Notions of semicomputability in topological algebras over the reals
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Bibliographic record
Abstract
Several results from classical computability theory (computability over discrete structures such as the natural numbers and strings over finite alphabets, due to Turing, Church, Kleene and others) have been shown to hold for generalisations of computability theory over total abstract algebras, usin g a computation model of a high level imperative (While) language. We present a number of results relating to computation on topological partial algebras using an abstract model of computation, While, based on high level imperative languages. We investigate the validity of several results from the classical theory in the context of topological algebras on the reals: closure of semicomputable sets under finite union, the equivalence of semicomputable and projectively (semi)computable sets, and Post’s Theorem. This research has significance in the field of scientific computation, which is underpinned by computability on the real numbers. By the Continuity Principle, computability of functions implies their continuity. Since equality, order, and other total boolean-valued functions on the reals are clearly discontinuous, we resolve this incompatibility by redefining such functions to be partial, leading us to consider topological partial algebras.
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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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.002 |
| 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