Relativizing Small Complexity Classes and their Theories
Bibliographic record
Abstract
Existing definitions of the relativizations of \NCOne, Ł and \NL\ do not preserve the inclusions $\NCOne \subseteq Ł$, $\NL\subseteq \ACOne$. We start by giving the first definitions that preserve them. Here for Ł and \NL\ we define their relativizations using Wilson's stack oracle model, but limit the height of the stack to a constant (instead of $\log(n)$). We show that the collapse of any two classes in $\{\ACZm, \TCZ, \NCOne, Ł, \NL\}$ implies the collapse of their relativizations. Next we exhibit an oracle $α$ that makes $\ACk(α)$ a proper hierarchy. This strengthens and clarifies the separations of the relativized theories in [Takeuti, 1995]. The idea is that a circuit whose nested depth of oracle gates is bounded by $k$ cannot compute correctly the $(k+1)$ compositions of every oracle function. Finally we develop theories that characterize the relativizations of subclasses of \Ptime\ by modifying theories previously defined by the second two authors. A function is provably total in a theory iff it is in the corresponding relativized class, and hence the oracle separations imply separations for the relativized theories.
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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 | 0.004 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.003 | 0.010 |
| Scholarly communication | 0.005 | 0.014 |
| Open science | 0.002 | 0.008 |
| Research integrity | 0.002 | 0.009 |
| 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".