HoTTSQL: proving query rewrites with univalent SQL semantics
Bibliographic record
Abstract
Every database system contains a query optimizer that performs query rewrites. Unfortunately, developing query optimizers remains a highly challenging task. Part of the challenges comes from the intricacies and rich features of query languages, which makes reasoning about rewrite rules difficult. In this paper, we propose a machine-checkable denotational semantics for SQL, the de facto language for relational database, for rigorously validating rewrite rules. Unlike previously proposed semantics that are either non-mechanized or only cover a small amount of SQL language features, our semantics covers all major features of SQL, including bags, correlated subqueries, aggregation, and indexes. Our mechanized semantics, called HoTT SQL, is based on K-Relations and homotopy type theory, where we denote relations as mathematical functions from tuples to univalent types. We have implemented HoTTSQL in Coq, which takes only fewer than 300 lines of code and have proved a wide range of SQL rewrite rules, including those from database research literature (e.g., magic set rewrites) and real-world query optimizers (e.g., subquery elimination). Several of these rewrite rules have never been previously proven correct. In addition, while query equivalence is generally undecidable, we have implemented an automated decision procedure using HoTTSQL for conjunctive queries: a well studied decidable fragment of SQL that encompasses many real-world queries.
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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 | 0.010 | 0.040 |
| Meta-epidemiology (narrow) | 0.001 | 0.002 |
| Meta-epidemiology (broad) | 0.001 | 0.005 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.006 | 0.011 |
| Open science | 0.005 | 0.007 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.007 | 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".