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## On the Monotonic Computability of Semi-Computable Real Numbers

### Citations

1627 |
On computable numbers, with an application to the entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...ally, we show a sufficient and necessary condition for the function h such that the h-monotonic computability is simply equivalent to the normal computability. 1 Introduction According to Alan Turing =-=[12]-=-, a real number x ∈ [0; 1] 1 is called computable if its decimal expansion is computable, i.e., x = � n∈N f(n) · 10−n for some computable function f : N → {0, 1, · · · , 9}. Let EC denote the class of... |

75 |
Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets
- Soare
- 1987
(Show Context)
Citation Context ...A → B denote a total function from A to B while f :⊆ A → B is a partial function with dom(f) ⊆ A and range(f) ⊆ B. We assume only very basic background on the classical computability theory (cf. e.g. =-=[10, 13]-=-). A function f :⊆ N → N is called (partial) computable if there is a Turing machine which computes f. Suppose that (Me) is an effective enumeration of all Turing machines. Let ϕe :⊆ N → N be the func... |

48 |
Nicht konstruktiv beweisbare Sätze der Analysis
- Specker
- 1949
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Citation Context ..., a computable real number can be effectively approximated with an effective error estimation. This effective error estimation is very essential for the computability of a real number because Specker =-=[11]-=- has shown that there is a computable increasing sequence of rational numbers which converges to a non-computable real number. The limit of a computable increasing sequence of rational numbers can be ... |

22 |
Computability, volume 9
- Weihrauch
- 1987
(Show Context)
Citation Context ...A → B denote a total function from A to B while f :⊆ A → B is a partial function with dom(f) ⊆ A and range(f) ⊆ B. We assume only very basic background on the classical computability theory (cf. e.g. =-=[10, 13]-=-). A function f :⊆ N → N is called (partial) computable if there is a Turing machine which computes f. Suppose that (Me) is an effective enumeration of all Turing machines. Let ϕe :⊆ N → N be the func... |

21 |
Weakly computable real numbers
- Ambos-Spies, Weihrauch, et al.
(Show Context)
Citation Context ...nt and necessary condition for the function h such that the h-monotonic computability is simply equivalent to the normal computability. 1 Introduction According to Alan Turing [12], a real number x ∈ =-=[0; 1]-=- 1 is called computable if its decimal expansion is computable, i.e., x = � n∈N f(n) · 10−n for some computable function f : N → {0, 1, · · · , 9}. Let EC denote the class of all computable real numbe... |

19 | Some Computability-Theoretical Aspects of Reals and Randomness
- Downey
- 2005
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Citation Context ... an n ∈ N such that y = x ± n and x, y are considered to have the same type of effectiveness. 2 Some authors use the notion computably enumerable (c.e. for short) instead of left computable. See e.g. =-=[2, 4]-=-.s2 Xizhong Zheng and George Barmpalias numbers are called semi-computable. The classes of all right and semi-computable real numbers is denoted by RC and SC, respectively. Notice that, for any semico... |

15 |
Recursive approximability of real numbers
- Zheng
- 2002
(Show Context)
Citation Context ...e if its decimal expansion is computable, i.e., x = � n∈N f(n) · 10−n for some computable function f : N → {0, 1, · · · , 9}. Let EC denote the class of all computable real numbers. Equivalently (see =-=[9, 5, 14]-=-), x is computable if and only if its Dedekind cut is computable and if and only if there is a computable sequence (xs) of rational numbers which converges to x effectively in the sense that |x−xn| ≤ ... |

13 | A characterization of c.e. random reals
- Calude
(Show Context)
Citation Context ...l number converges computably. More generally, Rettinger, Zheng, Gengler and von Braunmühl [8, 6] extended the condition (1) further to the following (∀n, m ∈ N)(n < m =⇒ h(n) · |x − xn| ≥ |x − xm|), =-=(2)-=- where h : N → Q is a function. That is, the ratios of error estimations are bounded by the function h. In this case, we call the sequence (xs) converges to x h-monotonically. A real number x is calle... |

12 |
Criteria of constructibility for real numbers
- Myhill
- 1953
(Show Context)
Citation Context ...e if its decimal expansion is computable, i.e., x = � n∈N f(n) · 10−n for some computable function f : N → {0, 1, · · · , 9}. Let EC denote the class of all computable real numbers. Equivalently (see =-=[9, 5, 14]-=-), x is computable if and only if its Dedekind cut is computable and if and only if there is a computable sequence (xs) of rational numbers which converges to x effectively in the sense that |x−xn| ≤ ... |

10 |
Braunmühl. Weakly computable real numbers and total computable real functions
- Rettinger, Zheng, et al.
- 2001
(Show Context)
Citation Context ...ch converges to x and a computable function g : N → N such that, for any n ∈ N, the numbers ofsMonotonic Computability 3 non-overlapped index pair (i, j) with |xi − xj| ≥ 2 −n is bounded by g(n) (see =-=[7, 14]-=- for the details). Besides, a dense hierarchy for k-mc real numbers is also shown there. The main results of [8, 6] are summarized as follows. Theorem 1.1 (Rettinger, Zheng, Gengler and von Braunmühl ... |

9 | Computable approximations of reals: An information-theoretic analysis
- Calude, Hertling
- 1998
(Show Context)
Citation Context ...≤ 1/2. Then the computable sequence (ys) defined by ys := xk0s converges effectively to x (remember x, x0 ∈ [0; 1] and hence |x − x0| ≤ 1) and hence x is a computable real number. Calude and Hertling =-=[3]-=- discussed the condition (1) for more general case, namely, without the restriction of 0 < c < 1. They call a sequence (xs) monotonically convergent if there is a constant c > 0 such that (1) holds. F... |

6 | Hierarchy of the monotonically computable real numbers
- Rettinger, Zheng
(Show Context)
Citation Context ...at |x−xs| ≤ 2 −n for any s ≥ g(n), although not every computable sequence which converges to a computable real number converges computably. More generally, Rettinger, Zheng, Gengler and von Braunmühl =-=[8, 6]-=- extended the condition (1) further to the following (∀n, m ∈ N)(n < m =⇒ h(n) · |x − xn| ≥ |x − xm|), (2) where h : N → Q is a function. That is, the ratios of error estimations are bounded by the fu... |

5 |
Review of “Peter, R., Rekursive Funktionen
- Robinson
- 1951
(Show Context)
Citation Context ...e if its decimal expansion is computable, i.e., x = � n∈N f(n) · 10−n for some computable function f : N → {0, 1, · · · , 9}. Let EC denote the class of all computable real numbers. Equivalently (see =-=[9, 5, 14]-=-), x is computable if and only if its Dedekind cut is computable and if and only if there is a computable sequence (xs) of rational numbers which converges to x effectively in the sense that |x−xn| ≤ ... |