Approximate Normalization and Eager Equality Checking for Gradual Inductive Families
Bibliographic record
Abstract
Harnessing the power of dependently typed languages can be difficult. Programmers must manually construct proofs to produce well-typed programs, which is not an easy task. In particular, migrating code to these languages is challenging. Gradual typing can make dependently-typed languages easier to use by mixing static and dynamic checking in a principled way. With gradual types, programmers can incrementally migrate code to a dependently typed language. However, adding gradual types to dependent types creates a new challenge: mixing decidable type-checking and incremental migration in a full-featured language is a precarious balance. Programmers expect type-checking to terminate, but dependent type-checkers evaluate terms at compile time, which is problematic because gradual types can introduce non-termination into an otherwise terminating language. Steps taken to mitigate this non-termination must not jeopardize the smooth transitions between dynamic and static. We present a gradual dependently-typed language that supports inductive type families, has decidable type-checking, and provably supports smooth migration between static and dynamic, as codified by the refined criteria for gradual typing proposed by Siek et al. (2015). Like Eremondi et al. (2019), we use approximate normalization for terminating compile-time evaluation. Unlike Eremondi et al., our normalization does not require comparison of variables, allowing us to show termination with a syntactic model that accommodates inductive types. Moreover, we design a novel a technique for tracking constraints on type indices, so that dynamic constraint violations signal run-time errors eagerly. To facilitate these checks, we define an algebraic notion of gradual precision, axiomatizing certain semantic properties of gradual terms.
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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 | 0.030 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.004 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.004 | 0.007 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.005 | 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".