Randomness and lowness notions via open covers

TL;DR

This paper introduces a unified framework using open covers to characterize various lowness notions in algorithmic randomness, simplifying proofs and deriving new results about the computational strength of random sequences.

Contribution

It provides a general approach to express lowness notions via open covers, unifying existing results and enabling new insights into randomness and computational strength.

Findings

Characterization of low for randomness via open covers.

Proof that Low(MLR;SR) and Low(W2R;SR) coincide.

New characterizations of highness notions and lowness for tests.

Abstract

One of the main lines of research in algorithmic randomness is that of lowness notions. Given a randomness notion R, we ask for which sequences A does relativization to A leave R unchanged (i.e., R^A = R)? Such sequences are call low for R. This question extends to a pair of randomness notions R and S, where S is weaker: for which A is S^A still weaker than R? In the last few years, many results have characterized the sequences that are low for randomness by their low computational strength. A few results have also given measure-theoretic characterizations of low sequences. For example, Kjos-Hanssen proved that A is low for Martin-L\"of randomness if and only if every A-c.e. open set of measure less than 1 can be covered by a c.e. open set of measure less than 1. In this paper, we give a series of results showing that a wide variety of lowness notions can be expressed in a similar way, i.e., via the ability to cover open sets of a certain type by open sets of some other type. This provides a unified framework that clarifies the study of lowness for randomness notions, and allows us to give simple proofs of a number of known results. We also use this framework to prove new results, including showing that the classes Low(MLR;SR) and Low(W2R;SR) coincide, answering a question of Nies. Other applications include characterizations of highness notions, a broadly applicable explanation for why low for randomness is the same as low for tests, and a simple proof that Low(W2R;S)=Low(MLR;S), where S is the class of Martin-L\"of, computable, or Schnorr random sequences. The final section gives characterizations of lowness notions using summable functions and convergent measure machines instead of open covers. We finish with a simple proof of a result of Nies, that Low(MLR) = Low(MLR; CR).