Bibliographic record
Abstract
We introduce a finitely axiomatizable second-order theory VTC 0 and show that it characterizes precisely the class uniform TC0. It is simply the theory V0 [12] together with the axiom NUMONES, which states the existence of a "counting array" Y for any string X : the ith row of Y contains only the number of 1 bits upto (excluding) bit i of X. First, we introduce the notion of "strong DB1 -definability" for relations in a theory, and use the recursive properties of TC0 relations (rather than functions) to show that TC0 relations are strongly DB1 -definable, and TC0 functions are SB1 -definable in VTC0. Then, we generalize the Witnessing Theorem for V0 [12], and obtain the witnessing theorem for VTC0 from this general result: exist;SB0+ SB1 theorems of VTC0 can be witnessed by TC0 functions (here, SB0+S B1 formulas are those obtained from SB1 formulas using and;, or; and bounded number quantifications). Finally, we show that VTC0 is RSUV isomorphic to the first-order theory Db1 -CR, which has been claimed the "minimal" theory for TC0 [20]. This isomorphism shows that VTC0 admits the SB0+D B1 comprehension rule. Hence, in VTC0, strong DB1 -definability and the usual DB1 -definability coincide. It also follows that Db1 - CR = Db1 - CRi, for some i. This answers affirmatively an open question from [20].
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".