Simulating Time With Square-Root Space

We show that for all functions t(n) ≥ n, every multitape Turing machine running in time t can be simulated in space only O(√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 √s · poly(log s) space, and that there are explicit problems solvable in $O(n)$ space which require n2-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].