New Results on the Complexity of the Middle Bit of Multiplication
Bibliographic record
Abstract
It is well known that the hardest bit of integer multiplication is the middle bit, i.e., MUL n−1,n . This paper contains several new results on its complexity. First, the size s of randomized read-k branching programs, or, equivalently, their space (log s) is investigated. A randomized algorithm for MUL n−1,n with $$k = {\mathcal{O}}(\hbox{log}\, n)$$ (implying time $${\mathcal{O}}(n\, \hbox{log}\, n))$$ , space $${\mathcal{O}}(\hbox{log}\, n)$$ and error probability n −c for arbitrarily chosen constants c is presented. Second, the size of general branching programs and formulas is investigated. Applying Nechiporuk’s technique, lower bounds of $$\Omega (n^{3/2}/ \hbox{log}\, n)$$ and Ω (n 3/2), respectively, are obtained. Moreover, by bounding the number of subfunctions of MUL n−1,n , it is proven that Nechiporuk’s technique cannot provide larger lower bounds than $${\mathcal{O}}(n^{5/3}/ \hbox{log}\, n)$$ and $${\mathcal{O}}(n^{5/3})$$ , respectively.
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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 | 0.002 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.005 | 0.011 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.007 |
| Insufficient payload (model declined to judge) | 0.014 | 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".