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CS-34-A2 · revision r1 · on CS-34

OpenAI claim summary for CS-34

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The equality with randomized logarithmic space handles the one-sided randomized subclass. Arbitrary nondeterministic logspace acceptance and directed reachability are not simulated by the claimed theorem.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. L versus NL
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

L versus NL

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 103; 2026-09-23), the asserted result is that polynomial-time randomized logarithmic space can be simulated deterministically in logarithmic space, giving $\mathrm L=\mathrm{RL}=\mathrm{BPL}$.

Mathematical statement of the claimed result

Manuscript 1.
For uniform polynomial-time, logarithmic-work-space Turing machines, the claimed complexity-class equality is \[ \mathrm L=\mathrm{RL}=\mathrm{BPL}. \] More quantitatively, fix any randomized machine $\mathcal M$ of polynomial worst-case time and $O(\log(n+2))$ work space. Given input $x$ of length $n$ and unary $1^q$, $q\ge1$, a deterministic algorithm is claimed to output an integer $0\le a\le2^{q+2}$ satisfying \[ |a2^{-(q+2)}-p_{\mathcal M}(x)|\le2^{-q} \] in $O(\log(n+2)+q)$ work space and $(n+2)^{O(1)}2^{O(q)}$ time. Constants depend only on the fixed machine.

Outline of the argument

For manuscript 1 (family 103; 2026-09-23), stochastic atoms are assembled with controlled numerical error and bounded suspended storage. Slice-mixing walks are generated without retaining their endpoints, and a fixed bounded-resource interpreter makes the deterministic simulation uniform over source machines.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 103; 2026-09-23).

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