Speedability of computably approximable reals and their approximations

Abstract

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.

Figures (2)

• Figure 1: $a_{f_1(u)}$, $a_{f_2(u)}$, $\dots$, $a_{f_k(u)}$ jump over the interval $[a_{s+1}-c(a_{s+1} - a_s), a_{s+1}]$, thus there exists $i\in\{1,\cdots,k\}$ such that $a_{f_i(u)} < a_s$.
• Figure 2: We first find $a_u$ to be the last one to the right of or equal to $a_{s+1}$. Then find $a_{f_i(u)}$ to the left of $a_s$, and then $a_v$ the last one to the left of or equal to $a_{f_i(u)}$, and then $a_{f_{i+j}(v)}$ to the right of $a_u$. This contradicts the fact that $a_u$ is the last one to the right of or equal to $a_{s+1}$.

Theorems & Definitions (36)

• Remark 1.1
• Theorem 1.2: Solovay, Kučera and Slaman
• Theorem 1.3: Barmpalias and Lewis-Pye
• Theorem 1.4: Barmpalias and Lewis-Pye, Miller
• Definition 1.5
• Lemma 1.6: Merkle and Titov
• Remark 1.7
• Definition 1.8
• Theorem 1.9: Merkle and Titov
• Remark 1.10