Shifts of finite type obtained by forbidding a single pattern
Bibliographic record
Abstract
Given a finite word $ w $, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981,183-208) showed that the autocorrelation polynomial $ \phi_w(t) $ of $ w $, which records the set of self-overlaps of $ w $, explicitly determines for each $ n $, the number $ |\mathcal{B}_n(w)| $ of words of length $ n $ that avoid $ w $. We consider this and related problems from the viewpoint of symbolic dynamics, focusing on the setting of $ X_{\{w\}} $, the space of all bi-infinite sequences that avoid $ w $. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981,183-208) and other work to show that the sequence $ |\mathcal{B}_n(w)| $ is equivalent to several invariants of $ X_{\{w\}} $. We then give a finite-state labeled graphical representation $ L_w $ of $ X_{\{w\}} $ and show that $ w $ can be recovered from the graph isomorphism class of the unlabeled version of $ L_w $. Using $ L_w $, we apply ideas from probability and Perron-Frobenius theory to obtain results comparing features of $ X_{\{w\}} $ for different $ w $. Next, we give partial results on the problem of classifying the spaces $ X_{\{w\}} $ up to conjugacy. Finally, we extend some of our results to spaces of multi-dimensional arrays that avoid a given finite pattern.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".