A second-order system for polytime reasoning using Gradel's theorem
Bibliographic record
Abstract
We introduce a second-order system V/sub 1/-Horn of bounded arithmetic formalizing polynomial-time reasoning, based on Gradel's (1992) second-order Horn characterization of P. Our system has comprehension over P predicates (defined by Gradel's second-order Horn formulas), and only finitely, many function symbols. Other systems of polynomial-time reasoning either allow induction on NP predicates (such as Buss's (1986) S/sub 2//sup 1/ or the second-order V/sub 1//sup 1/), and hence are more powerful than our system (assuming the polynomial hierarchy does not collapse), or use Cobham's theorem to introduce function symbols for all polynomial-time functions (such as Cook's PV and Zambella's P-def). We prove that our system is equivalent to QPV and Zambella's (1996) P-def. Using our techniques, we also show that V/sub 1/-Horn is finitely, axiomatizable, and, as a corollary, that the class of /spl forall//spl Sigma//sub 1//sup b/ consequences of S/sub 2//sup 1/ is finitely axiomatizable as well, thus answering an open question.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".