Bibliographic record
Abstract
We solve some fundamental problems in the number-on-forehead (NOF) $k$-player communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with one-sided false-positives error probability of 1/3, but which has linear communication complexity for deterministic protocols, and in fact, even for the more powerful nondeterministic protocols. The result holds for every $\epsilon > 0$ and every $k \le 2^{(1-\epsilon)n}$ players, where $n$ is the number of bits on each player's forehead. As a consequence, we obtain the NOF communication class separation $\mathsf{coRP} \not\subset \mathsf{NP}$. This in particular implies that $\mathsf{P} \neq \mathsf{RP}$ and $\mathsf{NP} \neq \mathsf{coNP}$. We also show that for every $\epsilon > 0$ and every $k \le n^{1-\epsilon}$, there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential; no explicit function is known to satisfy nontrivial lower bounds for $k \ge \log n$ players. However, for every $\epsilon > 0$ and every $k \le (1-\epsilon) \cdot \log n$ players, the $\mathsf{NP} \ne \mathsf{coNP}$ separation (and even the $\mathsf{coNP} \not\subset \mathsf{MA}$ separation) was obtained independently by Gavinsky and Sherstov (2010) using an explicit construction. In this work, for $k \le (1/9) \cdot \log n$ players, we exhibit an explicit function which has communication complexity $O(1)$ for public coin protocols and $\Omega(\log n)$ for deterministic protocols. This improves the best previously known deterministic lower bound for a function with efficient randomized protocols, which was $\Omega(\log \log n)$, given by Beigel, Gasarch, and Glenn (2006). It follows from our existential result that any function that is complete for the class of functions with polylogarithmic nondeterministic $k$-player communication complexity does not have polylogarithmic deterministic complexity. We show that the set intersection function, which is complete in the number-in-hand model, is not complete in the NOF model under cylindrical reductions.
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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 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.003 | 0.012 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".