Lowness and  nullsets

Downey, Rodney G.; Nies, André; Weber, Rebecca; Yu, Liang

doi:10.2178/jsl/1154698590

Cited by 43 publications

(53 citation statements)

References 5 publications

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“…At this point, the reader might be tempted to conjecture that the corresponding lowness notions for difference randomness might coincide with K-triviality as they do in the case of lowness for weak 2-randomness [4,8]. We show a connection between the corresponding lowness notions for difference randomness and two other well-known lowness classes arising in the study of K-triviality.…”

Section: Relationships With Lowness Classesmentioning

confidence: 77%

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Difference randomness

Franklin

2011

Proc. Amer. Math. Soc.

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Abstract. In this paper, we define new notions of randomness based on the difference hierarchy. We consider various ways in which a real can avoid all effectively given tests consisting of n-r.e. sets for some given n. In each case, the n-r.e. randomness hierarchy collapses for n ≥ 2. In one case, we call the resulting notion difference randomness and show that it results in a class of random reals that is a strict subclass of the Martin-Löf random reals and a proper superclass of both the Demuth random and weakly 2-random reals. In particular, we are able to characterize the difference random reals as the Turing incomplete Martin-Löf random reals. We also provide a martingale characterization for difference randomness.

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Section: Relationships With Lowness Classesmentioning

confidence: 77%

“…We recall that if R and S are classes of random reals, then Low(R,S) is the class of reals A such that R is a subset of S A . Then our A is in Low(W2R,ML) and thus it is K-trivial [4]. In particular, A must be low.…”

Section: Theorem 41 Suppose That a Is An Re Set Then A Is A Basementioning

confidence: 93%

Difference randomness

Franklin

2011

Proc. Amer. Math. Soc.

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“…of 'strong random sequences' still has measure 1 but none of its members have the defect 1. These developments are summarized in Downey et al (2006b). The goal of the present note is to offer a converging criterion for the strong random sequences.…”

Section: For Any Input I P Enters a Nonterminating Routine That Writmentioning

confidence: 98%

Recognizing Strong Random Reals

Osherson

Weinstein

2008

Rev. symb. logic

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Abstract. The class of strong random reals can be defined via a natural conception of effective null set. We show that the same class is also characterized by a learning-theoretic criterion of 'recognizability'.1. Characterizing randomness. Consider a physical process that, if suitably idealized, generates an indefinite sequence of independent random bits. One such process might be radioactive decay of a lump of uranium whose mass is kept at just the level needed to ensure that the probability is one-half that no alpha particle is emitted in the nth microsecond of the experiment. Let us think of the bits as drawn from {0, 1} and denote the resulting sequence by x with coordinates x 0 , x 1 , . . .. Now wouldn't it be odd if there were a computer program P with the following property? The program will not, in general, allow prediction of x i inasmuch as there is no requirement that the ultimate bit b m written by P(i) be marked as final. Nonetheless, shouldn't randomness exclude any computational process from having the kind of intimate knowledge of x i described in 1? For any input i, P enters a nonterminating routine that writes a nonemptyThe tension engendered by 1 afflicts a celebrated theory of randomness developed over the last half century. 2 The theory offers diverse criteria, each well motivated, for the concept 'infinite sequence of random bits'. Remarkably, the criteria yield the same collection of sequences -a collection, moreover, of measure 1 with respect to the 'coin flip' measure on the collection of infinite binary sequences. Despite this evidence for theoretical adequacy, some of the sequences labeled 'random' can be associated with a program P satisfying 1.One response to this state of affairs has been to modify (in a simple and satisfying way) the randomness criteria originally proposed by Martin-Löf (1966). The resulting collection Received xxxxx, 200x. 1 The sequences satisfying 1 correspond to the limit recursive sets of Gold (1965) and trial and error predicates of Putnam (1965); such limits were introduced earlier by Shoenfield (1959) to characterize the sets recursive in 0 .

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“…For example, it can be shown that X is weakly 2-random (i.e. in every Σ 0 2 class of measure 1) iff X is 1-random and its degree forms a minimal pair with ∅ (Downey, Nies, Weber, and Yu [48] plus Hirschfeldt and Miller (in [48]) for the hard direction). Hence no (weakly) 2-random real can bound a P A degree.…”

Section: Computability and Randomnessmentioning

confidence: 99%