Relative Randomness and Cardinality

George Barmpalias

Relative Randomness and Cardinality

2010, Notre Dame Journal of Formal Logic

Abstract

A set B ⊆ N is called low for Martin-Löf random if every Martin-Löf random set is also Martin-Löf random relative to B. We show that a ∆ 0 2 set B is low for Martin-Löf random iff the class of oracles which compress less efficiently than B, namely the class is countable (where K denotes the prefix free complexity and ≤ + denotes inequality modulo a constant). It follows that ∆ 0 2 is the largest arithmetical class with this property and if C B is uncountable, it contains a perfect Π 0 1 set of reals. The proof introduces a new method for constructing non-trivial reals below a ∆ 0 2 set which is not low for Martin-Löf random.

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 R which have the property that if α R 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. The authors' research was supported by the Marsden Fund for Basic Science. We work in Cantor space 2 ω with basic clopen sets [σ] = {σα : α ∈ 2 ω } for strings σ ∈ 2 <ω. The Lebesgue measure of a clopen set [σ] is 2 −|σ|. This space is measure-theoretically identical with the interval of reals (0, 1), though the two spaces are not homeomorphic. We identify a real with its binary expansion, which we may think of as an element of 2 ω , and hence with the set of natural numbers whose characteristic function is the same as that expansion. (Some reals have two binary expansions; for such a real, which is always rational, we choose the nonterminating expansion.) We also identify finite binary strings with rationals. Our computability-theoretic notation follows the standard of Soare [45]. Our main concern will be reals that are limits of computable increasing sequences of rationals. We call such reals computably enumerable (c.e.), though they have also been called recursively enumerable, left computable (by Ambos-Spies, Weihrauch, and Zheng [2]), left semicomputable, and lower semicomputable. If, in addition to the existence of a computable increasing sequence q 0 , q 1 ,. .. of rationals with limit α, there is a total computable function f such that α − q f (n) < 2 −n for all n, then α is called computable. These and related concepts have been widely studied. In addition to the papers and books mentioned elsewhere in this introduction, we may cite, among others, early work of Rice [41], Lachlan [31], Soare [43], and Ceȋtin [10], and more recent papers by Ko [24, 25], Calude, Coles, Hertling, and Khoussainov [7], Ho [23], and Downey and LaForte [20]. Several of the results mentioned below provide strong evidence that computably enumerable reals are natural objects in the study of effective randomness in the same way that computably enumerable sets are natural objects in classical computability theory. An alternate definition of c.e. reals can be given as follows. Definition 1.1. A set A ⊆ N is nearly computably enumerable if there is a computable approximation {A s } s∈ω such that A(x) = lim s A s (x) for all x and A s (x) > A s+1 (x) ⇒ ∃y < x(A s (y) < A s+1 (y)).

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