## On the Turing degrees of weakly computable real numbers (1986)

Venue: | Journal of Logic and Computation |

Citations: | 7 - 3 self |

### BibTeX

@ARTICLE{Zheng86onthe,

author = {Xizhong Zheng and Theoretische Informatik},

title = {On the Turing degrees of weakly computable real numbers},

journal = {Journal of Logic and Computation},

year = {1986},

volume = {13},

pages = {2003}

}

### OpenURL

### Abstract

The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semi-computable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1

### Citations

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(Show Context)
Citation Context ..., then this correspondence is even one-to-one. Naturally, the set A can be called a binary set of the number xA and the real number xA is called a binary real number of the set A. According to Turing =-=[18]-=-, a real number x is computable if x has a computable decimal expansion, i.e., x = � i∈N f(i) · 10−i for a computable function f : N → {0, 1, 2 · · · , 9}. The computability of real numbers is in fact... |

101 |
Trial and error predicates and the solution to a problem of Mostowski
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Citation Context ...utable if there is a computable function f : N → N such that En = Df(n) for all n ∈ N. In the following, A∆B := (A \ B) ∪ (B \ A) is the symmetrical difference of sets A and B. Definition 1.3 (Putnam =-=[11]-=-, Gold [8] and Ershov [7]) Let h : N → N be any function. A set A ⊆ N is called h.c.e. if there is a computable sequence (As) of finite subsets of N such that A0 = ∅, (∀n ∈ N)(|{s ∈ N : n ∈ As+1 △ As}... |

95 |
Limiting recursion
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Citation Context ...here is a computable function f : N → N such that En = Df(n) for all n ∈ N. In the following, A∆B := (A \ B) ∪ (B \ A) is the symmetrical difference of sets A and B. Definition 1.3 (Putnam [11], Gold =-=[8]-=- and Ershov [7]) Let h : N → N be any function. A set A ⊆ N is called h.c.e. if there is a computable sequence (As) of finite subsets of N such that A0 = ∅, (∀n ∈ N)(|{s ∈ N : n ∈ As+1 △ As}| ≤ h(n)) ... |

65 |
Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets
- Soare
- 1987
(Show Context)
Citation Context ...t B and k.c.e. set C. A Turing degree is called k.c.e. (or ω.c.e.) if it contains at least one k.c.e. (or ω.c.e.) set. In recursion theory, c.e. degrees and ω.c.e. degrees have been widely discussed (=-=[17, 3, 10]-=-). For the real numbers of c.e. degrees, Dunlop and Pour-El [5] have shown an interesting characterization as follows. Theorem 1.4 A real number x has a c.e. degree a iff there is a computable sequenc... |

47 |
A minimal degree less than 0
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Citation Context ... For any non-computable c.e. Turing degree a, there is a set A ∈ a such that xA is weakly computable but not semi-computable. Proof. Let a be a non-computable c.e. degree. By Sacks’ Splitting Theorem =-=[14]-=- there exist two incomparable c.e. degrees b0, b1 such that a = b0 ∪ b1. Choose two c.e. sets B0 ∈ b0 and B1 ∈ b1 and define set A =: B0 ⊕ B1. Then, deg T (A) = deg T (B0 ⊕B1) = deg T (B0)∪deg T (B1) ... |

44 |
On degrees of unsolvability
- Shoenfield
(Show Context)
Citation Context ...to that of the real numbers straightforwardly. For example, by definition, a real number x is computable iff it has the computable degree, i.e., degT (x) = 0. Similar to the Limit Lemma of Shoenfield =-=[15]-=- for ∆ 0 2 subsets of natural numbers, Ho [9] shows that, for any real number x, x is 0 ′ -computable if and only if there is a computable sequence (xs) of rational numbers which converges to x. In th... |

24 | Some computability-theoretic aspects of reals and randomness, in The Notre Dame Lectures
- Downey
- 2005
(Show Context)
Citation Context ...following relations hold: EC = LC ∩ RC � LC RC � SC = LC ∩ RC � WC � CA Because of the item 2 of Theorem 1.2, left computable real numbers are also called computably enumerable by some authors (e.g., =-=[4, 2]-=-). In the following, we will discuss the degree properties of real numbers from above classes. Especially, we are interested in the computable enumerability and ω-computable enumerability of real numb... |

21 |
Weakly computable real numbers
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(Show Context)
Citation Context ...here are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1 Introduction For simplicity, we consider only real numbers in the unit interval =-=[0; 1]-=-. For any such real number x ∈ [0; 1], there is a set A ⊆ N + such that x = xA := � i∈A 2−i . The set A consists of all 1-positions in the binary expansion of x. If we choose the finite set A to corre... |

15 |
Recursive approximability of real numbers
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Citation Context ...equence of rational numbers. For this reason, 0 ′ -computable real numbers are called computably approximable (c.a. for short). A lot of subclasses of c.a. real numbers are investigated in literature =-=[1, 12, 19, 20]-=-. First, as mentioned above, a real number x is computable if there is a computable sequence (xs) which converges to x effectively. x is called left (right) computable if there is an increasing (decre... |

14 |
Degrees of Unsolvability
- Cooper
- 1971
(Show Context)
Citation Context ...t B and k.c.e. set C. A Turing degree is called k.c.e. (or ω.c.e.) if it contains at least one k.c.e. (or ω.c.e.) set. In recursion theory, c.e. degrees and ω.c.e. degrees have been widely discussed (=-=[17, 3, 10]-=-). For the real numbers of c.e. degrees, Dunlop and Pour-El [5] have shown an interesting characterization as follows. Theorem 1.4 A real number x has a c.e. degree a iff there is a computable sequenc... |

