Complete and tractable machine-independent characterizations of second-order polytime
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Bibliographic record
Abstract
Abstract The class of Basic Feasible Functionals $$\mathtt{BFF}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>BFF</mml:mi> </mml:math> is the second-order counterpart of the class of first-order functions computable in polynomial time. We present several implicit characterizations of $$\mathtt{BFF}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>BFF</mml:mi> </mml:math> based on a typed programming language of terms. These terms may perform calls to imperative procedures, which are not recursive. The type discipline has two layers: the terms follow a standard simply-typed discipline and the procedures follow a standard tier-based type discipline. $$\mathtt{BFF}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>BFF</mml:mi> </mml:math> consists exactly of the second-order functionals that are computed by typable and terminating programs. The completeness of this characterization surprisingly still holds in the absence of lambda-abstraction. Moreover, the termination requirement can be specified as a completeness-preserving instance, which can be decided in time quadratic in the size of the program. As typing is decidable in polynomial time, we obtain the first tractable ( i.e. , decidable in polynomial time), sound, complete, and implicit characterization of $$\mathtt{BFF}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>BFF</mml:mi> </mml:math> , thus solving a problem opened for more than 20 years.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| 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