Kenshi Miyabe

Truth-table Schnorr randomness and truth-table reducible randomness

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness (defined in 6 too only by martingales) and truth-table reducible randomness, for which we prove that van Lambalgens Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgens Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness.

Schnorr Triviality and Its Equivalent Notions

We give some characterizations of Schnorr triviality. In concrete terms, we introduce a reducibility related to decidable prefix-free machines and show the equivalence with Schnorr reducibility. We also give a uniform-Schnorr-randomness version of the equivalence of LR-reducibility and LK-reducibility. Finally we prove a base-type characterization of Schnorr triviality.

Uniform 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 characterizations, such as those via complexity, martingales, Kurtz tt-traceability, and Kurtz dimensional measure.

Characterization of Kurtz Randomness by a Differentiation Theorem

Brattka, Miller and Nies (2012) showed that some major algorithmic randomness notions are characterized via differentiability. The main goal of this paper is to characterize Kurtz randomness by a differentiation theorem on a computable metric space. The proof shows that integral tests play an essential part and shows that how randomness and differentiation are connected.

Algorithmic randomness over general spaces

The study of Martin-Lof randomness on a computable metric space with a computable measure has seen much progress recently. In this paper we study Martin-Lof randomness on a more general space, that is, a computable topological space with a computable measure. On such a space, Martin-Lof randomness may not be a natural notion because there is no universal test, and Martin-Lof randomness and complexity randomness (defined in this paper) do not coincide in general. We show that SCT3 is a sufficient condition for the existence and coincidence, and study how much we can weaken this condition.

Reducibilities Relating to Schnorr Randomness

Some measures of randomness have been introduced for Martin-Löf randomness such as K-reducibility, C-reducibility and vL-reducibility. In this paper we study Schnorr-randomness versions of these reducibilities. In particular, we characterize the computably-traceable reducibility via relative Schnorr randomness, which was asked in Nies’ book [Nies 2009, Problem 8.4.22]. We also show that Schnorr reducibility implies uniform-Schnorr-randomness version of vL-reducibility, which is the Schnorr-randomness version of the result that K-reducibility implies vL-reducibility.

Unified characterizations of lowness properties via Kolmogorov complexity

Consider a randomness notion

Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

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