Calibrating Randomness

Abstract

We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level randomness notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set.

Algorithmically Independent Sequences

“…This is a notion that has been intensively studied recently (see [DHNT06]). Clearly every computable sequence is H-trivial, but the converse does not hold [Zam90,Sol75].…”

Turing incomparability in Scott sets

“…A short survey of it is also given in Ambos-Spies and Kučera [1]. Much deeper insight into the subject of algorithmic randomness can be found in a forthcoming book of Downey and Hirschfeldt [4]; a good survey is also in Downey, Hirschfeldt, Nies and Terwijn [5].…”

Strong jump-traceability II: K-triviality

“…All of these notions express feebleness of an oracle with respect to some notion of algorithmic randomness: for example, a real A is low for Martin-Löf randomness if every real which is Martin-Löf random remains Martin-Löf random relative to A; in other words, A cannot detect patterns in Martin-Löf random sets. We refer the reader to [7,9,23,24,25] for details of such results.…”