---
title: "Exact derandomization of logarithmic space: $\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}$"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 103; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "theoretical-computer-science"
  - "unformalized"
seed_rank: 1960
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 8
lineage: ai-mathematical-reasoning
cites:
  - title: "Exact derandomization of logarithmic space: L = RL = BPL"
    url: "https://github.com/openai/math/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

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

## One-sentence takeaway

OpenAI's result family 103 (Theoretical computer science) claims: Proves $\mathsf L=\mathsf{RL}=\mathsf{BPL}$, resolving derandomization for bounded-error logarithmic-space computation.

## Why it matters here

Closest to the systems side of the library: complexity, algorithms and computability statements here can change what is worth trying in ano, the engines or GRID COMMAND tooling. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): 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.
- *Exact derandomization of logarithmic space: L = RL = BPL*: We prove $\mathsf L=\mathsf{RL}=\mathsf{BPL}$, resolving the derandomization problem for polynomial-time randomized logarithmic space.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Exact derandomization of logarithmic space: L = RL = BPL](https://github.com/openai/math/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math