Source: J. Symbolic Logic
Volume 69, Issue 2
Schnorr randomness is a notion of algorithmic randomness for real numbers closely
related to Martin-Löf randomness. After its initial development in the 1970s
the notion received considerably less attention than Martin-Löf randomness, but
recently interest has increased in a range of randomness concepts. In this article,
we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr
random reals. We show that there are c.e. reals that are Schnorr random but
not Martin-Löf random, and provide a new characterization of Schnorr random
real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf
random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all
are in high Turing degrees. We use the machine characterization to define a notion
of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity
of reals. We define the class of “Schnorr trivial” reals, which are ones whose
initial segment complexity is identical with the computable reals, and demonstrate
that this class has non-computable members.
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