## Generalized high degrees have the complementation property

Venue: | Journal of Symbolic Logic |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Greenberg_generalizedhigh,

author = {Noam Greenberg and Antonio Montalbán and Richard and A. Shore},

title = {Generalized high degrees have the complementation property},

journal = {Journal of Symbolic Logic},

year = {},

volume = {69},

pages = {2004}

}

### OpenURL

### Abstract

Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order theoretic properties of a degree and its complexity of definition in arithmetic as expressed by the Turing jump operator which embodies a single step in the hierarchy of quantification. For example, there is a long history of results showing that 0 ′

### Citations

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- Lerman
- 1983
(Show Context)
Citation Context ... ) in Posner and Robinson [PR81] that proof was indirect (see [Ler83, IV.15]) and based on a nonuniform proof of the join theorem for GH1 (see [Ler83, IV.9]). It was also explicitly left open even in =-=[Ler83]-=- if the degree providing the join could be made GL1. This was answered by Lerman [Ler85] but again nonuniformly. We supply a uniform proof for this sharper version of the join theorem for GH1 which we... |

53 |
Classes of recursively enumerable sets and degrees of unsolvability
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Citation Context ...utcome and act appropriately. Case 2: In this case, a is no longer in GL2, hence d cannot approximate questions regarding totality. However, as was noted by Jockusch and Posner [JP78] (using Martin’s =-=[Mar66]-=- characterization of the domination properties of H1), this loss is offset by a’s ability to approximate bounds for searches conducted by 0 ′ : for every function f ≤T 0 ′ (even a ∨ 0 ′ ) there is a f... |

45 |
A minimal degree less than 0
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(Show Context)
Citation Context ... that 0 ′ has many special order theoretic properties. To cite just a few: every countable partial order can be embedded below 0 ′ (Kleene and Post [KP54]); there are minimal degrees below 0 ′ (Sacks =-=[Sac61]-=-); 0 ′ cups to every degree above it (and so has the cupping property)(Friedberg [Fri57]); every degree below 0 ′ joins up to 0 ′ (and so has the join property)(Robinson [Rob72], Posner and Robinson [... |

37 |
The upper semi-lattice of degrees of recursive unsolvability
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(Show Context)
Citation Context ... example, there is a long history of results showing that 0 ′ has many special order theoretic properties. To cite just a few: every countable partial order can be embedded below 0 ′ (Kleene and Post =-=[KP54]-=-); there are minimal degrees below 0 ′ (Sacks [Sac61]); 0 ′ cups to every degree above it (and so has the cupping property)(Friedberg [Fri57]); every degree below 0 ′ joins up to 0 ′ (and so has the j... |

34 | Interpretability and definability in the Recursively Enumerable Degrees
- Nies, Shore, et al.
- 1998
(Show Context)
Citation Context ...ferent methods involving coding models of arithmetic and other arguments by Shore and Slaman [SS99] as have all of the high/low classes in D(≤ 0 ′ ) with the exception of L1 by Nies, Shore and Slaman =-=[NSS98]-=-. The dream of a natural definition for any of these classes based on such order theoretic properties, however, still persists and can only be realized by investigations such as these. Moreover, the a... |

30 |
Pseudojump operators. II. Transfinite iterations, hierarchies and minimal covers
- Shore
- 1984
(Show Context)
Citation Context ... D) that codes D and to have the choices we make at each stage depend on A, so that the construction itself (and hence D) can be recovered from A ⊕ B. It combines the key idea of Jockusch and Shore’s =-=[JS84]-=- proof of the Posner-Robinson join theorem for 0 ′ with Jockusch’s [Joc77] use of the recursion theorem with degreess4 NOAM GREENBERG, ANTONIO MONTALBÁN, AND RICHARD A. SHORE in GH1. This approach is ... |

27 |
A criterion for completeness of degrees of unsolvability
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- 1957
(Show Context)
Citation Context ...le partial order can be embedded below 0 ′ (Kleene and Post [KP54]); there are minimal degrees below 0 ′ (Sacks [Sac61]); 0 ′ cups to every degree above it (and so has the cupping property)(Friedberg =-=[Fri57]-=-); every degree below 0 ′ joins up to 0 ′ (and so has the join property)(Robinson [Rob72], Posner and Robinson [PR81]). It was often hoped that some such property would distinguish either 0 ′ or some ... |

25 |
Minimal degrees and the jump operator
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- 1973
(Show Context)
Citation Context ... (n−1) . (We take GL0 = {0}.) All of the results mentioned above for d ≤T 0 ′ , for example, are true for all degrees as long as we use the generalized hierarchy. (Jockusch and Posner [JP78]; (Cooper =-=[Coo73]-=-) for H1 and Jockusch [Joc77] for GH1; Jockusch and Posner [JP78]; Posner [Pos77].) Once again approximations, rates of growth and domination properties play prominent roles in the constructions. It w... |

