Lowness for Weakly 1-generic and Kurtz-Random
نویسندگان
چکیده
It is shown that a set is low for weakly 1-generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, this answers negatively a recent question on the characterization of these sets. Furthermore, it is shown that every set which is low for weakly 1-generic is also low for Kurtz-random. In addition to this, it is shown that a set satisfies the notion “low for diagonally non-recursive” as introduced by Kjos-Hanssen and Nies iff it is recursive.
منابع مشابه
Higher Kurtz randomness
A real x is ∆1-Kurtz random (Π 1 1-Kurtz random) if it is in no closed null ∆1 set (Π 1 1 set). We show that there is a cone of Π 1 1-Kurtz random hyperdegrees. We characterize lowness for ∆1-Kurtz randomness as being ∆ 1 1-dominated and ∆ 1 1semi-traceable.
متن کاملLowness for Kurtz randomness
We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmune-free non-DNR degrees, which are also exactly the degrees that are low for weak 1-genericity. We also consider Low(M, Kurtz), the class of degrees a such that every element of M is a-Kurtz random. These are ...
متن کاملLowness for uniform Kurtz randomness
We propose studying uniform Kurtz randomness, which is the uniform relativization of Kurtz randomness. One advantage of this notion is that lowness for uniform Kurtz randomness has many characterizations, such as those via complexity, martingales, Kurtz tt-traceability, and Kurtz dimensional measure.
متن کاملNull-additivity in the Theory of Algorithmic Randomness
In this paper, we develop a general framework integrating algorithmic and higher randomness theories. We clarify the relationship of the notions of triviality and uniform-lowness in algorithmic randomness theory and null-additivity in set theory by effectivizing combinatorial characterizations of transitive additivity in set theory of the real line. For instance, we show that the following thre...
متن کاملUniform Kurtz randomness
We propose studying uniform Kurtz randomness, which is the uniform relativization of Kurtz randomness. This notion has more natural properties than the usual relativization. For instance, van Lambalgen’s theorem holds for uniform Kurtz randomness while not for (the usual relativization of) Kurtz randomness. Another advantage is that lowness for uniform Kurtz randomness has many characterization...
متن کامل