The NOF Multiparty Communication Complexity of Composed Functions.
Bibliographic record
Abstract
We study the k-party ‘number on the forehead ’ communication complexity of composed functions f ◦ g, where f: {0,1} n → {±1}, g: {0,1} k → {0,1} and for (x1,...,xk) ∈ ({0,1} n) k, f ◦g(x1,...,xk) = f (...,g(x1,i,...,xk,i),...). We show that there is an O(log 3 n) cost simultaneous protocol for SYM ◦ g when k> 1 + logn, SYM is any symmetric function and g is any function. Previously, an efficient protocol was only known for SYM ◦ g when g is symmetric and “compressible”. We also get a non-simultaneous protocol for SYM ◦ g of cost O(n/2 k · logn + k logn) for any k ≥ 2. In the setting of k ≤ 1 + logn, we study more closely functions of the form MAJORITY ◦g, MODm ◦g, and NOR ◦g, where the latter two are generalizations of the well-known and studied functions Generalized Inner Product and Disjointness respectively. We characterize the communication complexity of these functions with respect to the choice of g. In doing so, we answer a question posed by Babai et al. (SIAM Journal on Computing, 33:137–166, 2004) and determine the communication complexity of MAJORITY ◦ QCSBk, where QCSBk is the “quadratic character of the sum of the bits” function.
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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 | 0.006 | 0.031 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.015 |
| Open science | 0.004 | 0.004 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.007 | 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".