Exponential bounds for DPLL below the satisfiability threshold
Bibliographic record
Abstract
Abstract For each k> = 4, we give rk> 0 such that a random k-CNF formula F with n variables and brknc clausesis satisfiable with high probability, but ordered-dlltakes exponential time on F with uniformly positiveprobability. Using results of [2], this can be strengthened to a high probability result for certain natu-ral backtracking schemes and extended to many other DPLL algorithms. 1 Previous work In the last twenty years a significant amount of workhas been devoted to the study of randomly generated satisfiability instances and the performance of differentalgorithms on them. Historically, a major motivation for studying random instances has been the desire tounderstand the hardness of "typical " instances. Indeed, some of the better practical ideas in use today comefrom insights gained by studying the performance of algorithms on random k-SAT instances (defined below).Let Ck(n) denote the set of all possible disjunctionsof k distinct, non-complementary literals (k-clauses)from some canonical set of n Boolean variables. A ran-dom k-CNF formula Fk(n, m) is formed by selecting uni-formly, independently, and with replacement m clausesfrom Ck(n) and taking their conjunction. We will saythat a sequence of random events E n occurs with highprobability (w.h.p.) if lim n!1 Pr[En] = 1 and with uni-formly positive probability if lim inf n!1 Pr[En]> 0.It is widely believed that for each k> = 3, thereexists a constant ck such that Fk(n, m = cn) is w.h.p.satisfiable if c < ck and w.h.p. unsatisfiable if c> ck.Currently, the best general bounds are 2 k ln 2- O(k) < ck < 2k ln 2- O(1) , where by ck < c we mean that Fk(n, cn) is w.h.p. unsatisfiable (analogously for ck> c).Let res(F) denote the size of the minimal resolutionrefutation of a formula F (we define res(F) to be infinitewhen F is satisfiable). A celebrated result of Chv'ataland Szemer'edi [5] asserts that for all k> = 3 and every
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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 | 0.014 | 0.075 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.006 | 0.017 |
| Open science | 0.006 | 0.008 |
| Research integrity | 0.003 | 0.009 |
| Insufficient payload (model declined to judge) | 0.015 | 0.002 |
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".