25 |
Degrees joining to 0
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- 1981
(Show Context)
Citation Context ...]); 0 ′ cups to every degree above it (and so has the cupping property)(Friedberg [Fri57]); every degree below 0 ′ joins up to 0 ′ (and so has the join property)(Robinson [Rob72], Posner and Robinson =-=[PR81]-=-). It was often hoped that some such property would distinguish either 0 ′ or some class of degrees closely related to it. For degrees below 0 ′ , the notion of being close to 0 ′ (or 0 at the other e... |

19 |
Double jumps of minimal degrees
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- 1978
(Show Context)
Citation Context ...e high/low hierarchy. The sharpness of the first follows from the existence of a minimal degree a ∈ L2−L1: There is a minimal a < 0 ′ not in L1 by Sasso [Sas74] and it is in L2 by Jockusch and Posner =-=[JP78]-=-. This minimal degree is easily seen to be weakly recursive in the sense of Ishmukhametov [Ish99] and then, by that work, has a strong minimal cover b, i.e. {0, a, b} is an initial segment of the degr... |

19 | Definability in the Turing degrees
- Slaman, Woodin
- 1986
(Show Context)
Citation Context ...ery nonrecursive a < 0 ′ have a 1-generic complement? 3. Does every nonrecursive a < 0 ′ have a complement of minimal degree? 4. Does every d ∈ GH1 have the complementation property? Slaman and Steel =-=[SS89]-=- answered questions 1 and 2 simultaneously by providing a uniform proof that every nonrecursive a < 0 ′ has a 1-generic complement. Seetapun and Slaman circulated a sketch of a proposed affirmative an... |

16 |
The degree of hyperimmune sets
- Miller, Martin
- 1968
(Show Context)
Citation Context ...lso each nonempty. For (2) just take d ′ ≥ a ′′ . For (3) choose a ≥ 0 ′ and d any degree above a which is hyperimmune free with respect to a such as is given by relativizing either Miller and Martin =-=[MM68]-=- or the standard construction of a Spector minimal degree using total recursive binary trees to a. (The point here is that the usual forcing argument, even if only attempting to force minimality, deci... |

11 |
Weak recursive degrees and a problem of Spector, in Recursion Theory and Complexity
- Ishmukhametov
- 1999
(Show Context)
Citation Context ...a ∈ L2−L1: There is a minimal a < 0 ′ not in L1 by Sasso [Sas74] and it is in L2 by Jockusch and Posner [JP78]. This minimal degree is easily seen to be weakly recursive in the sense of Ishmukhametov =-=[Ish99]-=- and then, by that work, has a strong minimal cover b, i.e. {0, a, b} is an initial segment of the degrees. This proves the sharpness of the third fact. The existence of such an initial segment with b... |

10 | Defining the Turing jump
- Shore, Slaman
- 1999
(Show Context)
Citation Context ...e appropriate hierarchy in D or D(≤ 0 ′ ). In fact, the jump operator has been defined in D by entirely different methods involving coding models of arithmetic and other arguments by Shore and Slaman =-=[SS99]-=- as have all of the high/low classes in D(≤ 0 ′ ) with the exception of L1 by Nies, Shore and Slaman [NSS98]. The dream of a natural definition for any of these classes based on such order theoretic p... |

8 |
Weak recursive degrees and a problem of Spector
- Ishmukhametov
- 1997
(Show Context)
Citation Context ...∈ L2 − L1: There is a minimal a < 0 ′ not in L1 by Sasso [Sas74] and it is in L2 by Jockusch and Posner [JP78]. This minimal degree is easily seen to be weakly recursive in the sense of Ishmukhametov =-=[Ish99]-=- and then, by that work, has a strong minimal cover b, i.e. {0, a, b} is an initial segment of the degrees. This proves the sharpness of the third fact. The existence of such an initial segment with b... |

6 |
Minimal degrees of unsolvability and the full approximation construction
- Epstein
- 1975
(Show Context)
Citation Context ...plementation property. This result has a long history. R. W. Robinson [Rob72, cf. [PR81]] showed that every a ∈ L2 has a complement in D(≤ 0 ′ ); Posner [Pos77], that every a ∈ H1 has one and Epstein =-=[Eps75]-=-, that every r.e. a has one. Finally, Posner [Pos81] showed that every a /∈ L2 has a complement in D(≤ 0 ′ ) and so 0 ′ has the complementation property. By a further argument using relativization and... |

6 |
Simple proofs of some theorems on high degrees of unsolvability, Canadian
- Jockusch
- 1977
(Show Context)
Citation Context ... All of the results mentioned above for d ≤T 0 ′ , for example, are true for all degrees as long as we use the generalized hierarchy. (Jockusch and Posner [JP78]; (Cooper [Coo73]) for H1 and Jockusch =-=[Joc77]-=- for GH1; Jockusch and Posner [JP78]; Posner [Pos77].) Once again approximations, rates of growth and domination properties play prominent roles in the constructions. It was often hoped that these inv... |

