Redundancy analysis in lossless compression of a binary tree via its minimal DAG representation
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Bibliographic record
Abstract
Let T denote the set of all structurally inequivalent finite rooted ordered binary trees. For each t ϵ e T, let D(t) be the unique minimal DAG representation of t, and let r(t) ϵ (0,1] be the ratio of the number of vertices of D(t) to the number of leaves oft. A lossless prefix encoder φ on {D(t) : t ϵ T} is proposed, and then a two-step lossless encoder φ* on T is defined by φ*(t) = <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sup> φ(D(t)) for t ϵ T. Let γ be the function γ(x) = <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sup> (x/2)log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (2/x) for x ϵ (0,1]. It is shown that the normalized pointwise redundancy in encoding each t ϵ T via φ* is O(γ(r(t))). Furthermore, given a binary tree source whose output is a sequence of random trees growing in size, weak sufficient conditions on the source are presented under which the normalized average redundancy of φ* with respect to the source vanishes asymptotically. This result allows for the identification of some families of binary tree sources on which φ* acts as a universal code.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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 it