Higher Randomness Notions and Their Lowness Properties
نویسندگان
چکیده
We study randomness notions given by higher recursion theory, establishing the relationships Π1-randomness ⊂ Π1-Martin-Löf randomness ⊂ ∆1randomness = ∆1-Martin-Löf randomness. We characterize the set of reals that are low for ∆1 randomness as precisely those that are ∆ 1 1 -traceable. We prove that there is a perfect set of such reals.
منابع مشابه
CDMTCS Research Report Series Higher Randomness Notions and Their Lowness Properties
We study randomness notions given by higher recursion theory, establishing the relationships Π1-randomness ⊂ Π1-Martin-Löf randomness ⊂ ∆1randomness = ∆1-Martin-Löf randomness. We characterize the set of reals that are low for ∆1 randomness as precisely those that are ∆1 -traceable. We prove that there is a perfect set of such reals.
متن کاملLowness for Bounded Randomness Rod Downey and Keng
In [3], Brodhead, Downey and Ng introduced some new variations of the notions of being Martin-Löf random where the tests are all clopen sets. We explore the lowness notions associated with these randomness notions. While these bounded notions seem far from classical notions with infinite tests like Martin-Löf and Demuth randomness, the lowness notions associated with bounded randomness turn out...
متن کاملRandomness and lowness notions via open covers
One of the main lines of research in algorithmic randomness is that of lowness notions. Given a randomness notion R, we ask for which sequences A does relativization to A leave R unchanged (i.e., RA = R)? Such sequences are call low for R. This question extends to a pair of randomness notions R and S , where S is weaker: for which A is S A still weaker than R? In the last few years, many result...
متن کاملLowness for Computable Machines
Two lowness notions in the setting of Schnorr randomness have been studied (lowness for Schnorr randomness and tests, by Terwijn and Zambella [19], and by Kjos-Hanssen, Stephan, and Nies [7]; and Schnorr triviality, by Downey, Griffiths and LaForte [3, 4] and Franklin [6]). We introduce lowness for computable machines, which by results of Downey and Griffiths [3] is an analog of lowness for K. ...
متن کاملThema Algorithmic Randomness
We consider algorithmic randomness in the Cantor space C of the infinite binary sequences. By an algorithmic randomness concept one specifies a set of elements of C, each of which is assigned the property of being random. Miscellaneous notions from computability theory are used in the definitions of randomness concepts that are essentially rooted in the following three intuitive randomness requ...
متن کامل