Randomness and Reducibility

How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of reducibilities that act as measures of relative randomness, as embodied in the concept of initial-segment complexity. The initial segment complexity of a real is a natural measure of its relative randomness, and has been implicitly studied by many authors. For instance, by the work of Schnorr we know that a real α is Martin-Löf random if and only if its initial segment complexity is roughly speaking as big as it can be. (See below for the relevant definitions.) That is, if we denote prefix-free Kolmogorov complexity by H, then α is Martin-Löf random if and only if there is a constant c such that H(α n) > n− c for all n, where α n denotes the initial segment of α of length n. Furthermore, the work of Barzdins [3] shows that if a set is computably enumerable then its plain Kolmogorov complexity is bounded by 2 log n, and this bound can be sharp, as shown by Kummer [30]. Finally, recent work of Levin, Lutz, Mayordomo, Staiger, and others (e.g., [38, 52, 36, 34]) proves that effective Hausdorff dimension is essentially intertwined with initial segment complexity. We look at reducibilities 6R which have the property that if α 6R β then the prefix-free initial segment complexity of α is no greater than that of β (up to an additive constant), and hence act as measures of relative randomness. One such reducibility, called domination or Solovay reducibility, was introduced by Solovay [50], and has been studied by Calude, Hertling, Khoussainov, and Wang [8], Calude [4], Kučera and Slaman [29], and Downey, Hirschfeldt, and Nies [18], among others. Solovay reducibility has proved to be a powerful tool in the study of randomness of effectively presented reals. Motivated by certain shortcomings of Solovay reducibility, which we will discuss below, we introduce two new reducibilities and study, among other things, the relationships between these various measures of relative randomness.