Exponential bounds for DPLL below the satisfiability threshold
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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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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 it