Constructing Witnesses for Lower Bounds on Behavioural Distances
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Bibliographic record
Abstract
Behavioural distances provide a robust alternative to notions of equivalence such as bisimilarity in the context of probabilistic transition systems. They can be defined as least fixed points, whose universal property allows us to exhibit upper bounds on the distance between states, showing them to be at most some distance apart. In this paper, we instead consider the problem of bounding distances from below, showing states to be at least some distance apart. Contrary to upper bounds, it is possible to reason about lower bounds inductively. We exploit this by giving an inductive derivation system for lower bounds on an existing definition of behavioural distance for labelled Markov chains. This is inspired by recent work on apartness as an inductive counterpart to bisimilarity. Proofs in our system will be shown to closely match the behavioural distance by soundness and (approximate) completeness results. We further provide a constructive correspondence between our derivation system and formulas in a modal logic with quantitative semantics. This logic was used in recent work of Rady and van Breugel to construct evidence for lower bounds on behavioural distances. Our constructions provide smaller witnessing formulas in many examples.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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