Result 103, Theoretical computer science

Exact derandomization of logarithmic space: L=RL=BPL\mathsf L=\mathsf{RL}=\mathsf{BPL}

Proves L=RL=BPL\mathsf L=\mathsf{RL}=\mathsf{BPL}, resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds.

Proof

The bigger picture

Why it matters

The manuscript reports that every polynomial-time randomized computation using only logarithmic working memory can be made deterministic without changing that memory scale. If correct, randomness adds no power to decide problems in this tightly memory-limited setting.

What changes?

Logarithmic space means working memory growing at most proportionally to the logarithm of input length. L contains decision problems solvable deterministically within that space. RL and BPL allow randomness with bounded error on, respectively, one answer type or both. The claimed equality covers polynomial-time randomized machines in both classes. An effective compiler reportedly converts each into an always-correct deterministic logarithmic-space decider with explicit polynomial running-time bounds. The supplied summary and abstract do not specify those polynomials.

What does that help mathematicians do?

The claimed compiler would let researchers use randomness as an intermediate tool when designing memory-efficient decision procedures, then remove it without leaving logarithmic space. A randomized procedure meeting the stated time and error conditions would therefore establish deterministic solvability within the same space limit. Conversely, proving that a decision problem lies outside L would also rule out such randomized logarithmic-space procedures.

Are there practical applications?

The immediate value is foundational: the claim identifies the decision power of a sharply constrained computational model and offers an effective route from randomized machines to exact deterministic ones. This concerns working memory, not a demonstrated speed improvement. Polynomial running time alone does not establish practical efficiency, and the claim does not derandomize algorithms with unrestricted memory.

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

Exact derandomization of logarithmic space: L = RL = BPL

September 23, 2026 108 pages

We prove L=RL=BPL\mathsf L=\mathsf{RL}=\mathsf{BPL}, resolving the derandomization problem for polynomial-time randomized logarithmic space.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026,
  author = {{OpenAI}},
  title = {{Exact Derandomization of Logarithmic Space:
            $\mathsf{L}=\mathsf{RL}=\mathsf{BPL}$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf}{OAI:Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

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.