Pointed computations and Martin-Löf randomness

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 Fortnow (unpublished) and Nies, Stephan and Terwijn [NST05] observed that the condition K(X ↾n) ≥ n− c can be replaced with K(X ↾rn ) ≥ rn − c, for any fixed increasing computable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. George Barmpalias State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China. School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, New Zealand. E-mail: [email protected]. Web: http://barmpalias.net Andrew Lewis-Pye Department of Mathematics, Columbia House, London School of Economics, Houghton St., London, WC2A 2AE, United Kingdom. E-mail: [email protected]. Web: http://aemlewis.co.uk Angsheng Li State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China. E-mail: [email protected]. Date: August 9, 2016. Barmpalias was supported by the 1000 Talents Program for Young Scholars from the Chinese Government. Additional support was received by the Chinese Academy of Sciences (CAS) and the Institute of Software of the CAS. Lewis-Pye was supported by a Royal Society University Research Fellowship.