Barmpalias, George, LewisPye, Andrew and Li, Angsheng
(2018)
Pointed computations and MartinLöf randomnesss.
Computability, 7 (23).
pp. 171177.
ISSN 22113568
Abstract
Schnorr showed that a real X is MartinLöf random if and only if K(X �n) ≥ n − c for some constant c and all n, where K denotes the prefixfree complexity function. Fortnow (unpublished) and Nies, Stephan and Terwijn [NST05] observed that the condition K(X �n) ≥ n−c can be replaced with K(X �rn ) ≥ rn −c, for any fixed increasing computable sequence (rn), in this characterization. The purpose of this note is to establish the following generalisation of this fact. We show that X is MartinLöf random if and only if ∃c ∀n K(X �rn ) ≥ rn − c, where (rn) is any fixed pointedly Xcomputable sequence, in the sense that rn is computable from X in a selfdelimiting way, so that at most the first rn bits of X are queried in the computation of rn. On the other hand, we also show that there are reals X which are very far from being MartinLöf random, but for which there exists some Xcomputable sequence (rn) such that ∀n K(X �rn ) ≥ rn.
Actions (login required)

View Item 