Randomness via effective descriptive set theory

Abstract

An analog of ML-randomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π 1 1 counterparts. We prove the analogs of the Kraft-Chaitin Theorem and Schnorr's Theorem. In the new setting, while K-trivial sets exist that are not hyperarithmetical, each low for random set is. Finally, we begin to study a very strong yet effective randomness notion: Z is Π 1 1 random if Z is in no null Π 1 1 class. There is a greatest Π 1 1 null class, that is, a universal test for this notion.

Higher Randomness and Genericity

“…We mainly deal with what is called Π 1 1 -randomness and Σ 1 1 -genericity. The notion of Π 1 1 -randomness goes back to Sacks [Sac90] and Kechris [Kec75], and it started to be studied formally by Hjorth and Nies [HN07]. It is a notion of interest because of some remarkable properties shared with no other randomness notion.…”

“…Sacks (see [Sac90,IV2.5]) was the first to define the notion of Π 1 1 -randomness and show it is distinct from ∆ 1 1 -randomness. An important advance in the theory of 'higher randomness' was made by Hjorth and Nies in [HN07]. They used the analogy between computably enumerable and Π 1 randomness.…”

Randomness and degree theory for infinite time register machines1

“…We are pursuing the notion of ITRM-randomness in ongoing work. In contrast to strong Π 1 1 -randomness ( [8], [17]), it will be shown below that there is no universal ITRM-test.…”

“…Since its introduction, several variants of this general approach to defining randomness have been considered; a recent example is the work of Hjorth and Nies on Π 1 1 -randomness and a Π 1 1 version of ML-randomness, which led to interesting connections with descriptive set theory ( [8]). …”

Feedback Turing Computability, and Turing Computability as Feedback

“…In particular, to have this notion make sense, we need a notion of "machine" which takes in finite elements and outputs finite elements. One solution to this problem was proposed in [4], which introduced the notion of a Π 1 1 -machine and showed that the analogous notion of Kolmogorov randomness agrees with the notion of passing Π 1 1 Martin-Löf tests. Feedback machines provide another natural notion of a finite machine that performs a meta-computation.…”