Truly Supercritical Trade-Offs for Resolution, Cutting Planes, Monotone Circuits, and Weisfeiler–Leman
Bibliographic record
Abstract
We exhibit supercritical trade-off for monotone circuits, showing that there are functions computable by small circuits for which any small circuit must have depth superlinear or even super-polynomial in the number of variables, far exceeding the linear worst-case upper bound. We obtain similar trade-offs in proof complexity, where we establish the first size-depth trade-offs for cutting planes and resolution that are truly supercritical, i.e., in terms of formula size rather than number of variables, and also show supercritical trade-offs between width and size for treelike resolution. Our results build on a new supercritical width-depth trade-off for resolution, obtained by refining and strengthening the compression scheme for the cop-robber game in [Grohe, Lichter, Neuen & Schweitzer 2023]. This yields robust supercritical trade-offs for dimension versus iteration number in the Weisfeiler-Leman algorithm, which also translate into trade-offs between number of variables and quantifier depth in first-order logic. Our other results follow from improved lifting theorems that might be of independent interest.
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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 | 0.000 | 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".