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 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 | 0.001 |
| Open science | 0.001 | 0.001 |
| 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".