VTC/sup O/: a second-order theory for TC/sup 0/
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We introduce a finitely axiomatizable second-order theory, which is VTC/sup 0/ associated with the class FO-uniform TC/sup 0/. It consists of the base theory V/sup 0/ for AC/sup 0/ reasoning 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 up to (excluding) bit i of X. We introduce the notion of "strong /spl Delta//sub 1//sup B/-definability" for relations in a theory, and use a recursive characterization of the TC/sup 0/ relations (rather than functions) to show that the TC/sup 0/ relations are strongly /spl Delta//sub 1//sup B/-definable. It follows that the TC/sup 0/ functions are /spl Sigma//sub 1//sup B/-definable in VTC/sup 0/. We prove a general witnessing theorem for second-order theories and conclude that the/spl Sigma//sub 1//sup B/ theorems of VTC/sup 0/ are witnessed by TC/sup 0/ functions. We prove that VTC/sup 0/ is RSUV isomorphic to the first order theory /spl Delta//sub 1//sup b/-CR of Johannsen and Pollett (the "minimal theory for TC/sup 0/"), /spl Delta//sub 1//sup b/-CR includes the /spl Delta//sub 1//sup b/ comprehension rule, and J and P ask whether there is an upper bound to the nesting depth required for this rule. We answer "yes", because VTC/sup 0/ , and therefore /spl Delta//sub 1//sup b/-CR, are finitely axiomatizable. Finally, we show that /spl Sigma//sub 1//sup B/ theorems of VTC/sup 0/ translate to families of tautologies which have polynomial-size constant-depth TC/sup 0/-Frege proofs. We also show that PHP is a /spl Sigma//sub 0//sup B/ theorem of VTC/sup 0/. These together imply that the family of propositional tautologies associated with PHP has polynomial-size constant-depth TC/sup 0/-Frege proofs.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 it