Initial Segment Complexities of Randomness Notions
Schnorr famously proved that Martin-Löf-randomness of a sequence A can be characterised via the complexity of A’s initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that Kolmogorov randomness coincides with Martin-Löf randomness relative to the halting problem K; that is, a set A is Martin-Löf random relative to K iff there is no function f such that for all m and all n > f(m) it holds that C(A(0)A(1)...A(n)) ≤ n − m.
In the present work it is shown that characterisations of this style can also be given for other randomness criteria like strongly random, Kurtz random relative to K, PA-incomplete Martin-Löf random and strongly Kurtz random; here one does not just quantify over all functions f but over functions f of a specific form. For example, A is Martin-Löf random and PA-incomplete iff there is no A-recursive function f such that for all m and all n > f(m) it holds that C(A(0)A(1)...A(n)) ≤ n − m. The characterisation for strong randomness relates to functions which are the concatenation of an A-recursive function executed after a K-recursive function; this solves an open problem of Nies.
In addition to this, characterisations of a similar style are also given for Demuth randomness and Schnorr randomness relative to K. Although the unrelativised versions of Kurtz randomness and Schnorr randomness do not admit such a characterisation in terms of plain Kolmogorov complexity, Bienvenu and Merkle gave one in terms of Kolmogorov complexity defined by computable machines.
KeywordsInitial Segment Recursive Function Kolmogorov Complexity Strong Randomness Peano Arithmetic
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