Direct Product Theorems for Randomized Query Complexity
Why this work is in the frame
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Bibliographic record
Abstract
We establish two new direct product theorems for the randomized query complexity of Boolean functions. The first shows that computing n copies of a function f, even with a small success probability of $\gamma^{n}$, requires $\Theta(n)$ times the maximum distributional query complexity of f with success parameter $\gamma$. This result holds for all success parameters $\gamma$, even when $\gamma$ is very close to 1/2 or to 1. As a result, it unifies and generalizes Drucker’s direct product theorem (2012) for $\gamma$ bounded away from $\frac{1}{2}$ and 1 as well as the strong direct sum theorem of Blais and Brody (2019) for $\gamma \approx 1-1 / n$. The second establishes a general list decoding direct product theorem that captures many different variants of “partial computation” tasks related to the function $f^{n}$ consisting of n copies of f. Notably, our list decoding direct product theorem yields a new threshold direct product theorem and other new variants such as the labelled-threshold direct product theorem. Both of these direct product theorems are obtained by taking a new approach. Instead of directly analyzing the query complexity of algorithms, we introduce a new measure of complexity of functions that we call discounted score. We show that this measure satisfies a number of useful structural properties, including tensorization, that make it particularly suitable for the study of direct product questions.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it