14 |
Relatively recursive reals and real functions
- Ho
- 1994
(Show Context)
Citation Context ... For example, by definition, a real number x is computable iff it has the computable degree, i.e., degT (x) = 0. Similar to the Limit Lemma of Shoenfield [15] for ∆ 0 2 subsets of natural numbers, Ho =-=[9]-=- shows that, for any real number x, x is 0 ′ -computable if and only if there is a computable sequence (xs) of rational numbers which converges to x. In this paper we are interested mainly in the Turi... |

14 |
Cohesive sets and recursively enumerable Dedekind cuts
- Soare
(Show Context)
Citation Context ...t A ∈ a. Obviously, xA is a left computable real number. � Notice that, if we consider the binary expansion of a real number, the item 1 of Proposition 1.5 seems not so trivial, because Jockusch (see =-=[16]-=-) has observed that not every left computable real number has a c.e. binary representation. And in general, it can only be shown that, if xA is a left computable real number, then A is a h.c.e. set fo... |

13 | A characterization of c.e. random reals - Calude |

13 |
A certain hierarchy of sets
- Ershov
- 1970
(Show Context)
Citation Context ...table function f : N → N such that En = Df(n) for all n ∈ N. In the following, A∆B := (A \ B) ∪ (B \ A) is the symmetrical difference of sets A and B. Definition 1.3 (Putnam [11], Gold [8] and Ershov =-=[7]-=-) Let h : N → N be any function. A set A ⊆ N is called h.c.e. if there is a computable sequence (As) of finite subsets of N such that A0 = ∅, (∀n ∈ N)(|{s ∈ N : n ∈ As+1 △ As}| ≤ h(n)) and A = lims→∞ ... |

8 | Hierarchy of the monotonically computable real numbers
- Rettinger, Zheng
(Show Context)
Citation Context ...equence of rational numbers. For this reason, 0 ′ -computable real numbers are called computably approximable (c.a. for short). A lot of subclasses of c.a. real numbers are investigated in literature =-=[1, 12, 19, 20]-=-. First, as mentioned above, a real number x is computable if there is a computable sequence (xs) which converges to x effectively. x is called left (right) computable if there is an increasing (decre... |

7 |
A finite hierarchy of the recursively enumerable real numbers
- Weihrauch, Zheng
- 1998
(Show Context)
Citation Context ...y expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semi-computable and weakly computable real numbers introduced by Weihrauch and Zheng =-=[19]-=-. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1 Introduction For simplicity, we consider only real... |

6 |
Review of “Peter, R., Rekursive Funktionen
- Robinson
- 1951
(Show Context)
Citation Context ...expansion, i.e., x = � i∈N f(i) · 10−i for a computable function f : N → {0, 1, 2 · · · , 9}. The computability of real numbers is in fact independent of their representations as observed by Robinson =-=[13]-=-. In other words, x is computable, if and only if x has a computable Dedekind cut Lx := {r ∈ Q : r < x}, if and only if the binary set of x is recursive and if and only if there is a computable sequen... |

3 |
The degree of unsolvability of a real number
- Dunlop, Pour-El
- 2001
(Show Context)
Citation Context ...d by x ≡T y) if x ≤T y & y ≤T x. Accordingly, the Turing degree degT (x) of a real number x is the class of real numbers which are Turing equivalent to x, i.e., degT (x) := {y ∈ R : x ≡T y} (see also =-=[21, 5]-=-). Because of the corresponding between the set A and the real number xA, we do not distinguish the degrees degT (xA) and degT (A) explicitly in this paper. This should not cause confusion from the co... |

3 |
A certain hierarchy of sets. i, ii, iii
- Ershov
- 1968
(Show Context)
Citation Context ...mber of c.e. degree which is not weakly computable. 4sProof. By Theorem 1.6.2, it suffices to show that there is a subset A ⊆ N of c.e. degree such that A is not ω.c.e. By Hierarchy Theorem of Ershov =-=[6]-=-, there is a ∆0 2-set B ⊆ N which is not ω.c.e. Let A := B ⊕K where K is the halting problem. Then A is obviously not ω.c.e. too. Since A ⊕ ∅ ≡T A ≡T K, the real number xA⊕∅ has the c.e. degree 0 ′ . ... |

3 |
Classical Recursion Theory, volume 129
- Odifreddi
- 1989
(Show Context)
Citation Context ...t B and k.c.e. set C. A Turing degree is called k.c.e. (or ω.c.e.) if it contains at least one k.c.e. (or ω.c.e.) set. In recursion theory, c.e. degrees and ω.c.e. degrees have been widely discussed (=-=[17, 3, 10]-=-). For the real numbers of c.e. degrees, Dunlop and Pour-El [5] have shown an interesting characterization as follows. Theorem 1.4 A real number x has a c.e. degree a iff there is a computable sequenc... |

2 |
On the definition of degrees of unsolvability for reals
- Zheng, Ding, et al.
- 1993
(Show Context)
Citation Context ...d by x ≡T y) if x ≤T y & y ≤T x. Accordingly, the Turing degree degT (x) of a real number x is the class of real numbers which are Turing equivalent to x, i.e., degT (x) := {y ∈ R : x ≡T y} (see also =-=[21, 5]-=-). Because of the corresponding between the set A and the real number xA, we do not distinguish the degrees degT (xA) and degT (A) explicitly in this paper. This should not cause confusion from the co... |