Derandomization of logarithmic space: is L = RL = BPL?

Claims $\mathsf L=\mathsf{RL}=\mathsf{BPL}$ : every language decided with bounded one-sided or two-sided error by a randomized machine with polynomial worst-case running time and $O(\log n)$ space is decided by a deterministic $O(\log n)$ -space machine. Quantitatively, acceptance probabilities can be approximated to within $2^{-q}$ in space $O(\log(n+2)+q)$ and time $(n+2)^{O(1)}2^{O(q)}$ . The method builds a hierarchy of copies of the configuration graph with a shared finite random environment and evaluates a family of $O(\log n)$ -bit estimators, more than three quarters of them accurate, by exhaustive enumeration. It does not give a pseudorandom generator with logarithmic seed, and it does not address randomized logspace without a polynomial time bound.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Derandomization of logarithmic space: is L = RL = BPL? · L = RL = BPL

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Unreleased internal OpenAI model
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VibeMathed record: Derandomization of logarithmic space: is L = RL = BPL?

specialist registry · passed

No independent mathematician has checked this yet. Theorem 1.1 was read against the posed question and states the full equality L = RL = BPL for the standard model (polynomial worst-case running time on every random tape, logarithmic space counting all live counters). Lean-checked on the release's Comparator challenge LogspaceEquality (declaration OAI.ExactDerandomization.exact_logarithmic_space_derandomization, solution module OAI.Computability.Logspace.Equality), added in the 7 October 2026 update and listed in lean/formalization.yaml. Its statement was read here: for finite transition-table machines over binary words, with read-only input heads, binary work tapes, one coin bit per step and space at most $c\lceil\log_2(n+2)\rceil$ on every coin tape, the classes L, RL (polynomial clock, $\ge1/2$ versus $0$ ) and BPL ( $\ge2/3$ versus $\le1/3$ ) satisfy L = RL and RL = BPL. That is the headline. Not rebuilt here. The paper's quantitative approximation ( $2^{-q}$ accuracy in space $O(\log n+q)$ ) and its explicit compiler are not part of the Comparator statement. The paper concerns polynomial-time randomized logspace; it does not claim that randomized logspace machines without a time bound are derandomized.