Lowness for Computable Machines

Journal of Logic and Computation

2013

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.

“…Lowness for Schnorr randomness has previously been studied in the literature such as [9,18,8]. However, lowness for uniform Schnorr randomness has more natural properties [15,22,21].…”

Section: Introductionmentioning

confidence: 99%

Uniform Kurtz randomness

Kihara

Journal of Logic and Computation

2013

“…Similarly, the lowness notions for Schnorr randomness have been studied [7,8,15,6]. Unlike the case of Martin-Löf randomness, lowness for Schnorr randomness is not equivalent to Schnorr triviality, which is a Schnorr-randomness version of K-triviality [8].…”

Section: Introductionmentioning

confidence: 99%

Schnorr Triviality and Its Equivalent Notions

2013

Theory Comput Syst

Truth-table Schnorr randomness and truth-table reducible randomness

Mathematical Logic Quarterly

2011

MSC (2010) 68Q30, 03D15, 03D25Schnorr 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 Lambalgen's 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 Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness.