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RANDOMNESS VIA INFINITE COMPUTATION AND EFFECTIVE DESCRIPTIVE SET THEORY

Published online by Cambridge University Press:  01 August 2018

MERLIN CARL  and
PHILIPP SCHLICHT
MERLIN CARL
Affiliation:
FACHBEREICH MATHEMATIK UND STATISTIK UNIVERSITÄT KONSTANZ 78457 KONSTANZ, GERMANYE-mail:merlin.carl@uni-konstanz.de
PHILIPP SCHLICHT
Affiliation:
MATHEMATISCHES INSTITUT, UNIVERSITÄT BONN ENDENICHER ALLEE 60 53115 BONN, GERMANYE-mail:schlicht@math.uni-bonn.de
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Abstract

We study randomness beyond ${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between ${\rm{\Pi }}_1^1$ and ${\rm{\Sigma }}_2^1$ that is given by the infinite time Turing machines (ITTMs) introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined via effective descriptive set theory such as ${\rm{\Pi }}_1^1$-randomness. For instance, mutual randoms do not share information and a version of van Lambalgen’s theorem holds.

Towards these results, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool towards this result, we prove facts of independent interest about random forcing over increasing unions of admissible sets, which allow efficient proofs of some classical results about hyperarithmetic sets.

Keywords

03D6003D3203E15algorithmic randomnessinfinite time Turing machineseffective descriptive set theory
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 2 , June 2018 , pp. 766 - 789
Copyright
Copyright © The Association for Symbolic Logic 2018 

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