Result 137, Theoretical computer science

One-tape time simulation in two-fifths-power space

Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using O(T2/5log⁡C(T+2))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.

New or sharp bound

The bigger picture

Why it matters

How much memory is needed to determine what a long computation has done by a deadline? The manuscript reports a smaller bound for machines whose writable memory is a single tape, a row of storage cells.

What changes?

For a fixed deterministic machine with one writable tape and head, the reported workspace bound is T to the two-fifths power times a fixed power of log(T+2) bits, up to constant factors. The binary time cap T is at least two. A fixed number of read-only input heads is allowed; heads move at most one cell per step. Initial contents are T-independent; a fixed accessor supplies initial tape and input symbols within distance T of head origins in polylogarithmic space. Simulation time is unrestricted.

What does that help mathematicians do?

The output is whether the machine has halted by T and its finite-control state, the small internal record directing its actions, rather than the entire tape contents. Replacing the square-root exponent with two-fifths lowers the guaranteed memory growth rate, up to logarithmic factors. This gives researchers a stronger connection between computation time and the memory needed to determine an outcome, specifically for this one-tape model.

Are there practical applications?

The immediate value is foundational: the result sharpens our understanding of how computation time can be traded for storage under explicit restrictions on memory access. It is not a demonstrated practical speedup or low-memory implementation strategy. Because the simulator's running time is unrestricted, the memory saving alone gives no guarantee of usable performance.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Simulating One-Tape Time in Two-Fifths-Power Space

September 25, 2026 17 pages

We show that a fixed deterministic Turing machine with one writable tape and head can be simulated in O(T2/5polylog(T+2))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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026,
  author = {{OpenAI}},
  title = {{Simulating One-Tape Time in Two-Fifths-Power Space}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026/article.pdf}{OAI:Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.