## Benign cost functions and lowness properties

Citations: | 9 - 5 self |

### BibTeX

@MISC{Greenberg_benigncost,

author = {Noam Greenberg and André Nies},

title = {Benign cost functions and lowness properties},

year = {}

}

### OpenURL

### Abstract

Abstract. We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost function, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of Diamondstone’s and Ng’s regarding cupping with supwerlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. 1.

### Citations

93 | Computability and Randomness
- Nies
- 2009
(Show Context)
Citation Context .... 1 The robustness of this class is expressed by its coincidence with several notions indicating that the set is computationally feeble (Nies; Hirschfeldt and Nies; Hirschfeldt, Nies and Stephan; see =-=[16, 9, 15]-=-): • lowness for Martin-Löf randomness: as an oracle, the set A ∈ 2 ω cannot detect any patterns in a Martin-Löf random set. • lowness for K: as an oracle, the set A cannot compress any strings beyond... |

78 | Lowness properties and randomness
- Nies
(Show Context)
Citation Context ... K(n) + b, where K denotes prefix-free Kolmogorov complexity. The robustness of this class is expressed by its coincidence with several notions indicating that the set is computationally feeble (Nies =-=[18]-=-; Hirschfeldt and Nies; Hirschfeldt, Nies and Stephan [10]; for more background on these coincidences see [17, Ch. 5]). • lowness for Martin-Löf randomness: as an oracle, the set A ∈ 2 ω cannot detect... |

56 | Trivial reals
- Downey, Hirschfeldt, et al.
- 2003
(Show Context)
Citation Context ...is so feeble that some A-random set can compute A. The key for this equivalence is the notion of cost functions and obeying them. The by-now standard constructions of a promptly simple K-trivial set (=-=[6]-=-) shows that a set which is low for Martin-Löf randomness ([11]) has similar dynamic properties as a set which is low for K (Mučnik, see [2]). The requirements, which want to enumerate numbers into th... |

36 |
Soare, An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees
- Ambos-Spies, Jockusch, et al.
(Show Context)
Citation Context ... turned out to be quite difficult to solve, was whether superlow cuppability coincided with low cuppability (which in turn was shown to be equivalent to having a promptly simple degree in the classic =-=[1]-=-). This question was recently settled in the negative by Diamondstone [5]. In parallel, Ng [13] showed that in analogy with the almost deep degree of [4], there is a c.e. degree which joins every supe... |

34 | Using random sets as oracles
- Hirschfeldt, Nies, et al.
(Show Context)
Citation Context .... 1 The robustness of this class is expressed by its coincidence with several notions indicating that the set is computationally feeble (Nies; Hirschfeldt and Nies; Hirschfeldt, Nies and Stephan; see =-=[16, 9, 15]-=-): • lowness for Martin-Löf randomness: as an oracle, the set A ∈ 2 ω cannot detect any patterns in a Martin-Löf random set. • lowness for K: as an oracle, the set A cannot compress any strings beyond... |

28 | Reals which compute little
- Nies
- 2002
(Show Context)
Citation Context ...owness for Schnorr randomness; a variant was also used by Ishmukhametov [10] in his study of strong minimal covers in the Turing degrees. A third variant, jump-traceability, was introduced by Nies in =-=[17]-=-. The strong version of jump-traceability was defined by Figueira, Nies and Stephan [7]. They showed that a non-computable strongly jump-traceable c.e. set exists. For the formal definitions, recall t... |

21 | Lowness properties and approximations of the jump
- Figueira, Nies, et al.
- 2006
(Show Context)
Citation Context ...y of strong minimal covers in the Turing degrees. A third variant, jump-traceability, was introduced by Nies in [17]. The strong version of jump-traceability was defined by Figueira, Nies and Stephan =-=[7]-=-. They showed that a non-computable strongly jump-traceable c.e. set exists. For the formal definitions, recall that a c.e. trace for a partial function ψ is a uniformly c.e. sequence 〈Tx〉 of finite s... |

21 | and André Nies. Randomness and computability: open questions - Miller |

18 | Almost everywhere domination and superhighness
- Simpson
(Show Context)
Citation Context ...at ∅ ′ ≤LR X, or LR-hard sets, is also known as the class of “high for random” sets. A set is LR-hard if and only if it is (uniformly) almost everywhere dominating (Kjos-Hanssen, Miller, Solomon; see =-=[18, 15]-=-). Theorem 1.4. Every strongly jump-traceable c.e. set is computable from any LRhard random set. As an immediate corollary we see that in the c.e. degrees, the collection of strongly jump-traceable de... |

15 | An almost deep degree
- Cholak, Groszek, et al.
(Show Context)
Citation Context ...having a promptly simple degree in the classic [1]). This question was recently settled in the negative by Diamondstone [5]. In parallel, Ng [13] showed that in analogy with the almost deep degree of =-=[4]-=-, there is a c.e. degree which joins every superlow c.e. degree to a superlow degree; he called such degrees almost superdeep. We show in Section 5, using the class (ω-c.e.) ♦ , that every strongly ju... |

10 | Mass problems and almost everywhere domination
- Simpson
(Show Context)
Citation Context ... FUNCTIONS AND LOWNESS PROPERTIES 5 the inclusions and SJT ⊆ LRH ♦ SJT ⊆ (ω-c.e.) ♦ , where SJT denotes the collection of strongly jump-traceable c.e. sets. We recall that Hirschfeldt and Miller (see =-=[19]-=- or [15]) showed, via a cost function construction, that if C is a null Σ 0 3 class, then C ♦ contains a promptly simple c.e. set. Thus the content of Theorems 1.4 and 1.5 is that if for a given C, Hi... |

