Affiliations: [a] State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China. School of Mathematics and Statistics, Victoria University of Wellington, New Zealand. barmpalias@gmail.com. http://barmpalias.net | [b] Department of Mathematics, Columbia House, London School of Economics, Houghton St., London, WC2A 2AE, United Kingdom. A.Lewis7@lse.ac.uk. http://aemlewis.co.uk | [c] State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China. angsheng@ios.ac.cn
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 Fortnow (unpublished) and Nies, Stephan and Terwijn [J. Symbolic Logic 70 (2005), 515–535] observed that the condition K(X↾n)⩾n−c can be replaced with K(X↾rn)⩾rn−c, for any fixed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∀nK(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 ∀nK(X↾rn)⩾rn.
