Low uniform versions of NC1.
Bibliographic record
Abstract
In the setting known as DLOGTIME-uniformity, the fundamental complexity classes AC ⊂ ACC ⊆ TC ⊆ NC have several robust characterizations. In this paper we refine uniformity further and examine the impact of these refinements on NC and its subclasses. When applied to the logarithmic circuit depth characterization of NC, some refinements leave NC unchanged while others collapse NC to NC. Thus we study refinements of other circuit characterizations of NC. In the case of the AC(A5) characterization of NC , where A5 is the NC-complete word problem of the group A5, our refinements collapse NC to a subset of the regular languages. For the AC(D+) characterizations of NC, where D+ is the NC-complete language capturing the formula value problem, interestingly, these refinements scale down to circuits with linear fan-in. In particular, the latter refinements bring to the fore two classes, denoted FO[<]-uniform AC(D+)LIN and FO[<]uniform TCLIN , whose separation may be within the reach of current lower bound techniques, and whose separation would amount to distinguishing the power of a MAJ gate from that of a D+ gate.
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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 | 0.001 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 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".