Dynamic Boolean Formula Evaluation.
Bibliographic record
Abstract
We present a linear space data structure for Dynamic Evaluation of k-CNF Boolean Formulas which achieves O(m^{1-1/k}) query and variable update time where m is the number of clauses in the formula and clauses are of size at most a constant k. Our algorithm is additionally able to count the total number of satisfied clauses. We then show how this data structure can be parallelized in the PRAM model to achieve O(log m) span (i.e. parallel time) and still O(m^{1-1/k}) work. This parallel algorithm works in the stronger Binary Fork model. We then give a series of lower bounds on the problem including an average-case result showing the lower bounds hold even when the updates to the variables are chosen at random. Specifically, a reduction from k-Clique shows that dynamically counting the number of satisfied clauses takes time at least n^{(2ω-3)/6 √{2k} -1 -o(√k)}, where 2 ≤ ω < 2.38 is the matrix multiplication constant. We show the Combinatorial k-Clique Hypothesis implies a lower bound of m^{(1-k^{-1/2})(1-o(1))} which suggests our algorithm is close to optimal without involving Matrix Multiplication or new techniques. We next give an average-case reduction to k-clique showing the prior lower bounds hold even when the updates are chosen at random. We use our conditional lower bound to show any Binary Fork algorithm solving these problems requires at least Ω(log m) span, which is tight against our algorithm in this model. Finally, we give an unconditional linear space lower bound for Dynamic k-CNF Boolean Formula Evaluation.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.004 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.022 | 0.004 |
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".