Satisfiability Threshold for Power Law Random 2-SAT in Configuration\n Model
Bibliographic record
Abstract
The Random Satisfiability problem has been intensively studied for decades.\nFor a number of reasons the focus of this study has mostly been on the model,\nin which instances are sampled uniformly at random from a set of formulas\nsatisfying some clear conditions, such as fixed density or the probability of a\nclause to occur. However, some non-uniform distributions are also of\nconsiderable interest. In this paper we consider Random 2-SAT problems, in\nwhich instances are sampled from a wide range of non-uniform distributions.\n The model of random SAT we choose is the so-called configuration model, given\nby a distribution $\\xi$ for the degree (or the number of occurrences) of each\nvariable. Then to generate a formula the degree of each variable is sampled\nfrom $\\xi$, generating several \\emph{clones} of the variable. Then 2-clauses\nare created by choosing a random paritioning into 2-element sets on the set of\nclones and assigning the polarity of literals at random.\n Here we consider the random 2-SAT problem in the configuration model for\npower-law-like distributions $\\xi$. More precisely, we assume that $\\xi$ is\nsuch that its right tail $F_{\\xi}(x)$ satisfies the conditions\n$W\\ell^{-\\alpha}\\le F_{\\xi}(\\ell)\\le V\\ell^{-\\alpha}$ for some constants $V,W$.\nThe main goal is to study the satisfiability threshold phenomenon depending on\nthe parameters $\\alpha,V,W$. We show that a satisfiability threshold exists and\nis determined by a simple relation between the first and second moments of\n$\\xi$.\n
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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 | 0.002 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 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".