Speedability of computably approximable reals and their approximations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917364518354944 |
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| author | Barmpalias, George Fang, Nan Merkle, Wolfgang Titov, Ivan |
| author_facet | Barmpalias, George Fang, Nan Merkle, Wolfgang Titov, Ivan |
| contents | An approximation of a real is a sequence of rational numbers that converges to the real. An approximation is left-c.e. if it is computable and nondecreasing and is d.c.e. if it is computable and has bounded variation. A real is computably approximable if it has some computable approximation, and left-c.e. and d.c.e. reals are defined accordingly.
An approximation $\{a_s\}_{s \in ω}$ is speedable if there exists a nondecreasing computable function $f$ such that the approximation $\{a_{f(s)}\}_{s \in ω}$ converges in a certain formal sense faster than $\{a_s\}_{s \in ω}$. This leads to various notions of speedability for reals, e.g., one may require for a computably approximable real that either all or some of its approximations of a specific type are speedable.
Merkle and Titov established the equivalence of several speedability notions for left-c.e. reals that are defined in terms of left-c.e. approximations. We extend these results to d.c.e. reals and d.c.e. approximations, and we prove that in this setting, being speedable is equivalent to not being Martin-Löf random. Finally, we demonstrate that every computably approximable real has a computable approximation that is speedable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26484 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Speedability of computably approximable reals and their approximations Barmpalias, George Fang, Nan Merkle, Wolfgang Titov, Ivan Logic Logic in Computer Science An approximation of a real is a sequence of rational numbers that converges to the real. An approximation is left-c.e. if it is computable and nondecreasing and is d.c.e. if it is computable and has bounded variation. A real is computably approximable if it has some computable approximation, and left-c.e. and d.c.e. reals are defined accordingly. An approximation $\{a_s\}_{s \in ω}$ is speedable if there exists a nondecreasing computable function $f$ such that the approximation $\{a_{f(s)}\}_{s \in ω}$ converges in a certain formal sense faster than $\{a_s\}_{s \in ω}$. This leads to various notions of speedability for reals, e.g., one may require for a computably approximable real that either all or some of its approximations of a specific type are speedable. Merkle and Titov established the equivalence of several speedability notions for left-c.e. reals that are defined in terms of left-c.e. approximations. We extend these results to d.c.e. reals and d.c.e. approximations, and we prove that in this setting, being speedable is equivalent to not being Martin-Löf random. Finally, we demonstrate that every computably approximable real has a computable approximation that is speedable. |
| title | Speedability of computably approximable reals and their approximations |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2603.26484 |