Randomness via infinite computation and effective descriptive set theory

TL;DR

This paper explores randomness notions related to infinite time Turing machines, establishing their properties and differences from classical and hyperarithmetic randomness, and introduces technical results on forcing and admissible sets.

Contribution

It introduces and analyzes randomness notions associated with infinite time Turing machines, showing they share properties with classical randomness and differ from hyperarithmetic cases.

Findings

Randomness notions for infinite time Turing machines have properties similar to classical randomness.

Mutual randoms do not share information and satisfy van Lambalgen's theorem.

If a real is computable relative to all reals in a positive measure set, it is already computable.

Abstract

We study randomness beyond $\Pi^1_1$-randomness and its Martin-L\"of type variant, introduced in \cite{MR2340241} and further studied in \cite{Continuous-higher-randomness}. The class given by the infinite time Turing machines (\ITTM s), introduced by Hamkins and Kidder, is strictly between $\Pi^1_1$ and $\Sigma^1_2$. We prove that the natural randomness notions associated to this class have several desirable properties resembling those of the classical random notions such as Martin-L\"of randomness, and randomness notions defined via effective descriptive set theory such as $\Pi^1_1$-randomness. For instance, mutual randoms do not share information and can be characterized as in van Lambalgen's theorem. We also obtain some differences to the hyperarithmetic setting. Already at the level of $\Sigma^1_2$, some properties of randomness notions are independent \cite{Infinite-computations}. Towards the results about randomness, 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, we prove facts of independent interest about random forcing over admissible sets and increasing unions of admissible sets. These results are also useful for more efficient proofs of some classical results about hyperarithmetic sets.