Adam R. Day

Randomness for non-computable measures

Different approaches have been taken to defining randomness for non-computable probability measures. We will explain the approach of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas. We will show that these approaches are fundamentally equivalent. Having clarified what it means to be random for a non-computable probability measure, we turn our attention to Levin’s neutral measures, for which all sequences are random. We show that every PA degree computes a neutral measure. We also show that a neutral measure has no least Turing degree representation and explain why the framework of the continuous degrees (a substructure of the enumeration degrees studied by Miller) can be used to determine the computational complexity of neutral measures. This allows us to show that the Turing ideals below neutral measures are exactly the Scott ideals. Since X ∈ 2 is an atom of a neutral measure μ if and only if it is computable from (every representation of) μ, we have a complete understanding of the possible sets of atoms of a neutral measure. One simple consequence is that every neutral measure has a Martin-Lof random atom. 1. Defining randomness Let X be an element of Cantor space and μ a Borel probability measure on Cantor space. What should it mean for X to be random with respect to μ? In the case that μ is the Lebesgue measure, then the theory of μrandomness is well developed (for recent treatises on the subject the reader is referred to Downey and Hirschfeldt, and Nies [2, 13]). In fact if μ is a computable measure, then early work of Levin showed that μ-randomness can be seen as essentially a variant on randomness for Lebesgue measure [10]. This leaves the question of how to define randomness if μ is non-computable. We will show that the two approaches that have previously been used to define μ-randomness, for non-computable μ, are equivalent. Later, in Theorem 4.12, we will provide another characterization of μ-randomness using the enumeration degrees. Last compilation: September 8, 2011 Last time the following date was changed: November 22, 2010. 2010 Mathematics Subject Classification. Primary 03D32; Secondary 68Q30, 03D30. The second author was supported by the National Science Foundation under grants DMS-0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness.

Cupping with random sets

We prove that a set is K-trivial if and only if it is not weakly ML-cuppable. Further, we show that a set below zero jump is K-trivial if and only if it is not ML-cuppable. These results settle a question of Ku\v{c}era, who introduced both cuppability notions.

Ergodic-type characterizations of algorithmic randomness

A theorem of Kucera states that given a Martin-Lof random infinite binary sequence ω and an effectively open set A of measure less than 1, some tail of ω is not in A. We show that this result can be seen as an effective version of Birkhoffs ergodic theorem (in a special case). We prove several results in the same spirit and generalize them via an effective ergodic theorem for bijective ergodic maps.

On the computational power of random strings

Abstract There are two fundamental computably enumerable sets associated with any Kolmogorov complexity measure. These are the set of non-random strings and the overgraph. This paper investigates the computational power of these sets. It follows work done by Kummer, Muchnik and Positselsky, and Allender and co-authors. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not t t -complete. This paper answers this question in the negative by proving that the overgraph of any optimal monotone machine, or any optimal process machine, is t t -complete. The monotone results are shown for both descriptional complexity K m and K M , the complexity measure derived from algorithmic probability. A distinction is drawn between two definitions of process machines that exist in the literature. For one class of process machines, designated strict process machines, it is shown that there is a universal machine whose set of non-random strings is not t t -complete.

Limits to joining with generics and randoms

Posner and Robinson [4] proved that if S ⊆ ω is non-computable, then there exists a G ⊆ ω such that S ⊕ G ≥T G′. Shore and Slaman [7] extended this result to all n ∈ ω, by showing that if S T ∅(n−1) then there exists a G such that S ⊕ G ≥T G(n). Their argument employs KumabeSlaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing in any way. We answer the question of whether this is a necessary complication by showing that for all n ≥ 1, the set G of the Shore-Slaman theorem cannot be chosen to be even weakly 2generic. Our result applies to several other effective forcing notions commonly used in computability theory, and we also prove that the set G cannot be chosen to be 2-random.

Process and truth-table characterisations of randomness

This paper uses quick process machines to provide characterisations of computable randomness, Schnorr randomness and weak randomness. The quick process machine is a type of process machine first considered in work of Levin and Zvonkin. A new technique for building process machines and quick process machines is presented. This technique is similar to the KC theorem for prefix-free machines. Using this technique, a method of translating computable martingales to quick process machines is given. This translation forms the basis for these new randomness characterisations. Quick process machines are also used to provide characterisations of computable randomness, Schnorr randomness, and weak randomness in terms of truth-table reducibility.

Notes on Computable Analysis

Computable analysis has been part of computability theory since Turing’s original paper on the subject (Turing, Proc. London Math. Sc. 42:230–265, 1936). Nevertheless, it is difficult to locate basic results in this area. A first goal of this paper is to give some new simple proofs of fundamental classical results (highlighting the role of Π10

Increasing the Gap between Descriptional Complexity and Algorithmic Probability

{{\Pi }_{1}^{0}}

A constructive version of Birkhoff's ergodic theorem for Martin-Löf random points

classes). Naturally this paper cannot cover all aspects of computable analysis, but we hope that this gives the reader a completely self-contained ingress into this area. A second goal is to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpiński, and the Blaschke Selection Theorem.

Density, forcing, and the covering problem

The coding theorem is a fundamental result of algorithmic information theory. A well known theorem of Gács shows that the analog of the coding theorem fails for continuous sample spaces. This means that descriptional monotonic complexity does not coincide within an additive constant with the negative logarithm of algorithmic probability. Gácss proof provided a lower bound on the difference between these values. He showed that for infinitely many finite binary strings, this difference was greater than a version of the inverse Ackermann function applied to string length. This paper establishes that this lower bound can be substantially improved. The inverse Ackermann function can be replaced with a function O(log(log(x))). This shows that in continuous sample spaces, descriptional monotonic complexity and algorithmic probability are very different. While this proof builds on the original work by Gács, it does have a number of new features, in particular, the algorithm at the heart of the proof works on sets of strings as opposed to individual strings.