM decomposition. The result yields notable corollaries, including sublinear space evaluation of bounded-fanin circuits and near-polynomial time lower bounds for linear-space problems, thereby contributing progress toward separating ${\sf P}$ from ${\sf PSPACE}$. The paper also discusses potential refinements, higher-dimensional extensions, and the broader implications for time-space tradeoffs and complexity theory.
Abstract
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 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 $n^{2-\varepsilon}$ time on a multitape Turing machine for all $\varepsilon > 0$, thereby making a little progress on the $P$ versus $PSPACE$ problem. Our simulation reduces the problem of simulating time-bounded multitape Turing machines to a series of implicitly-defined Tree Evaluation instances with nice parameters, leveraging the remarkable space-efficient algorithm for Tree Evaluation recently found by Cook and Mertz [STOC 2024].