## Mass problems and almost everywhere domination (2007)

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Venue: | Mathematical Logic Quarterly |

Citations: | 10 - 7 self |

### BibTeX

@INPROCEEDINGS{Simpson07massproblems,

author = {Stephen G. Simpson},

title = {Mass problems and almost everywhere domination},

booktitle = {Mathematical Logic Quarterly},

year = {2007},

pages = {53--483}

}

### OpenURL

### Abstract

We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω. Let 1 and 0 be the top and bottom elements of Pw. We show that inf(b1,1) and inf(b2,1) and inf(b3,1) belong to Pw and that 0 < inf(b1,1) < inf(b2,1) < inf(b3,1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, we show that inf(b1,1) and inf(b3,1) but not inf(b2,1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′. In order to make this paper more self-contained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan.

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Citation Context ...in Theorem 4.5 which are �≥T 0 ′ . On the other hand, a result of Hirschfeldt/Nies/Stephan [13, Corollary 3.6] says that if X is random and �≥T 0 ′ then any recursively enumerable A ≤T X is K-trivial =-=[24]-=-, i.e., low-for-random [22, 24, 33]. In particular, any A as in Theorem 4.5 is low-for-random. 5 A theorem of Nies In this section we present a new proof of a theorem of Nies [26, Theorem VI.18] refin... |

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Citation Context ... � Lemma 2.6. MLR ≤w PA. Proof. Since MLR is Σ0 2 and nonempty, we can find a nonempty Π01 set P ⊆ MLR. Since P is a nonempty Π0 1 subset of 2ω ,wehaveP≤wPA in view of Scott/Tennenbaum [30] and Scott =-=[29]-=-. The lemma follows. � By Lemmas 2.4 and 2.5 and 2.6 we have MLR ∪ PA �≤w AED. From this it follows trivially that AED ∪ PA <w (MLR × AED) ∪ PA. In other words, inf(b1, 1) < inf(b2, 1). Next we prove ... |

36 | problems and randomness
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(Show Context)
Citation Context ...blem inequalities 3 3 Comparison with r.e. Turing degrees 6 4 A theorem of Hirschfeldt and Miller 7 5 A theorem of Nies 9 6 A theorem of Stephan 11 References 12 1 Introduction In our previous papers =-=[3, 31, 34, 32, 7]-=- we studied the lattice Pw of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω .We showed that Pw contains many specific, natural degrees in addition to 1 and 0,... |

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Citation Context ...it and discuss some relatively new examples of specific, natural degrees in Pw. The new examples arise from almost everywhere domination, a concept which was originally introduced by Dobrinen/Simpson =-=[9]-=-. Let B beaTuringoracle. WesaythatB is almost everywhere dominating if, for all X ∈ 2 ω except a set of measure 0, each function computable from X is dominated by some function computable from B. It i... |

25 |
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(Show Context)
Citation Context ...ve Theorem 2.2, it suffices to show that AED and MLR are Σ 0 3 . After Dobrinen/Simpson [9], the concept of almost everywhere domination was subsequently explored by Binns/Kjos-Hanssen/Lerman/Solomon =-=[2]-=-, Cholak/Greenberg/Miller [5], Kjos-Hanssen [18], Kjos-Hanssen/Miller/Solomon [19], and Simpson [33]. We now know [18, 19] that B is almost everywhere dominating if and only if 0 ′ ≤LR B. Here0 ′ deno... |

25 | Uniform almost everywhere domination
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(Show Context)
Citation Context ...o show that AED and MLR are Σ 0 3 . After Dobrinen/Simpson [9], the concept of almost everywhere domination was subsequently explored by Binns/Kjos-Hanssen/Lerman/Solomon [2], Cholak/Greenberg/Miller =-=[5]-=-, Kjos-Hanssen [18], Kjos-Hanssen/Miller/Solomon [19], and Simpson [33]. We now know [18, 19] that B is almost everywhere dominating if and only if 0 ′ ≤LR B. Here0 ′ denotes the Halting Problem, and ... |

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(Show Context)
Citation Context ...blem inequalities 3 3 Comparison with r.e. Turing degrees 6 4 A theorem of Hirschfeldt and Miller 7 5 A theorem of Nies 9 6 A theorem of Stephan 11 References 12 1 Introduction In our previous papers =-=[3, 31, 34, 32, 7]-=- we studied the lattice Pw of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω .We showed that Pw contains many specific, natural degrees in addition to 1 and 0,... |

24 |
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(Show Context)
Citation Context ...∪PA �≤w (MLR×AED)∪PA. In other words, inf(a, 1) �≤ inf(b2, 1). On the other hand, since A is recursively enumerable and not Turing complete, we have DNR �≤w {A} by the Arslanov Completeness Criterion =-=[14]-=-, hence MLR ∪ PA �≤w {A} by Lemmas 2.5 and 2.6. From this it follows trivially that (MLR × AED) ∪ PA �≤w {A}∪PA. In other words, inf(b2, 1) �≤ inf(a, 1). This completes the proof. � Theorem 3.2. There... |

