Simulating Time with Square-Root Space

Abstract

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 logt). This is a substantial improvement over Hopcroft, Paul, and Valiant’s simulation of time t in O(t/logt) 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(logs) space, and that there are explicit problems solvable in O(n) space which require at least n2−ε time on every 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].