Bibliographic record
Abstract
As we have seen (Theorem 5.4.2), every ultimately periodic sequence is k -automatic for all integers k ≥ 2. In this chapter we prove a beautiful and deep theorem due to Cobham, which states that if a sequence s = ( s ( n )) n ≥0 is both k -automatic and l -automatic and k and l are multiplicatively independent, then S is ultimately periodic. (Recall that Theorem 2.5.7 discusses when two integers are multiplicatively independent.) Syndetic and Right Dense Sets In this section, we prove some useful preliminary results. We say that a set X ⊆ Σ* is right dense if for any word u ∈ Σ* there exists a υ ∈ Σ* such that u υ ∈ X (that is, any word appears as a prefix of some word in X ). Lemma 11.1.1 Let k, l ≥ 2 be multiplicatively independent integers, and let X be an infinite k-automatic set of integers . Then 0*( X ) l = 0*{( n ) l : n ∈ X } is right dense . Proof . Since X is infinite and k -automatic, by the pumping lemma there exist strings t , u , υ with u nonempty such that tu *υ ⊆ ( X ) k . Let x ∈ {0, 1, …, l — 1}*. Our goal is to construct y such that xy ∊ 0*( X ) l .
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.015 | 0.004 |
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".