Cookies?
Library Header Image
LSE Research Online LSE Library Services

Pointed computations and Martin-Löf randomnesss

Barmpalias, George and Lewis-Pye, Andrew and Li, Angsheng (2017) Pointed computations and Martin-Löf randomnesss. Computability. ISSN 2211-3568 (In Press)

[img] PDF - Accepted Version
Restricted to Repository staff only

Download (118kB) | Request a copy

Abstract

Schnorr showed that a real X is Martin-Löf random if and only if K(X �n) ≥ n − c for some constant c and all n, where K denotes the prefix-free 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 Fortnow (unpublished) and Nies, Stephan and Terwijn [NST05] observed that the condition K(X �n) ≥ n−c can be replaced with K(X �rn ) ≥ rn −c, for any fixed increasing computable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 Martin-Löf random if and only if ∃c ∀n K(X �rn ) ≥ rn − c, where (rn) is any fixed pointedly X-computable sequence, in the sense that rn is computable from X in a self-delimiting 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 Martin-Löf random, but for which there exists some X-computable sequence (rn) such that ∀n K(X �rn ) ≥ rn.

Item Type: Article
Official URL: http://www.computability.de/journal/
Additional Information: © 2017 IOS Press
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 17 Mar 2017 15:44
Last Modified: 05 Sep 2017 09:14
Projects: D1101130, 2014CB340302
Funders: 1000 Talents Program for Young Scholars, Chinese Academy of Sciences, Institute of Software of the CAS, Royal Society University Research Fellowship, National Basic Science Program (973 Program Group)
URI: http://eprints.lse.ac.uk/id/eprint/69854

Actions (login required)

View Item View Item

Downloads

Downloads per month over past year

View more statistics