Bibliographic record
Abstract
Well-known theorems of Hanf and Gaifman establishing locality of first-order definable properties have been used in many applications. These theorems were recently generalized to other logics, which led to new applications in descriptive complexity and database theory. However, a logical characterization of local properties that correspond to Hanf's and Gaifman's theorems is still lacking. Such a characterization only exists for structures of bounded valence. In this paper, we give logical characterizations of local properties behind Hanf's and Gaifman's theorems. We first deal with an infinitary logic with counting terms and quantifiers that is known to capture Hanf-locality on structures of bounded valence. We show that testing isomorphism of neighborhoods can be added to it without violating Hanf-locality, while increasing its expressive power. We then show that adding local second-order quantification to it caputures precisely all Hanf-local properties. To capture Gaifman-locality, one must also add a (potentially infinite) case statement. We further show that the hierarchy based on the number of variants in the case statement is strict.
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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 | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".