## Bounded Immunity and Btt-Reductions (1999)

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Venue: | MLQ Math. Log. Q |

Citations: | 8 - 3 self |

### BibTeX

@ARTICLE{Fenner99boundedimmunity,

author = {Stephen Fenner and Marcus Schaefer},

title = {Bounded Immunity and Btt-Reductions},

journal = {MLQ Math. Log. Q},

year = {1999},

volume = {45},

pages = {3--21}

}

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### Abstract

We define and study a new notion called k-immunity that lies between immunity and hyperimmunity in strength. Our interest in k-immunity is justified by the result that # # does not k-tt reduce to a k-immune set, which improves a previous result by Kobzev [7, 13]. We apply the result to show that # # does not btt-reduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new characterization of e#ectively simple sets and show that simple sets are not btt-cuppable. Keywords: Computability, Recursion Theory, bounded reducibilities, minimal programs, immunity, k-immune, regressive, retraceable, e#ectively simple, cuppable. 1 Introduction There seems to be a large gap between immunity and hyperimmunity (h-immunity) that is waiting to be filled. What happens, one wonders if the disjoint strong arrays that try to witness that a set is not h-immune are subjected to additional conditions...

### Citations

497 |
Recursively Enumerable Sets and Degrees
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- 1987
(Show Context)
Citation Context ... simple sets. Finally we show that simple sets are neither btt-cuppable nor d-cuppable. For notation and definitions of the standard concepts of computability we refer the reader to the usual sources =-=[13, 16]-=-. In the following, degree always means Turing degree, unless specified otherwise. 2 A concise history of immunity and completeness We collect the known results that connect notions of immunity with c... |

322 |
Classical Recursion Theory
- Odifreddi
- 1989
(Show Context)
Citation Context ... lies between immunity and hyperimmunity in strength. Our interest in k-immunity is justified by the result that # # does not k-tt reduce to a k-immune set, which improves a previous result by Kobzev =-=[7, 13]-=-. We apply the result to show that # # does not btt-reduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We... |

66 |
The Undecidable
- Davis, ed
- 1965
(Show Context)
Citation Context ... have played in computability theory 1 . In 1944 Post published his seminal paper on Recursively enumerable sets of positive integers and their decision problems (reprinted in Davis's The Undecidable =-=[2]-=-). With this paper he initiated what has since become known as Post's program: relating the thinness of a set to its degree. His goal was to show that there was a noncomputable c.e. set with a complem... |

35 | Computability and recursion
- Soare
- 1996
(Show Context)
Citation Context ...e. set with a complement # Partially supported by NSF Grant CCR-9501794 + Partially supported by NSF Grants CCR 92-53582 and CCR-9501794 1 In this paper we will use the terminology suggested by Soare =-=[15]-=-. In particular we will denote a computably (recursively) enumerable set by c.e. and talk about computable partial functions instead of partial recursive functions. 1 so thin that it could not be Turi... |

22 | Postâ€™s Program and incomplete recursively enumerable sets
- Harrington, Soare
- 1991
(Show Context)
Citation Context ...ss which is not, strictly speaking, a thinness properties. Since then several other such properties---including E-definable ones---implying Turing incompleteness have been found (Harrington and Soare =-=[5]-=-). 3 The notion of k-immunity Let (D x ) x## be a canonical numbering of the finite sets. A family (D f(x) ) x## of finite sets is a disjoint strong array if f is a computable function, and all the D ... |

13 | A guided tour of minimal indices and shortest descriptions - Schaefer - 1998 |

12 |
jump classes and strong reducibilities
- Downey, Jockusch
- 1987
(Show Context)
Citation Context ...r hand the Friedberg and Rogers result remains true with hyperimmunity instead of hypersimplicity; the usual proof shows that. (iv) The Friedberg and Rogers result was improved by Downey and Jockusch =-=[4]-=-. By their result no hypersimple set is wtt-cuppable, namely if H is hypersimple there is no c.e. set B such that H # B is wtt-complete, unless B itself is already wtt-complete. A similar result has b... |

7 |
Uniformly introreducible sets
- Jockusch
- 1968
(Show Context)
Citation Context ...folklore result will be proved in Lemma 3.2). The concepts of k-immunity and bounded immunity have been used implicitly several times in the past, for example by Appel and McLaughlin [1] and Jockusch =-=[6]-=- in their work on retraceable and regressive sets. We begin the paper with a short historical account of the success of Post's program. It contains no information essential to the rest of the paper, a... |

6 |
A note on universal sets
- Lachlan
- 1966
(Show Context)
Citation Context ...that # # # r A # B and # # ## r B, where r is a class of reductions (like m, 1, btt, c, d, tt, wtt, T, e) The following three results are known: Theorem 7.2 . An immune set is not m-cuppable (Lachlan =-=[10]-=-). . A hypersimple set is not wtt-cuppable (Downey, Jockusch [4]). . A # # -hypersimple set is not e-cuppable (Nies, Sorbi [12]). We will now show that an analogous statement is true for btt-reduction... |

2 |
On Properties of Regressive Sets
- Appel, McLaughlin
- 1965
(Show Context)
Citation Context ...complement (this folklore result will be proved in Lemma 3.2). The concepts of k-immunity and bounded immunity have been used implicitly several times in the past, for example by Appel and McLaughlin =-=[1]-=- and Jockusch [6] in their work on retraceable and regressive sets. We begin the paper with a short historical account of the success of Post's program. It contains no information essential to the res... |

2 |
Three theorems on elementary theories and tt-reducibility
- Denisov
- 1974
(Show Context)
Citation Context ...[2] simple, and B is part of a computably B ## btt A. Kobzev, 1973 [7] inseparable pair of c.e. sets, hypersimple, # # ## tt A. Post, 1944 [2] hyperimmune, and B is part of a B ## tt A. Denisov, 1974 =-=[3]-=- computably inseparable pair of c.e. sets, hypersimple, # # ## wtt A. Friedberg, Rogers, 1959 hyperhypersimple, # # ## Q A. Soloviev, 1974, and Gill, Morris, 1974 noncomputable, semirecursive, # # ## ... |

2 |
On Complete Btt-degrees
- Kobzev
- 1974
(Show Context)
Citation Context ...ote that the proposition implies that immune sets are not m-cuppable. Kobzev proved the following variant of Lachlan's result by making some slight modifications to the proof. Proposition 7.5 (Kobzev =-=[8]-=-) (i) If A is productive and B is c.e., then either A # B or A # B is productive. (ii) If A is creative and B is computable, then either A # B or A # B is creative. The second item easily follows from... |

2 |
On the Complexity of Random
- Kummer
- 1996
(Show Context)
Citation Context ...here is a k-immune set which is (k + 1)-disjunctively equivalent to # # . In particular, for the sets R and MIN as defined in the last section we have the following corollaries. Corollary 4.6 (Kummer =-=[9]-=-) # # ## btt R. Concerning weaker reductions Kummer proved the surprising fact that R is truth-table complete. Corollary 4.7 # # ## btt MIN. This result is optimal in the sense that there is a Godel n... |