Schnorr randomness

Abstract

Abstract.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.

On Schnorr and computable randomness, martingales, and machines

“…Since effective martingales correspond to computably enumerable betting strategies rather than computable betting strategies, Schnorr argued that 1-randomness is too strong to capture the intuitive notion of effective randomness. He suggested two alternative notions Definition 4 (Schnorr [14]) (i) We say that a real α is computably random iff no computable martingale 3) F : 2 <ω −→ Q + ∪ {0} succeeds.…”

“…It was a longstanding open question of van Lambalgen and others, for instance, to give a machine characterization of Schnorr or of computable randomness. In [3], Downey and Griffiths gave the first machine characterization of Schnorr randomness.…”

“…Nies has later shown that the Turing degrees of K-trivial reals form a Σ 0 3 ideal in the degrees, and are all low. In [3], Downey and Griffiths began the study of Schnorr trivial reals, where now we ask that α ≤ Sch 1 ∞ . Downey and Griffiths proved that noncomputable Schnorr trivials exist.…”

“…It was shown in Downey and Griffiths [3] that Schnorr randomness is closely related to another pre-theoretic notion of randomness: one involving the difficulty of describing initial segments of a random real via prefix-free machines with a computable domain. If M is a prefix-free machine, then µ(M) = M(σ)↓ 2 −|σ| is a c. e. real.…”

“…(If τ is not in the range of M, then its M-complexity is ∞.) Schnorr and Chaitin showed that a real x is Martin-Löf random if and only if there is no way for any prefix-free machine to describe its initial segments succinctly: for every prefix-free machine M there is a c such that for every n, K M (x n) > n − c. Downey and Griffiths [3] showed requiring the measure of the domain of a prefix-free machine to be a computable real leads to an alternative characterization of Schnorr randomness. Notice that no universal prefix-free machine can be computable, since such a machine must have measure equal to a Martin-Löf random real, which must have degree 0 .…”