24 |
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(Show Context)
Citation Context ...ble to Y . A weak degree is an equivalence class of subsets of 2 ω under mutual weak reducibility. The weak degree of P is denoted deg w(P ). Weak degrees have sometimes been known as Muchnik degrees =-=[23]-=-. Note that for p =deg w(P )andq =deg w(Q) wehaveinf(p, q) =deg w(P ∪ Q) and sup(p, q) = deg w(P × Q). Note also that for X, Y ∈ 2 ω we have X ≤T Y if and only if {X} ≤w {Y }. Here {X} denotes the sin... |

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(Show Context)
Citation Context ...blem inequalities 3 3 Comparison with r.e. Turing degrees 6 4 A theorem of Hirschfeldt and Miller 7 5 A theorem of Nies 9 6 A theorem of Stephan 11 References 12 1 Introduction In our previous papers =-=[3, 31, 34, 32, 7]-=- we studied the lattice Pw of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω .We showed that Pw contains many specific, natural degrees in addition to 1 and 0,... |

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(Show Context)
Citation Context ...work in this paper owes much to conversations with Bjørn Kjos-Hanssen, Antonín Kučera, and Joseph S. Miller. In particular, the fact that inf(b1, 1) belongs to Pw was already implicit in Kjos-Hanssen =-=[18]-=-, and Miller corrected an error in one of our early proofs of the inequality inf(b2, 1) < inf(b3, 1). The reader who is familiar with the basics of recursion theory will find that this paper is largel... |

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(Show Context)
Citation Context |

18 |
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(Show Context)
Citation Context ...b1, 1) < inf(c, 1). � 4 A theorem of Hirschfeldt and Miller In this section we exposit the proof of the following theorem of Hirschfeldt and Miller 2006, generalizing a much earlier theorem of Kučera =-=[21]-=-. Theorem 4.1 (Hirschfeldt/Miller). Let S ⊆ 2ω be Σ0 3 of measure 0. Then we can find a nonrecursive, recursively enumerable set A such that A ≤T X for all random X ∈ S. Proof. We follow the writeup o... |

18 | Almost everywhere domination and superhighness
- Simpson
(Show Context)
Citation Context ...E1 and E2 are two such expressions, we write E1 � E2 to mean that E1 and E2 are both defined and equal, or both undefined. Throughout this paper, a convenient background reference is our recent paper =-=[33]-=-, which includes a fairly thorough exposition of almost everywhere domination and Martin-Löf randomness. 2 Some mass problem inequalities The purpose of this section is to prove our mass problem inequ... |

15 | Lowness notions, measure and domination
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- 2012
(Show Context)
Citation Context ...impson [9], the concept of almost everywhere domination was subsequently explored by Binns/Kjos-Hanssen/Lerman/Solomon [2], Cholak/Greenberg/Miller [5], Kjos-Hanssen [18], Kjos-Hanssen/Miller/Solomon =-=[19]-=-, and Simpson [33]. We now know [18, 19] that B is almost everywhere dominating if and only if 0 ′ ≤LR B. Here0 ′ denotes the Halting Problem, and ≤LR denotes LR-reducibility: A ≤LR B if and only if ∀... |

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5 |
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(Show Context)
Citation Context ...d Simpson [31]. � Lemma 2.6. MLR ≤w PA. Proof. Since MLR is Σ0 2 and nonempty, we can find a nonempty Π01 set P ⊆ MLR. Since P is a nonempty Π0 1 subset of 2ω ,wehaveP≤wPA in view of Scott/Tennenbaum =-=[30]-=- and Scott [29]. The lemma follows. � By Lemmas 2.4 and 2.5 and 2.6 we have MLR ∪ PA �≤w AED. From this it follows trivially that AED ∪ PA <w (MLR × AED) ∪ PA. In other words, inf(b1, 1) < inf(b2, 1).... |

5 | Marin-Löf random and PA-complete sets
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- 2006
(Show Context)
Citation Context ...exists B ∈ MLR ∩ AED such that 0 ′ �≤T B. Proof. See Nies [26, Theorem VI.18]. Alternatively, see Section 5 below. � Lemma 2.11 (Stephan). If B ∈ MLR and 0 ′ �≤T B, thenPA �≤w {B}. Proof. See Stephan =-=[35]-=-. Alternatively, see Section 6 below. � By Lemma 2.10 let B ∈ MLR ∩ AED be such that 0 ′ �≤T B. By Lemma 2.11 we have PA �≤w {B}. Thus PA �≤w MLR ∩ AED. It follows trivially that (MLR ∩ AED) ∪ PA <w P... |

2 |
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- 2006
(Show Context)
Citation Context ...o show that AED and MLR are Σ 0 3 . After Dobrinen/Simpson [9], the concept of almost everywhere domination was subsequently explored by Binns/Kjos-Hanssen/Lerman/Solomon [2], Cholak/Greenberg/Miller =-=[5]-=-, Kjos-Hanssen [18], Kjos-Hanssen/Miller/Solomon [19], and Simpson [33]. We now know [18, 19] that B is almost everywhere dominating if and only if 0 ′ ≤LR B. Here 0 ′ denotes the Halting Problem, and... |

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