## Almost everywhere domination and superhighness (2007)

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Venue: | MATHEMATICAL LOGIC QUARTERLY |

Citations: | 21 - 9 self |

### BibTeX

@ARTICLE{Simpson07almosteverywhere,

author = {Stephen G. Simpson},

title = {Almost everywhere domination and superhighness},

journal = {MATHEMATICAL LOGIC QUARTERLY},

year = {2007},

volume = {53},

pages = {2007}

}

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### Abstract

Let ω denote the set of natural numbers. For functions f,g: ω → ω, we say that f is dominated by g if f(n) <g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X ∈ 2 ω, each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of Kjos-Hanssen, Kjos-Hanssen/Miller/Solomon, and others concerning LR-reducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0 ′ ′ is truth-table computable from B ′ , the Turing jump of B.

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Citation Context ...ept for some peripheral remarks. Throughout this paper we give full proofs and strive for simplicity and clarity. 22 Notation We use standard recursion-theoretic notation and terminology from Rogers =-=[25]-=-. We write r.e. as an abbreviation for “recursively enumerable”. If C is a Turing oracle, we write C-recursive for “recursive relative to C”, C-r.e. for “recursively enumerable relative to C”, etc. We... |

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Citation Context ...)↓ 1/2|σ| ≤ 1, and this is known as the Kraft Inequality. Conversely we have the following theorem attributed by Nies [21, Theorem 3.2] and Downey/Hirschfeldt/Nies/Stephan [9, Theorem 2.1] to Chaitin =-=[3]-=-. Theorem 10.3 (Kraft/Chaitin Theorem). Let L be an r.e. subset of ω × 2 <ω such that ∑ (m,τ)∈L 1/2m ≤ 1. Then we can effectively find a prefix-free machine M such that for all (m, τ) ∈ L there exists... |

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Citation Context ..., we may take F to be prefix-free. 3 Randomness Our work in this paper is based on a robust concept of randomness relative to a Turing oracle. The original, unrelativized concept is due to Martin-Löf =-=[17]-=- and has been studied by Kučera [15] and many others. Definition 3.1 (Martin-Löf 1966). Let C be a Turing oracle. We say that X ∈ 2ω is C-random if X /∈ ⋂ for all uniformly Σ0,C 1 sequences of sets n ... |

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Citation Context ...illuminated this concept and established its close relationship to the decisive results on K-triviality and low-for-randomness which are due to Downey, Hirschfeldt, Kučera, Nies, Stephan, and Terwijn =-=[16, 9, 21, 11]-=-. The purpose of this paper is to update the Dobrinen/Simpson account of almost everywhere domination by expositing this subsequent research. We provide introductory accounts of Martin-Löf randomness,... |

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Citation Context ...Moreover UC n 4⋂ n UC n is uniformly Π0,C 2 , hence RC is uniformly Σ 0,C 2 . Also µ(UC n ) ≤ 1/2n for all n, hence µ( ⋂ n UC n ) = 0, hence µ(RC) = 1. □ We now present van Lambalgen’s Theorem, from =-=[30]-=-. Lemma 3.4. Assume that A ⊕ B is random. Then A is B-random, and B is A-random. In particular, A and B are random. Proof. Suppose for instance that B is not A-random. Then B ∈ ⋂ n V A n where is unif... |

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Citation Context ...andomness Our work in this paper is based on a robust concept of randomness relative to a Turing oracle. The original, unrelativized concept is due to Martin-Löf [17] and has been studied by Kučera =-=[15]-=- and many others. Definition 3.1 (Martin-Löf 1966). Let C be a Turing oracle. We say that X ∈ 2ω is C-random if X /∈ ⋂ n U C n for all uniformly Σ 0,C 1 sequences of sets UCn ⊆ 2 ω, n = 0, 1, 2, . . ... |

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Citation Context ...illuminated this concept and established its close relationship to the decisive results on K-triviality and low-for-randomness which are due to Downey, Hirschfeldt, Kučera, Nies, Stephan, and Terwijn =-=[16, 9, 21, 11]-=-. The purpose of this paper is to update the Dobrinen/Simpson account of almost everywhere domination by expositing this subsequent research. We provide introductory accounts of Martin-Löf randomness,... |

