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CS-148 · computer science · mathematics · Automata and formal languages

Two-fifths-power space simulation for one writable tape

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For deterministic machines with one writable tape and head, finitely many read-only input heads moving by unit steps from fixed origins, cap-independent symbol functions and the prescribed local-access condition, can the finite-control and halting outcome by binary time cap T be simulated in $\widetilde O(T^{2/5})$ workspace bits, excluding read-only input storage, with unrestricted simulation time and no requirement to materialize the final tape?

Short English statement

For deterministic machines with one writable tape and head, finitely many read-only input heads moving by unit steps from fixed origins, cap-independent symbol functions and the prescribed local-access condition, can the finite-control and halting outcome by binary time cap T be simulated in $\widetilde O(T^{2/5})$ workspace bits, excluding read-only input storage, with unrestricted simulation time and no requirement to materialize the final tape?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-148-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-148

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 137; 2026-09-25).

Two-fifths-power space simulation for one writable tape

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 137; 2026-09-25), the asserted result is that a fixed one-writable-tape machine run for a supplied cap $T$ can be simulated in $O(T^{2/5}\operatorname{polylog}(T+2))$ work bits, with the stated initial-symbol accessor and unrestricted time.

Mathematical statement of the claimed result

Manuscript 1.
Fix a deterministic finite-control machine with one integer-indexed writable tape/head and finitely many read-only heads on the input, with their number fixed, all moving by at most one cell per step from fixed origins. Initial symbol functions $\sigma_h(x,q)$ are fixed independently of time caps. The access premise requires one fixed deterministic accessor and constants $C_0,C_1$ such that, for every tag $h$ and $|q|\le T$, it returns $\sigma_h(x,q)$ using at most $C_0\log^{C_1}(T+2)$ work bits. Given binary cap $T\ge2$ satisfying that premise, the claimed simulator determines the control state and whether the machine halts within $T$ steps, using \[ O(T^{2/5}\log^C(T+2)) \] work bits for a fixed $C$, excluding read-only input storage. There is no asserted bound on simulation time. Outputting the final contents of the writable tape is not required. If the access premise holds for every cap and the original machine halts after $t$ steps, the analogous outcome uses $O((t+2)^{2/5}\log^C(t+2))$ space without a supplied running time.

Outline of the argument

For manuscript 1 (family 137; 2026-09-25), block visits and epoch checkpoints create dependencies with different storage weights. An interpolation evaluator borrows and restores registers from arbitrary contents, so only designated dependency edges save large indices; balancing three scales gives the two-fifths exponent.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 137; 2026-09-25).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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