Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using O(T^{2/5}\log^C(T+2)) space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted. (Lean)
Manuscript
Simulating One-Tape Time in Two-Fifths-Power Space
September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026}
We show that a fixed deterministic Turing machine with one writable tape and head can be simulated in O(T^{2/5}\mathop{\mathrm{polylog}}\nolimits (T+2)) work-space bits when a binary time cap T ≥ 2 is supplied. The simulator computes the finite-control and halting outcome by time T; its running time is unrestricted. The result allows a fixed number of read-only input heads and requires a fixed accessor that supplies every initial writable and read-only symbol within distance T of the relevant head origin in polylogarithmic space. This improves the square-root space exponent for one-tape machines, answering Williams's question for this model.
Result 137 · Theoretical computer science. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md