Recognizing Lexicographically Smallest Words and Computing Successors in Regular Languages
Bibliographic record
Abstract
For a formal language [Formula: see text], the problem of language enumeration asks to compute the length-lexicographically smallest word in [Formula: see text] larger than a given input [Formula: see text] (henceforth called the [Formula: see text]-successor of [Formula: see text]). We investigate this problem for regular languages from a computational complexity and state complexity perspective. We first show that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are (in general) necessary and sufficient for an unambiguous finite-state transducer to compute [Formula: see text]-successors. As a byproduct, we obtain that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are sufficient for a DFA to recognize the subset [Formula: see text] of [Formula: see text] composed of its lexicographically smallest words. We give a matching lower bound that holds even if [Formula: see text] is represented as an NFA. It has been known that [Formula: see text]-successors can be computed in polynomial time, even if the regular language is given as part of the input (assuming a suitable representation of the language, such as a DFA). In this paper, we refine this result in multiple directions. We show that if the regular language is given as part of the input and encoded as a DFA, the problem is in [Formula: see text]. If the regular language [Formula: see text] is fixed, we prove that the enumeration problem of the language is reducible to deciding membership to the Myhill-Nerode equivalence classes of [Formula: see text] under [Formula: see text]-uniform [Formula: see text] reductions. In particular, this implies that fixed star-free languages can be enumerated in [Formula: see text], arbitrary fixed regular languages can be enumerated in [Formula: see text] and that there exist regular languages for which the problem is [Formula: see text]-complete.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 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".