Logics with aggregate operators
Bibliographic record
Abstract
We study adding aggregate operators, such as summing up elements of a column of a relation, to logics with counting mechanisms. The primary motivation comes from database applications, where aggregate operators are present in all real life query languages. Unlike other features of query languages, aggregates are not adequately captured by the existing logical formalisms. Consequently, all previous approaches to analyzing the expressive power of aggregation were only capable of producing partial results, depending on the allowed class of aggregate and arithmetic operations.We consider a powerful counting logic, and extend it with the set of all aggregate operators. We show that the resulting logic satisfies analogs of Hanf's and Gaifman's theorems, meaning that it can only express local properties. We consider a database query language that expresses all the standard aggregates found in commercial query languages, and show how it can be translated into the aggregate logic, thereby providing a number of expressivity bounds, that do not depend on a particular class of arithmetic functions, and that subsume all those previously known. We consider a restricted aggregate logic that gives us a tighter capture of database languages, and also use it to show that some questions on expressivity of aggregation cannot be answered without resolving some deep problems in complexity theory.
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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 | 0.004 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.013 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 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".