Bibliographic record
Abstract
Abstract We study the space complexity of refuting unsatisfiable randomk‐CNFs in the Resolution proof system. We prove that for Δ ≥ 1 and any ϵ > 0, with high probability a randomk‐CNF overnvariables and Δnclauses requires resolution clause space of Ω(n/Δ1+ϵ). For constant Δ, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3‐CNFs with clause density Δ ≫n. This bound is nearly tight. Specifically, we show that with high probability, a random 3‐CNF with Δnclauses requires treelike refutation size of exp(Ω(n/Δ1+ϵ)), for any ϵ > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2‐player Matching Game on bipartite graphsGto prove that there are no perfect matchings inG. (2) We reduce lower bounds for the clause space of a formulaFin Resolution to lower bounds for the complexity of the game played on the bipartite graphG(F) associated withF. (3) We prove that the complexity of the game is large wheneverGis an expander graph. Finally, a simple probabilistic analysis shows that for a random formulaF, with high probabilityG(F) is an expander. We also extend our result to the case ofG‐PHP, a generalization of the Pigeonhole principle based on bipartite graphsG. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 92–109, 2003
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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 | 0.011 | 0.071 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.005 | 0.011 |
| Open science | 0.004 | 0.004 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 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".