PROPERTIES OF PSEUDO-PRIMITIVE WORDS AND THEIR APPLICATIONS
Bibliographic record
Abstract
A pseudo-primitive word with respect to an antimorphic involution θ is a word which cannot be written as a catenation of occurrences of a strictly shorter word t and θ(t). Properties of pseudo-primitive words are investigated in this paper. These properties link pseudo-primitive words with essential notions in combinatorics on words such as primitive words, (pseudo)-palindromes, and (pseudo)-commutativity. Their applications include an improved solution to the extended Lyndon-Schützenberger equation u 1 u 2 ⋯ u ℓ = v 1 ⋯ v n w 1 ⋯ w m , where u 1 , … , u ℓ ∈ {u,θ(u)}, v 1 , …, v n ∈ {v, θ(v)}, and w 1 , … , w m ∈ {w, θ(w)} for some words u, v, w, integers ℓ, n, m ≥ 2, and an antimorphic involution θ. We prove that for ℓ ≥ 4, n, m ≥ 3, this equation implies that u, v, w can be expressed in terms of a common word t and its image θ(t). Moreover, several cases of this equation where ℓ = 3 are examined.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 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".