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Citation Context ..., it follows that A ⊕ C is random, hence A is C-random. □ 6Remarkably, the previous corollary holds even without the assumption C ≥T 0 ′ . We mention without proof the following theorem of Miller/Yu =-=[19]-=-. Theorem 3.10 (Miller/Yu 2004). If A is random and A ≤T B where B is C-random, then A is C-random. 4 LR-reducibility In this section we study the following reducibility notion, which was originally i... |

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Citation Context ... 18 9 Counterexamples via duality 20 10 Prefix-free Kolmogorov complexity 22 References 26 1 Introduction The concept of almost everywhere domination was originally introduced by Dobrinen and Simpson =-=[7]-=- with applications to the reverse mathematics of measure theory [26, Section X.1]. Subsequent work by Binns, Cholak, Greenberg, Kjos-Hanssen, Lerman, Miller, and Solomon [2, 5, 13, 14] has greatly ill... |

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Citation Context ...illuminated this concept and established its close relationship to the decisive results on K-triviality and low-for-randomness which are due to Downey, Hirschfeldt, Kučera, Nies, Stephan, and Terwijn =-=[16, 9, 21, 11]-=-. The purpose of this paper is to update the Dobrinen/Simpson account of almost everywhere domination by expositing this subsequent research. We provide introductory accounts of Martin-Löf randomness,... |

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Citation Context ...ed by Dobrinen and Simpson [7] with applications to the reverse mathematics of measure theory [26, Section X.1]. Subsequent work by Binns, Cholak, Greenberg, Kjos-Hanssen, Lerman, Miller, and Solomon =-=[2, 5, 13, 14]-=- has greatly illuminated this concept and established its close relationship to the decisive results on K-triviality and low-for-randomness which are due to Downey, Hirschfeldt, Kučera, Nies, Stephan,... |

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Citation Context ...for-random relative to C. To obtain our examples, we combine a construction of Kučera/Terwijn [16] with the Pseudojump Inversion Theorem of Jockusch/Shore [12] and the Join Theorem of Posner/Robinson =-=[24]-=-. □ 14Theorem 6.1 (Kučera/Terwijn 1997). We can find a nonrecursive r.e. set A ⊆ ω such that A ≤LR 0, i.e., A is low-for-random. Proof. By Corollary 3.3 we know that {X ∈ 2ω | X is random} is Σ0 2 an... |

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Citation Context ...ed by Dobrinen and Simpson [7] with applications to the reverse mathematics of measure theory [26, Section X.1]. Subsequent work by Binns, Cholak, Greenberg, Kjos-Hanssen, Lerman, Miller, and Solomon =-=[2, 5, 13, 14]-=- has greatly illuminated this concept and established its close relationship to the decisive results on K-triviality and low-for-randomness which are due to Downey, Hirschfeldt, Kučera, Nies, Stephan,... |

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25 |
On a question of Dobrinen and Simpson concerning almost everywhere domination
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(Show Context)
Citation Context ...ed by Dobrinen and Simpson [7] with applications to the reverse mathematics of measure theory [26, Section X.1]. Subsequent work by Binns, Cholak, Greenberg, Kjos-Hanssen, Lerman, Miller, and Solomon =-=[2, 5, 13, 14]-=- has greatly illuminated this concept and established its close relationship to the decisive results on K-triviality and low-for-randomness which are due to Downey, Hirschfeldt, Kučera, Nies, Stephan,... |

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Citation Context ...m 10.10, Nies proved the equivalence A ≤LK 0 ⇐⇒ A ≤LR 0 (see [21, Corollary 5.3]) and noted that A ≤LK B implies A ≤LR B. The full equivalence A ≤LK B ⇐⇒ A ≤LR B is due to Kjos-Hanssen/Miller/Solomon =-=[14]-=-. Remark 10.12. By means of relativized prefix-free Kolmogorov complexity, one can prove a number of interesting results concerning ≤LR for which no other proofs are presently known. Among these resul... |

