## Monotonically computable real numbers (2002)

Venue: | Math. Log. Quart |

Citations: | 8 - 5 self |

### BibTeX

@ARTICLE{Zheng02monotonicallycomputable,

author = {Xizhong Zheng and Robert Rettinger and George Barmpalias},

title = {Monotonically computable real numbers},

journal = {Math. Log. Quart},

year = {2002},

volume = {48},

pages = {2002}

}

### OpenURL

### Abstract

Key words h-monotone computable real, ω-monotone computable real.

### Citations

1280 |
On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...are called computable. Here a sequence (xs) converges effectively to x means |x − xs| ≤2−s for all s (see [10, 5, 9]). This is also equivalent to Turing’s original definition of computable numbers in =-=[13]-=- that x ∈ [0; 1] is computable if it has a computable decimal expansion, i. e., x = �∞ n=0 f(n) · 10−n for some computable function f : N −→ {0, 1,... ,9}. On the other hand, the limits of computable ... |

41 |
Nicht konstruktiv beweisbare Satze der Analysis
- Specker
- 1949
(Show Context)
Citation Context ...onable sense. In some sense, a sequence (xs) converges optimally if it converges effectively, because we have a full and effective control on the error-estimation to its limit. As observed by Specker =-=[12]-=-, even monotone sequences do not necessarily converge effectively. This leads to a weaker version of computable real numbers. We call a real number x left computable (right computable) if it is the li... |

40 |
Recursive Real Numbers
- Rice
- 1954
(Show Context)
Citation Context ...esult, only the limits of computable sequences of rational numbers which converge effectively are called computable. Here a sequence (xs) converges effectively to x means |x − xs| ≤2−s for all s (see =-=[10, 5, 9]-=-). This is also equivalent to Turing’s original definition of computable numbers in [13] that x ∈ [0; 1] is computable if it has a computable decimal expansion, i. e., x = �∞ n=0 f(n) · 10−n for some ... |

24 | Some computability-theoretic aspects of reals and randomness, in The Notre Dame Lectures
- Downey
- 2005
(Show Context)
Citation Context ...ft computable if and only if it has a c. e. left Dedekind cut Lx := {r ∈ Q : r<x}. Thus, left computable real numbers are also called computably enumerable or c. e. for short in literature (see e.g., =-=[4, 2]-=-). The difference of any two left computable real numbers is called weakly computable or d-c. e. (see Rodney G. Downey [4]). Ambos-Spies, Weihrauch and Zheng [1] showed that x is weakly computable if ... |

21 |
Weakly computable real numbers
- Ambos-Spies, Weihrauch, et al.
(Show Context)
Citation Context ...putable. Here a sequence (xs) converges effectively to x means |x − xs| ≤2−s for all s (see [10, 5, 9]). This is also equivalent to Turing’s original definition of computable numbers in [13] that x ∈ =-=[0; 1]-=- is computable if it has a computable decimal expansion, i. e., x = �∞ n=0 f(n) · 10−n for some computable function f : N −→ {0, 1,... ,9}. On the other hand, the limits of computable sequences of rat... |

15 |
Recursive approximability of real numbers
- Zheng
- 2002
(Show Context)
Citation Context ...l numbers ω-MC. On the other hand, the class ω-MC is not even contained in the class DBC, a proper superclass of WC, which is the class of all divergence bounded computable real numbers introduced in =-=[7, 15]-=-. Namely, x is divergence bounded computable means that there is a computable sequence (xs) of rational numbers and a computable function f such that, for any n, there are at most f(n) non-overlapping... |

14 |
Criteria of Constructibility for Real Numbers
- Myhill
- 1953
(Show Context)
Citation Context ...esult, only the limits of computable sequences of rational numbers which converge effectively are called computable. Here a sequence (xs) converges effectively to x means |x − xs| ≤2−s for all s (see =-=[10, 5, 9]-=-). This is also equivalent to Turing’s original definition of computable numbers in [13] that x ∈ [0; 1] is computable if it has a computable decimal expansion, i. e., x = �∞ n=0 f(n) · 10−n for some ... |

11 | Hertling Computable approximations of reals: An information-theoretic analysis
- Calude, P
(Show Context)
Citation Context ...n effective approximation (xs) with continuously improving approximation-errors. That is, |x − xn| ≥|x− xm| hold for any natural numbers n<m. As a generalization of this property, Calude and Hertling =-=[3]-=- introduced the notion of monotonical computability. According to Calude and Hertling, a sequence (xs) converges to x monotonically if there is a constant c such that (1) (∀n, m ∈ N)(n<m⇒ c ·|x − xn| ... |

11 |
Braunmühl. Weakly computable real numbers and total computable real functions
- Rettinger, Zheng, et al.
- 2001
(Show Context)
Citation Context ...l numbers ω-MC. On the other hand, the class ω-MC is not even contained in the class DBC, a proper superclass of WC, which is the class of all divergence bounded computable real numbers introduced in =-=[7, 15]-=-. Namely, x is divergence bounded computable means that there is a computable sequence (xs) of rational numbers and a computable function f such that, for any n, there are at most f(n) non-overlapping... |

6 |
On the hierarchy and extension of monotonically computable real numbers
- Rettinger, Zheng
- 2003
(Show Context)
Citation Context ...rges to x. Thus, a real number x is computable if and only if it is c-mc for some c<1 and x is semi-computable if and only if it is 1-mc. More interesting properties of c-mc real numbers are shown in =-=[8, 6]-=-. For example, the dense hierarchy theorem of c-mc real numbers holds, i. e., for any constants c1 >c2 ≥ 1 there exists a c1-mc real number which is not c2-mc. And every c-mc real number is also weakl... |

5 | A Review of
- Robinson, Jones, et al.
- 2003
(Show Context)
Citation Context ...esult, only the limits of computable sequences of rational numbers which converge effectively are called computable. Here a sequence (xs) converges effectively to x means |x − xs| ≤2−s for all s (see =-=[10, 5, 9]-=-). This is also equivalent to Turing’s original definition of computable numbers in [13] that x ∈ [0; 1] is computable if it has a computable decimal expansion, i. e., x = �∞ n=0 f(n) · 10−n for some ... |

2 |
Soare, Recursively Enumerable Sets and Degrees (Springer-Verlag
- I
- 1987
(Show Context)
Citation Context ...y f : A −→ B a total function from A to B and by f :⊆ A −→ B 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. =-=[11, 14]-=-). A (partial) function f :⊆ N −→ N is called 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 fu... |

1 | Calude, A characterization of c. e. random reals - S - 2002 |