Bibliographic record
Abstract
The replacement (or collection or choice,) axiom scheme BB(/spl Gamma/) asserts bounded quantifier exchange as follows: /spl forall/I < |a| /spl exist/x < ao(i, x) /spl rarr/ /spl exist/w /spl forall/i < |a| o (i, [w]/sub i/) where o is in the class /spl Gamma/ of formulas. The theory S/sub 2//sup 1/ proves the scheme BB(/spl Sigma//sub 1//sup b/), and thus in S/sub 2//sup 1/ every /spl Sigma//sub 1//sup b/ formula is equivalent to a strict /spl Sigma//sub 1//sup b/ formula (in which all non-sharply-bounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S/sub 2//sup 1/ do not prove either BB(/spl Sigma//sub 1//sup b/) or BB(/spl Sigma//sub 0//sup b/). We show (unconditionally) that V/sup 0/ does not prove BB(/spl Sigma//sub 1//sup B/), where V/sup 0/ (essentially I/spl Sigma//sub 0//sup 1,b/) is the two-sorted theory associated with the complexity class AC/sup 0/. We show that PV does not prove BB(/spl Sigma//sub 0//sup b/), assuming that integer factoring is not possible in probabilistic polynomial time. Johannsen and Pollet introduced the theory C/sub 2//sup 0/ associated with the complexity class TC/sup 0/, and later introduced an apparently weaker theory /spl Delta//sub 1//sup b/ - CR for the same class. We use our methods to show that /spl Delta//sub 1//sup b/ - CR is indeed weaker than C/sub 2//sup 0/, assuming that RSA is secure against probabilistic polynomial time attack. Our main tool is the KPT witnessing theorem.
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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 | 0.006 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.012 |
| Scholarly communication | 0.005 | 0.010 |
| Open science | 0.002 | 0.008 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 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".