Simulating Time With Square-Root Space

TL;DR

This paper presents a new simulation method that reduces the space needed to simulate time-bounded multitape Turing machines, improving previous bounds and impacting complexity theory and circuit evaluation.

Contribution

It introduces a space-efficient simulation of Turing machines running in time t, improving upon the classical O(t/ log t) space bound from 1975, and connects this to circuit evaluation and complexity class separations.

Findings

Simulation in O(√t log t) space for all t ≥ n

Implication for evaluating bounded fan-in circuits in sublinear space

Progress on separating P and PSPACE via explicit problems

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].