Bibliographic record
Abstract
We revisit the basic concept of quotient of a regular language [Formula: see text] by a language [Formula: see text] that is not necessarily regular. We revise the deterministic complexity upper bounds of the quotient operation, and we also address the nondeterministic case. Specifically, the revised deterministic upper bound is shown to be more accurate for all hard streams of regular languages. When both languages [Formula: see text] and [Formula: see text] are regular, we give algorithms to construct their quotient in four ways. If the two languages are given via NFAs, the first algorithm produces an NFA for the desired quotient. If the two languages are given via regular expressions, we present an algorithm that produces a regular expression for the quotient both using ordinary derivatives and using partial derivatives. Finally, we consider regular expressions with a quotient operator and define the set of partial derivatives for regular expressions with this operator. Thus, using the partial derivative automaton, one can directly produce an NFA for the quotient. We have implemented all algorithms and present here experimental results.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.004 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".