Notions of semicomputability in topological algebras over the reals
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.012 |
| Scholarly communication | 0.005 | 0.012 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 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 source (direct Gemma or distilled Codex), 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".