SAS

Abstract

We present some new characterizations of Schnorr triviality. Schnorr triviality is defined using complexity via a computable measure machine, with which Schnorr randomness has a characterization. Since we have a characterization of Schnorr randomness via decidable prefix-free machine, we also have a characterization of Schnorr triviality using complexity via a decidable prefix-free machine. It should be noted that numerous characterizations of Schnorr triviality have the following form: for any computable object, there exists another computable object such that the real is in some object. By defining a basis for Schnorr randomness in a similar manner, we can show the equivalence to Schnorr triviality while Franklin and Stephan (2010) showed that there exists a Schnorr trivial set that is not truth-table reducible to any Schnorr random set.