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 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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".