Randomness via Infinite Computation and Effective Descriptive Set Theory

Abstract

We study randomness beyond Π 1 1 -randomness and its Martin-Löf type variant, introduced in [HN07] and further studied in [BGM].The class given by the infinite time Turing machines (ITTMs), introduced by Hamkins and Kidder, is strictly between Π 1 1 and Σ 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-Löf randomness, and randomness notions defined via effective descriptive set theory such as Π 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 Σ 1 2 , some properties of randomness notions are independent [CS17]. 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.

Measure-theoretic uniformity and the Suslin functional

“…Fourth, Carl and Schlicht proved the analogue of Proposition 1.4 for infinite time Turing machine computability in [2,Theorem 4.5]. In Section 4 we will look closer at how their results connect to ours.…”

“…the basis theorem is not mere theory. Finally, we discuss similar generalisations for infinite time Turing machines, originally due to Carl and Schlicht [2].…”

“…Using the above results, we show that the computability theory induced by infinite time Turing machines (ITTMs) satisfies similar properties of measure-theoretic uniformity as for the hyperarithmetical sets (Tanaka-Sacks; see Proposition 1.3) and computability relative to S (Section 3). The key result of this section is already known, namely as [2,Theorem 4.5], where it is proved in the context of random reals and forcing. In this section, we adjust Theorem 3.9 to obtain a generalisation to ITTMs.…”

“…Remark 3.6. Lemma 3.7 and Theorem 3.9 below are consequences of Lemma 2.11 in Carl and Schlicht [2]. They work in the context of random reals and forcing.…”

Some Observations on Infinitary Complexity

“…in [wITRM], [KS], [ITRM], [OTM], [ORM], analogues of several central topics in classical computability theory were developed for these machine types, among them degree theory [W1], computable model theory [? ], randomness ( [CS], [C14], [CS2]) and complexity theory. Complexity theory was first studied by Schindler in the case of ITTMs, who proved that P = NP for ITTMs [Schindler], which was later refined in various ways [DHS], [HW].…”