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Citation Context ...e to B then A ≤T B ′ , hence for each B there are only countably many such A. But Miller and Yu [19, 18] have constructed a B such that {A | A ≤LR B} is uncountable. Recently Barmpalias/Lewis/Soskova =-=[1]-=- have shown that this holds for any B which is generalized superhigh. Remark 4.5. On the other hand, consider the equivalence relation ≡LR defined by letting A ≡LR B if and only if A ≤LR B and B ≤LR A... |

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Citation Context ...omness is as follows. It can be shown (see Lemma 7.4 and Remark 10.12) that if A is low-for-random relative to B then A ≤T B′, hence for each B there are only countably many such A. But Miller and Yu =-=[18, 19]-=- have constructed a B such that {A | A ≤LR B} is uncountable. Recently Barmpalias/Lewis/Soskova [1] have shown that this holds for any B which is generalized superhigh. Remark 4.5. On the other hand, ... |

10 | Mass problems and almost everywhere domination
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(Show Context)
Citation Context ...tudy of weak degrees (a.k.a., Muchnik degrees) of mass problems associated with nonempty Π 0 1 subsets of 2ω . This aspect has been examined in Kjos-Hanssen [13] and in Simpson [28]. See also Simpson =-=[27]-=- for some newer results. We now turn to the proof of Theorem 5.7. The proof (see Remark 5.14 below) will be based on the following lemma and theorem. Lemma 5.11 (Kjos-Hanssen/Miller/Solomon 2006). Ass... |

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Citation Context ...s not recursive. Thus ≤LR does not coincide with ≤T. Remark 4.3. Evidently the reducibility relation ≤LR is closely related to the notion of low-for-randomness, which was first introduced by Zambella =-=[31]-=- and has been studied extensively by Kučera/Terwijn [16], Terwijn/Zambella [29], Downey/Hirschfeldt/Nies/Stephan [9], Hirschfeldt/Nies/Stephan [11], and Nies [21]. By definition, A is low-for-random i... |

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Citation Context ... significance for the study of weak degrees (a.k.a., Muchnik degrees) of mass problems associated with nonempty Π 0 1 subsets of 2ω . This aspect has been examined in Kjos-Hanssen [13] and in Simpson =-=[28]-=-. See also Simpson [27] for some newer results. We now turn to the proof of Theorem 5.7. The proof (see Remark 5.14 below) will be based on the following lemma and theorem. Lemma 5.11 (Kjos-Hanssen/Mi... |

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Citation Context ...t a C such that 0 ′ ≤LR C yet 0 ′ is not low-for-random relative to C. To obtain our examples, we combine a construction of Kučera/Terwijn [16] with the Pseudojump Inversion Theorem of Jockusch/Shore =-=[12]-=- and the Join Theorem of Posner/Robinson [24]. □ 14Theorem 6.1 (Kučera/Terwijn 1997). We can find a nonrecursive r.e. set A ⊆ ω such that A ≤LR 0, i.e., A is low-for-random. Proof. By Corollary 3.3 w... |

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Citation Context ...examples showing that neither of these implications can be reversed. In order to obtain our counterexamples, we use a duality technique which has been used previously by Jockusch/Shore [12], Mohrherr =-=[20]-=-, Nies [21], and Kjos-Hanssen [13]. The technique is based on the following theorem due to Jockusch/Shore [12] which we call the Duality Theorem. 20 Theorem 9.1 (Duality Theorem). Given a pseudojump o... |

1 |
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Citation Context ...reducibility relation ≤LR is closely related to the notion of low-for-randomness, which was first introduced by Zambella [31] and has been studied extensively by Kučera/Terwijn [16], Terwijn/Zambella =-=[29]-=-, Downey/Hirschfeldt/Nies/Stephan [9], Hirschfeldt/Nies/Stephan [11], and Nies [21]. By definition, A is low-for-random if and only if A ≤LR 0. Relativizing to B, we see that A is low-for-random relat... |