Uniform Hardness Amplification in NP via Monotone Codes
Bibliographic record
Abstract
We consider the problem of amplifying uniform average-case hardness of languages in NP, where hardness is with respect to BPP algorithms. We introduce the notion of monotone errorcorrecting codes, and show that hardness amplification for NP is essentially equivalent to constructing efficientlylocally encodable and locally list-decodable monotone codes. The previous hardness amplification results for NP [Tre03, Tre05] focused on giving a direct construction of some locally encodable/decodable monotone codes, running into the problem of large amounts of nonuniformity used by the decoding algorithm. In contrast, we propose the indirect approach to constructing locally encodable/decodable monotone codes, combining the uniform Direct Product Lemma of [IJK06] and arbitrary, not necessarily locally encodable, monotone codes. The latter codes have fewer restrictions, and so may be easier to construct. We study what parameters are achievable by monotone codes in general, giving negative and positive results. We present two constructions of monotone codes. Our first code is a uniquely decodable code based on the Majority function, and has an efficient decoding algorithm. Our second code is combinatorially list-decodable, but we do not have an efficient decoding algorithm. In conjunction with an appropriate Direct Product Lemma, our first code yields uniform hardness amplification for NP from inverse polynomial to constant average-case hardness. Our second code, even with a brute-force decoding algorithm, yields further hardness amplification to 1/2 −log (1) n. Together, these give an alternative proof of Trevisan’s result [Tre03, Tre05]. Getting any non-brute-force decoding algorithm for our second code would imply improved parameters for the problem of hardness amplification in NP.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.027 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.003 | 0.007 |
| Research integrity | 0.002 | 0.008 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".