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CS-130-A1 · revision r1 · on CS-130

OpenAI claim summary for CS-130

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 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 121; 2026-09-24).

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

Manuscript

Contents
  1. Almost-linear edit-distance approximation scheme
  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

Almost-linear edit-distance approximation scheme

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 121; 2026-09-24), the asserted result is that fixed-accuracy edit distance has a uniform randomized $(1+\varepsilon)$ estimator with almost-linear expected work, polynomially bounded integer alphabets, and certain zero output on identical strings.

Mathematical statement of the claimed result

Manuscript 1.
For explicitly encoded strings $x,y$ of total length $N$ over a finite alphabet, let $D=\operatorname{ED}(x,y)$ with insertions, deletions and substitutions each of unit cost. Given rational $0<\varepsilon<1$, the asserted uniform randomized algorithm outputs an integer $\widehat D\ge0$ satisfying \[ \Pr[D\le\widehat D\le(1+\varepsilon)D]\ge2/3. \] For each fixed rational $\varepsilon$, the worst-case expected time is $N^{1+o(1)}$ on a logarithmic-word RAM, including all input, preprocessing, arithmetic and random-sampling costs. Equal strings yield zero deterministically; the time bound is an expectation, not a bound on every random path.

Outline of the argument

For manuscript 1 (family 121; 2026-09-24), a coarse random table is defined simultaneously for all interval states but evaluated only when requested. Precision sampling bounds adaptive requests, and cost-cone Bellman refinement uses predicted bands to turn the integer seeds into the final estimate.

Formal verification

For manuscript 1 (family 121; 2026-09-24), OpenAI lists Lean statements for this assertion: fixed-accuracy edit distance has a uniform randomized $(1+\varepsilon)$ estimator with almost-linear expected work, polynomially bounded integer alphabets, and certain zero output on identical strings.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are EditApproximation.lean.

Their named declarations include OAI.EditApproximation.almostLinearEditDistanceRawBand.

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