Bibliographic record
Abstract
One fundamental question in database theory is the following: Given a Boolean conjunctive query Q , what is the best complexity for computing the answer to Q in terms of the input database size N ? When restricted to the class of combinatorial algorithms, it is known that the best known complexity for any query Q is captured by the submodular width of Q . However, beyond combinatorial algorithms, certain queries are known to admit faster algorithms that often involve a clever combination of fast matrix multiplication and data partitioning. Nevertheless, there is no systematic way to derive and analyze the complexity of such algorithms for arbitrary queries Q . In this work, we introduce a general framework that captures the best complexity for answering any Boolean conjunctive query Q using matrix multiplication. Our framework unifies both combinatorial and non-combinatorial techniques under the umbrella of information theory. It generalizes the notion of submodular width to a new stronger notion called the ω- submodular width that naturally incorporates the power of fast matrix multiplication. We describe a matching algorithm that computes the answer to any query Q in time corresponding to the ω- submodular width of Q . We show that our framework recovers the best known complexities for Boolean queries that have been studied in the literature, to the best of our knowledge, and also discovers new algorithms for some classes of queries that improve upon the best known complexities.
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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 | 0.006 | 0.034 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.001 | 0.004 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.007 | 0.027 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.006 | 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".