6 |
A minimal degree not realizing least possible jump
- SASSO
- 1974
(Show Context)
Citation Context ...esults are all known to be sharp in terms of the high/low hierarchy. The sharpness of the first follows from the existence of a minimal degree a ∈ L2−L1: There is a minimal a < 0 ′ not in L1 by Sasso =-=[Sas74]-=- and it is in L2 by Jockusch and Posner [JP78]. This minimal degree is easily seen to be weakly recursive in the sense of Ishmukhametov [Ish99] and then, by that work, has a strong minimal cover b, i.... |

6 |
The upper semilattice of degrees 0 is complemented
- Posner
- 1981
(Show Context)
Citation Context ...y. R. W. Robinson [Rob72, cf. [PR81]] showed that every a ∈ L2 has a complement in D(≤ 0 ′ ); Posner [Pos77], that every a ∈ H1 has one and Epstein [Eps75], that every r.e. a has one. Finally, Posner =-=[Pos81]-=- showed that every a /∈ L2 has a complement in D(≤ 0 ′ ) and so 0 ′ has the complementation property. By a further argument using relativization and an additional division into cases along with some k... |

6 |
Defining the Turing
- Shore, Slaman
- 1999
(Show Context)
Citation Context ...e appropriate hierarchy in D or D(≤ 0 ′ ). In fact, the jump operator has been defined in D by entirely different methods involving coding models of arithmetic and other arguments by Shore and Slaman =-=[SS99]-=- as have all of the high/low classes in D(≤ 0 ′ ) with the exception of L1 by Nies, Shore and Slaman [NSS98]. The dream of a natural definition for any of these classes based on such order theoretic p... |

4 |
High Degrees
- Posner
- 1977
(Show Context)
Citation Context ...or example, are true for all degrees as long as we use the generalized hierarchy. (Jockusch and Posner [JP78]; (Cooper [Coo73]) for H1 and Jockusch [Joc77] for GH1; Jockusch and Posner [JP78]; Posner =-=[Pos77]-=-.) Once again approximations, rates of growth and domination properties play prominent roles in the constructions. It was often hoped that these investigations would lead to a definition of 0 ′ in D o... |

4 |
On the ordering of classes in high/low hierarchies
- Lerman
- 1984
(Show Context)
Citation Context ...on a nonuniform proof of the join theorem for GH1 (see [Ler83, IV.9]). It was also explicitly left open even in [Ler83] if the degree providing the join could be made GL1. This was answered by Lerman =-=[Ler85]-=- but again nonuniformly. We supply a uniform proof for this sharper version of the join theorem for GH1 which we then modify to get this case of the complementation theorem. The basic idea of our proo... |

3 |
Initial segments of degrees below 0
- Epstein
- 1981
(Show Context)
Citation Context ... third question around 1992 and that sketch has recently been extended and made into a proof by Lewis ([Lew03]). There have been several partial and related results about the fourth question. Epstein =-=[Eps81]-=-, for example, showed that if a < h are r.e. and h ∈ H1 then a has a complement in D(≤ h). In this paper we supply a full affirmative answer: Theorem 1.1. Every degree d ∈ GH1 has the complementation ... |

3 |
Degrees which do not bound minimal degrees
- Lerman
- 1986
(Show Context)
Citation Context ...s the sharpness of the third fact. The existence of such an initial segment with b also below 0 ′ follows by Lerman’s methods as outlined in [Ler83, XII.5.8]. The sharpness of the second is by Lerman =-=[Ler86]-=-.) The techniques used to prove all these positive results are tied up with approximation methods, rates of growth conditions and domination properties. Thus they are of independent interest for relat... |

1 |
segments of degrees below 0
- Initial
- 1981
(Show Context)
Citation Context ... third question around 1992 and that sketch has recently been extended and made into a proof by Lewis ([Lew03]). There have been several partial and related results about the fourth question. Epstein =-=[Eps81]-=-, for example, showedsGENERALIZED HIGH DEGREES HAVE THE COMPLEMENTATION PROPERTY 3 that if a < h are r.e. and h ∈ H1 then a has a complement in D(≤ h). In this paper we supply a full affirmative answe... |

1 |
which do not bound minimal degrees, Ann
- Degrees
- 1986
(Show Context)
Citation Context ...100035. 1 c○ 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00s2 NOAM GREENBERG, ANTONIO MONTALBÁN, AND RICHARD A. SHORE in [Ler83, XII.5.8]. The sharpness of the second is by Lerman =-=[Ler86]-=-.) The techniques used to prove all these positive results are tied up with approximation methods, rates of growth conditions and domination properties. Thus they are of independent interest for relat... |

1 |
Aspects of complementing in the turing degrees
- Lewis
- 2003
(Show Context)
Citation Context ...neric complement. Seetapun and Slaman circulated a sketch of a proposed affirmative answer to the third question around 1992 and that sketch has recently been extended and made into a proof by Lewis (=-=[Lew03]-=-). There have been several partial and related results about the fourth question. Epstein [Eps81], for example, showedsGENERALIZED HIGH DEGREES HAVE THE COMPLEMENTATION PROPERTY 3 that if a < h are r.... |