On Kurtz randomness

Sub-computable Boundedness Randomness

“…The above proofs are similar to those that have already appeared in the literature. For example Downey, Griffiths and Reid [16] proved the equivalence of analogues of (1) and (3) of Theorem 2.7 in the setting of Kurtz randomness, and their proof could be modified to prove the equivalence of (1) and (3) in Theorem 2.7. Similarly, Bienvenu and Merkle [6] gave a proof the equivalence of parts (1) and (2) in the setting of Kurtz randomness and their proof can be modified to work in our setting.…”

Algorithmic randomness and computability

“…Downey, Griffith and Reid [21] gave a machine characterization of Kurtz randomness, and showed that each computably enumerable non-zero degree contained a Kurtz random left-c.e. real.…”

“…Yu [104] (also Miller and Greenberg (unpublished)) proved that there are no sets low for 1-genericity. Sets low for Kurtz randomness were first constructed by Downey, Griffiths and Reid [21]. They were shown there to be all hyperimmune-free and were implied by Schnorr lowness.…”

“…We will look at various methods of calibration by initial segment complexity such as those introduced by Solovay [89], Downey, Hirschfeldt, and Nies [26], Downey, Hirschfeldt, and LaForte [23], Downey [16], as well as other methods such as lowness notions of Kučera and Terwijn [47], Terwijn and Zambella [97], Nies [75], [76], Downey, Griffiths and Reid [21], and methods such as higher level randomness notions going back to the work of Kurtz [50], Kautz [38], and Solovay [89], and other calibrations of randomness based on changing definitions along the lines of Schnorr, computable, s-randomness, etc. Particularly fascinating is the recent work on lowness, which began with Downey, Hirschfeldt, Nies and Stephan, and developed in a series of deep papers by Nies [75], [76] and his co-authors.…”

Higher Kurtz randomness

“…Chong, Nies and Yu [1] proved a higher analog of this result, characterizing lowness for ∆ 1 1 randomness by ∆ 1 1 traceability. Our goal is to carry out similar investigations for higher analogs of Kurtz randomness [3]. A real x is Kurtz random if avoids each Π 0 1 null class.…”