[Wil25] Simulating Time in Square-Root Space

Authors: Ryan Williams | Venue: STOC 2025 | Source

Abstract

We show that for all functions $t(n)\ge n$, 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 $O(t^{\frac{1}{2}}\log t)$ 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 $\sqrt{s}\text{poly}(\log s)$ space, and that there are explicit problems solvable in $O(n)$ space which require $n^{2-ϵ}$ time on a multitape Turing machine for all $ϵ>0$, 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].

BibTeX

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@inproceedings{Wil25,
  author    = {R. Ryan Williams},
  title     = {Simulating Time in Square-Root Space},
  booktitle = {Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC 2025)},
  year      = {2025},
  url       = {https://arxiv.org/abs/2502.17779}
}