Abstract versus concrete computation on metric partial algebras
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Bibliographic record
Abstract
In the theory of computation on topological algebras there is a considerable gap between so-called abstract and concrete models of computation. In concrete models, unlike abstract models, the computations depend on the representation of the algebra. First, we show that with abstract models, one needs algebras with <i>partial operations</i>, and computable functions that are both <i>continuous</i> and <i>many-valued</i>. This many-valuedness is needed even to compute single-valued functions, and so <i>abstract models must be nondeterministic even to compute deterministic problems</i>. As an abstract model, we choose the "while"-array programming language, extended with a nondeterministic "countable choice" assignment, called the <i><b>WhileCC*</b></i> model. Using this, we introduce the concept of <i>approximable many-valued computation</i> on metric algebras. For our concrete model, we choose metric algebras with <i>effective representations</i>. We prove:(1) for any metric algebra <i>A</i> with an effective representation α, <i><b>WhileCC*</b></i> approximability implies computability in α, and (2) also the converse, under certain reasonable conditions on <i>A</i>. From (1) and (2) we derive an equivalence theorem between abstract and concrete computation on metric partial algebras. We give examples of algebras where this equivalence holds.
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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 | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it