A Tight Composition Theorem for the Randomized Query Complexity of Partial Functions: Extended Abstract
Bibliographic record
Abstract
We prove two new results about the randomized query complexity of composed functions. First, we show that the randomized composition conjecture is false: there are families of partial Boolean functions f and g such that R(f°g) ≪ R(f)R(g). In fact, we show that the left hand side can be polynomially smaller than the right hand side (though in our construction, both sides are polylogarithmic in the input size of f). Second, we show that for all f and g, R(f°g) = Ω(noisyR(f) R(g)), where noisyR(f) is a measure describing the cost of computing f on noisy oracle inputs. We show that this composition theorem is the strongest possible of its type: for any measure M(·) satisfying R(f°g)=Ω(M(f)R(g)) for all f and g, it must hold that noisyR(f)=Ω(M(f)) for all f. We also give a clean characterization of the measure noisyR(f): it satisfies noisyR(f)=Θ(R(f°GapMajn)/R(GapMajn)), where n is the input size of f and GapMajn is the √n-gap majority function on n bits.
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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 | 0.020 | 0.087 |
| Meta-epidemiology (narrow) | 0.003 | 0.003 |
| Meta-epidemiology (broad) | 0.005 | 0.007 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.004 | 0.009 |
| Scholarly communication | 0.011 | 0.033 |
| Open science | 0.006 | 0.015 |
| Research integrity | 0.004 | 0.013 |
| Insufficient payload (model declined to judge) | 0.021 | 0.003 |
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".