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Computer Science > Computational Complexity

arXiv:2502.17779 (cs)
[Submitted on 25 Feb 2025]

Title:Simulating Time With Square-Root Space

Authors:R. Ryan Williams
View a PDF of the paper titled Simulating Time With Square-Root
Space, by R. Ryan Williams
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    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].

Comments: To appear in ACM STOC 2025
Subjects: Computational Complexity (cs.CC)
Cite as:  arXiv:2502.17779 [cs.CC]
          (or arXiv:2502.17779v1 [cs.CC] for this version)
          https://doi.org/10.48550/arXiv.2502.17779
          Focus to learn more
          arXiv-issued DOI via DataCite

Submission history

From: Ryan Williams [view email]
[v1] Tue, 25 Feb 2025 02:13:49 UTC (34 KB)
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