Speedability of computably approximable reals and their approximations
This work addresses foundational questions in computability theory for researchers in mathematical logic and theoretical computer science, but it is incremental as it builds on prior results.
The paper extends speedability notions from left-c.e. reals to d.c.e. reals, proving equivalence to non-Martin-Löf randomness and showing every computably approximable real has a speedable approximation.
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 $\{a_s\}_{s \in Ï}$ is speedable if there exists a nondecreasing computable function $f$ such that the approximation $\{a_{f(s)}\}_{s \in Ï}$ converges in a certain formal sense faster than $\{a_s\}_{s \in Ï}$. 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.