Derandomizing polynomial identity tests means proving circuit lower bounds
Bibliographic record
Abstract
We show that derandomizing Polynomial Identity Testing is, essentially, equivalent to proving circuit lower bounds for NEXP. More precisely, we prove that if one can test in polynomial time (or, even, nondeterministic subexponential time, infinitely often) whether a given arithmetic circuit over integers computes an identically zero polynomial, then either (i) NEXP ⊄ P/poly or (ii) Permanent is not computable by polynomial-size arithmetic circuits. We also prove a (partial) converse: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic formula computes an identically zero polynomial.Since Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If RP=P (or, even, coRP⊆ ∩ε > 0NTIME(2(nε)), infinitely often), then NEXP is not computable by polynomial-size arithmetic circuits. Thus, establishing that RP=coRP or BPP=P would require proving superpolynomial lower bounds for Boolean or arithmetic circuits. We also show that any derandomization of RNC would yield new circuit lower bounds for a language in NEXP.
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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 | 0.040 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.005 | 0.011 |
| Open science | 0.004 | 0.006 |
| Research integrity | 0.002 | 0.008 |
| Insufficient payload (model declined to judge) | 0.007 | 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".