On initial segment complexity and degrees of randomness

Annals of Pure and Applied Logic, 2004

We investigate the initial segment complexity of random reals. Let K() denote preÿx-free Kolmogorov complexity. A natural measure of the relative randomness of two reals and ÿ is to compare complexity K(n) and K(ÿ n). It is well-known that a real is 1-random i there is a constant c such that for all n, K(n) ¿ n − c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the ÿne behaviour of K(n) for random. Following work of Downey, Hirschfeldt and LaForte, we say that 6 K ÿ i there is a constant O(1) such that for all n, K(n) 6 K(ÿ n) + O(1). We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real so that lim sup n K(n) − K(n) = ∞ where is Chaitin's random real. One is based upon (unpublished) work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that ({ÿ : ÿ 6 K }) = 0. As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if n = m then the initial segment complexities of the natural n-and m-random sets (namely ∅(n−1) and ∅(m−1)) are di erent. The techniques introduced in this paper have already found a number of other applications.

Bulletin of Symbolic Logic, 2013

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

Journal of Computer and System Sciences, 2004

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)).

Theoretical Computer Science, 2002

We consider for a real number the Kolmogorov complexities of its expansions with respect to di erent bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the preÿxes of their expansions with respect to di erent bases r and b are related in a way that depends only on the relative information of one base with respect to the other.

Advances in Mathematics, 2011

We study oscillation in the prefix-free complexity of initial segments of 1-random reals. For upward oscillations, we prove that P n∈ω 2 −g(n)

2012

The aim of this expository paper is to present a nice series of results, obtained in the papers of Chaitin [3], Solovay [8], Calude et al. [2], Kučera and Slaman [5]. This joint effort led to a full characterization of lower semicomputable random reals, both as those that can be expressed as a "Chaitin Omega" and those that are maximal for the Solovay reducibility. The original proofs were somewhat involved; in this paper, we present these results in an elementary way, in particular requiring only basic knowledge of algorithmic randomness. We add also several simple observations relating lower semicomputable random reals and busy beaver functions.

Annals of Pure and Applied Logic, 2006

If x = x 1 x 2 · · · x n · · · is a random sequence, then the sequence y = 0x 1 0x 2 · · · 0x n · · · is clearly not random; however, y seems to be "about half random". L. Staiger [Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 (1993) 159-194 and A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory Comput. Syst. 31 (1998) 215-229] and K. Tadaki [A generalisation of Chaitin's halting probability Ω and halting self-similar sets, Hokkaido Math. J. 31 (2002) 219-253] have studied the degree of randomness of sequences or reals by measuring their "degree of compression". This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain a characterisation of Σ 1 -dimension (as defined by Schnorr and Lutz in terms of martingales) in terms of strong Martin-Löf ε-tests (a variant of Martin-Löf tests), and we show that ε-randomness for ε ∈ (0, 1) is different (and more difficult to study) than the classical 1-randomness.

Lecture Notes in Computer Science, 2004

Recently there has been exciting progress in our understanding of algorithmic randomness for reals, its calibration, and its connection with classical measures of complexity such as degrees of unsolvability. In this paper, I will give a biased review of (some of) this progress. In particular, I will concentrate upon randomness for reals. In this paper "real" will mean a member of Cantor space 2 ω. This space is equipped with the topology where the basic clopen sets are [σ] = {σα : α ∈ 2 ω }. Such clopen sets have measure 2 −|σ|. This space is measure-theoretically identical with the rational interval (0, 1), without being homeomorphic spaces. An important program which began in the early 20th Century was to give a proper mathematical foundation to notion of randomness. In terms of understanding this for probability theory, the work of Kolmogorov and others provides an adequate foundation. However, another key direction is to attempt to answer this question via notion of randomness in terms of algorithmic randomness. Here we try to capture the nature of randomness in terms of algorithmic considerations. (This is implicit in the work on Kollektivs in the fundamental paper of von Mises [88].) There are three basic approaches to algorithmic randomness. They are to characterize randomness in terms of algorithmic predictability ("a random real should have bits that are hard to predict"), algorithmic compressibility ("a random real should have segments that are hard to describe with short programs"), and measure theory ("a random real should pass all reasonable algorithmic statistical tests"). A classic example of the relationship between these three is given by the emergence of what is now called Martin-Löf randomness. For a real α = .a 1 a 2 • • • ∈ 2 ω , a consequence of the law of large numbers is that if α is to be random then lim s a1+•••+as s = 1 2. Consider the null set of reals that fail such a test. Then Martin-Löf argued that a real α can only be random if it was not in such a null set. He argued that a random real should pass all such "effectively presented" statistical tests. Thus we define a Martin-Löf test as a computable collection ⋆ Research supported by the Marsden Fund of New Zealand.

The Journal of Symbolic Logic, 2008

We say that A ≤ LR B if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever ∝ is not GL 2 the LR degree of ∝ bounds degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees.

SIAM Journal on Computing, 2002

We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [32] and studied by Calude, Hertling, Khoussainov, and Wang [6], Calude [2], Kučera and Slaman [20], and Downey, Hirschfeldt, and LaForte [15], among others. This measure is called domination or Solovay reducibility, and is defined by saying that α dominates β if there are a constant c and a partial computable function ϕ such that for all positive rationals q < α we have ϕ(q) ↓< β and β − ϕ(q) c(α − q). The intuition is that an approximating sequence for α generates one for β whose rate of convergence is not much slower than that of the original sequence. It is not hard to show that if α dominates β then the initial segment complexity of α is at least that of β. In this paper we are concerned with structural properties of the degree structure generated by Solovay reducibility. We answer a natural question in this area of investigation by proving the density of the Solovay degrees. We also provide a new characterization of the random c.e. reals in terms of splittings in the Solovay degrees. Specifically, we show that the Solovay degrees of computably enumerable reals are dense, that any incomplete Solovay degree splits over any lesser degree, and that the join of any two incomplete Solovay degrees is incomplete, so that the complete Solovay degree does not split at all. The methodology is of some technical interest, since it includes a priority argument in which the injuries are themselves controlled by randomness considerations.