State Complexity of the Quotient Operation on Input-Driven Pushdown Automata
Bibliographic record
Abstract
The quotient of a formal language [Formula: see text] by another language [Formula: see text] is the set of all strings obtained by taking a string from [Formula: see text] that ends with a suffix of a string from [Formula: see text], and removing that suffix. The quotient of a regular language by any language is always regular, whereas the context-free languages and many of their subfamilies, such as the linear and the deterministic languages, are not closed under the quotient operation. This paper establishes the closure of the family of languages recognized by input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the quotient operation. A construction of automata representing the result of the operation is given, and its state complexity with respect to nondeterministic IDPDA is shown to be exactly [Formula: see text], where [Formula: see text] and [Formula: see text] are the numbers of states in the automata recognizing [Formula: see text] and [Formula: see text], respectively.
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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 | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.004 | 0.001 |
| 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".