## Simple Sets Are Not Btt-Cuppable (1997)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Schaefer97simplesets,

author = {Marcus Schaefer},

title = {Simple Sets Are Not Btt-Cuppable},

institution = {},

year = {1997}

}

### OpenURL

### Abstract

We extend Post's result that a simple set cannot be btt-complete by showing that in fact it cannot be btt-cuppable, i.e. if the join of a c.e. set and a simple set is btt-complete, then the non-simple set is btt-complete itself. The proof also yields that simple sets are not d-cuppable (i.e. not cuppable with regard to disjunctive reductions). Post showed that a simple set cannot be btt-complete. In a paper by Stephen Fenner and the author [3] this was generalized to non-c.e. sets by isolating the immunity property which is responsible for the incompleteness. Another approach to the btt-incompleteness of simple sets would have been through degrees. How incomplete are simple sets? Putting it dioeerently: can the join of a btt-incomplete degree with a simple degree be btt-complete? We will show that the answer is no. Deønition 1 A set A is called r-cuppable, if there is a c.e. set B such that ; 0 r A \Phi B and ; 0 6 r B, where r is a class of reductions (like m, 1, btt, c, d, tt...

### Citations

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326 | Classical Recursion Theory
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(Show Context)
Citation Context ...sult was proven independently by Lachlan and Kobzev, the former using the technique of Proposition 4, the latter giving a proof using his result on productive sets. Proposition 7 (Kobzev [6], Lachlan =-=[9]-=-) A btt-complete set is bd-complete. With this we are now ready to prove our result. Proof of Theorem 3. Suppose ; 0sbtt A \Phi B where A is simple, and B is c.e. By Proposition 7 we know that there a... |

12 |
jump classes and strong reducibilities
- Downey, Jockusch
- 1987
(Show Context)
Citation Context ... B, where r is a class of reductions (like m, 1, btt, c, d, tt, wtt, T, etc.) The following three results are known: Theorem 2 ffl (Lachlan [7]) An immune set is not m-cuppable. ffl (Downey, Jockusch =-=[2]-=-) A hypersimple set is not wtt-cuppable. ffl (Nies, Sorbi [8]) A ; 0 -hypersimple set is not e-cuppable. We will now show that an analogous statement is true for btt-reductions as well. Theorem 3 A si... |

6 | Posts’s program and incomplete recursively enumerable sets - Harrington, Soare - 1991 |

6 |
A note on universal sets
- Lachlan
- 1966
(Show Context)
Citation Context ...f there is a c.e. set B such that ; 0sr A \Phi B and ; 0 6 r B, where r is a class of reductions (like m, 1, btt, c, d, tt, wtt, T, etc.) The following three results are known: Theorem 2 ffl (Lachlan =-=[7]-=-) An immune set is not m-cuppable. ffl (Downey, Jockusch [2]) A hypersimple set is not wtt-cuppable. ffl (Nies, Sorbi [8]) A ; 0 -hypersimple set is not e-cuppable. We will now show that an analogous ... |

2 | Three theorems on elementary theories and tt-reducibility - Denisov - 1974 |

2 |
On Complete Btt-degrees
- Kobzev
- 1974
(Show Context)
Citation Context ...B. Note that the proposition implies that immune sets are not m-cuppable. Kobzev proved the following variant of Lachlan's result by making some slight modications to the proof. Proposition 5 (Kobzev =-=[6]) (i) If A is produc-=-tive 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... |

1 |
Bounded Immunity and Btt-Reductions. Submitted to Mathematical Logic Quarterly
- Fenner, Schaefer
- 1997
(Show Context)
Citation Context ...yields that simple sets are not d-cuppable (i.e. not cuppable with regard to disjunctive reductions). Post showed that a simple set cannot be btt-complete. In a paper by Stephen Fenner and the author =-=[3]-=- this was generalized to non-c.e. sets by isolating the immunity property which is responsible for the incompleteness. Another approach to the btt-incompleteness of simple sets would have been through... |

1 |
Branching in the \Sigma 2 Enumeration Degrees. unpublished manuscript
- Nies, Sorbi
- 1997
(Show Context)
Citation Context ...t, wtt, T, etc.) The following three results are known: Theorem 2 ffl (Lachlan [7]) An immune set is not m-cuppable. ffl (Downey, Jockusch [2]) A hypersimple set is not wtt-cuppable. ffl (Nies, Sorbi =-=[8]-=-) A ; 0 -hypersimple set is not e-cuppable. We will now show that an analogous statement is true for btt-reductions as well. Theorem 3 A simple set is not btt-cuppable. For the proof we need to have a... |