9 |
Characterizing the strongly jump traceable sets via randomness
- Greenberg, Hirschfeldt, et al.
(Show Context)
Citation Context ...ogether with Hirschfeldt shows the converse of the foregoing theorem, and hence the coincidence of the c.e. strongly jump-traceable sets and the sets which are computable from every ω-c.e. random set =-=[9]-=-. Theorem 1.5 can be improved as follows. In [8], the authors define a binary relation which is a very strong version of weak truth-table reducibility: Y ≤ T (tu) X (Y is reducible to X with tiny use)... |

8 |
Weak recursive degrees and a problem of Spector
- Ishmukhametov
- 1997
(Show Context)
Citation Context ...ility was introduced into computability theory by Terwijn and Zambella [20] for their study of another lowness notion, that of lowness for Schnorr randomness; a variant was also used by Ishmukhametov =-=[10]-=- in his study of strong minimal covers in the Turing degrees. A third variant, jump-traceability, was introduced by Nies in [17]. The strong version of jump-traceability was defined by Figueira, Nies ... |

7 |
Terwijn and Domenico Zambella. Computational randomness and lowness
- Sebastiaan
(Show Context)
Citation Context ... outside its interaction with algorithmic randomness. A one-time candidate was the class of strongly jump-traceable sets. Traceability was introduced into computability theory by Terwijn and Zambella =-=[20]-=- for their study of another lowness notion, that of lowness for Schnorr randomness; a variant was also used by Ishmukhametov [10] in his study of strong minimal covers in the Turing degrees. A third v... |

6 | Calculus of cost functions
- Nies
(Show Context)
Citation Context ...2 set is K-trivial if and only if it obeys the standard cost function cK. We note, by the way, that if A is a c.e. set which obeys a cost function c, then A has a computable enumeration which obeys c =-=[20]-=-. We usually require our cost functions to satisfy the limit condition limx→∞ c(x) = 0, where again c(x) = lims c(x, s). In this paper we introduce a class of cost function that satisfy the limit cond... |

4 | On strongly jump traceable reals
- Ng
(Show Context)
Citation Context ...s is strictly contained in the ideal of K-trivial degrees. Together with Proposition 2.2, which is a uniform version of the proof of the easy direction of Theorem 1.2, Theorem 1.3 implies Ng’s result =-=[12]-=- that no single order function h can ensure strong jump-traceability. Indeed, in the same paper, Ng went on to show that the index-set for the collection of strongly jump-traceable c.e. sets is Π 0 4-... |

4 | Beyond strong jump traceability
- Ng
- 2010
(Show Context)
Citation Context ...that each strongly jump-traceable c.e. set is in the diamond class. However, no natural classes which lie strictly between the strongly jumptraceable and the K-trivial degrees have yet been found. Ng =-=[16]-=- has defined and investigated a class which is strictly contained in the strongly jump-traceable degrees. This class is obtained by partially relativising strong jump-traceability to all c.e. sets, an... |

3 |
Strong jump-traceabilty I: The computably enumerable case
- Cholak, Downey, et al.
(Show Context)
Citation Context ...x, |Tx| ≤ h(x). Finally, a set A is strongly jump-traceable if for every order function h, every partial function ψ : ω → ω which is partial computable in A has a c.e. trace which is bounded by h. In =-=[3]-=-, Cholak, Downey and Greenberg showed that the attempt to define Ktriviality using strong jump-traceability fails, but that in fact, restricted to the c.e. degrees, the strongly jump-traceable degrees... |

2 | Torenvliet. Enumerations of the Kolmogorov function
- Beigel, Buhrman, et al.
(Show Context)
Citation Context ... standard constructions of a promptly simple K-trivial set ([6]) shows that a set which is low for Martin-Löf randomness ([11]) has similar dynamic properties as a set which is low for K (Mučnik, see =-=[2]-=-). The requirements, which want to enumerate numbers into the set A which is being built, are restrained from doing so not by discrete negative requirements, such as in the standard Friedberg construc... |

2 |
Subclasses of the c.e. strongly jump traceable sets
- Nies
(Show Context)
Citation Context ... -c.e. if it can be computably approximated while counting down the natural well-order of ω × ω at each change. Nies has recently shown that (ω 2 -c.e.) ♦ is a proper subideal of (ω-c.e.) ♦ = SJTc.e. =-=[21]-=-. Not much is known otherwise. 2. Proof of the main theorem In this section we prove Theorem 1.2. We first prove the easy direction: if A is a c.e. set which obeys every benign cost function, then A i... |

1 |
Promptness does not imply superlow cuppability
- Diamondstone
(Show Context)
Citation Context ...ity coincided with low cuppability (which in turn was shown to be equivalent to having a promptly simple degree in the classic [1]). This question was recently settled in the negative by Diamondstone =-=[5]-=-. In parallel, Ng [13] showed that in analogy with the almost deep degree of [4], there is a c.e. degree which joins every superlow c.e. degree to a superlow degree; he called such degrees almost supe... |

1 |
Reducibilities with tiny use
- Franklin, Greenberg, et al.
(Show Context)
Citation Context ...together with D. Hirschfeldt shows the coincidence of the c.e. strongly jump-traceable sets and the sets which are computable from every ω-c.e. random sets. Theorem 1.5 can be improved as follows. In =-=[8]-=-, the authors define a binary relation which is a very strong version of weak truth-table reducibility: Y ≤ T (tu) X (Y is reducible to X with tiny use) if for every order function h, there is a reduc... |