Bibliographic record
Abstract
Computable analysis studies problems involving real numbers, sets and functions from the viewpoint of computability. Elements of uncountable sets (such as real numbers) are represented through approximation and processed by Turing machines. However, application of this approach to computational complexity has been limited in generality. In this thesis, we present a refined framework that is suitable for discussing computational complexity. The key idea is to use (a certain class of) string functions as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An important advantage of using string functions is that we can define their size in the way inspired by higher-type complexity theory. This enables us to talk about computation on string functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions. Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of several problems whose inputs and outputs are real numbers, real sets, and real functions. The latter two cannot be represented succinctly using existing approaches based on infinite sequences, so ours is the first treatment of functions on them. As an interesting example, the task of numerical algorithms for solving the initial value problem of differential equations is naturally viewed as an operator taking real functions to real functions. Because there was no complexity theory for operators, previous results could only state how complex the solution can be. We now reformulate them and show that the operator itself is polynomial-space complete. We survey some of such complexity results involving real numbers and cast them in our framework.
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.010 |
| Scholarly communication | 0.005 | 0.009 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.006 | 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 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".