Optimal redundancy in computations from random oracles

A classic result in algorithmic information theory is that every infinite binary sequence is computable from a Martin-Löf random infinite binary sequence. Proved independently by Kučera [Kuč85] and Gács [Gác86], this result answered a question by Charles Bennett and has seen numerous applications in the last 30 years. The optimal redundancy in such a coding process has, however, remained unknown. If the computation of the first n bits of a sequence requires n + g(n) bits of the random oracle, then g is the redundancy of the computation. Kučera implicitly achieved redundancy n log n while Gács used a more elaborate block-coding procedure which achieved redundancy √ n log n. Different approaches to coding such as the one by Merkle and Mihailović [MM04] have not improved this redundancy bound. In this paper we devise a new coding method that achieves optimal logarithmic redundancy. In particular, for any computable non-decreasing function g such that ∑ i 2 −g(i) < 1 we show that there is a uniform coding process that codes every infinite binary sequence into a Martin-Löf infinite binary sequence with redundancy g. This redundancy bound is exponentially smaller1 than the previous bound of √ n log n and is known to be the best possible by recent work [BLPT15], where it was shown that if ∑ i 2 −g(i) diverges then there exists an infinite binary sequence X which cannot be computed by any Martin-Löf random infinite binary sequence with redundancy g. It follows that redundancy ǫ · log n in computation from a random oracle is possible for every stream, if and only if ǫ > 1. 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 Street, London, WC2A 2AE, United Kingdom. E-mail: [email protected]. Web: http://aemlewis.co.uk ∗Barmpalias was supported by the 1000 Talents Program for Young Scholars from the Chinese Government, and the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative No. 2010Y2GB03. Additional support was received by the CAS and the Institute of Software of the CAS. Partial support was also received from a Marsden grant of New Zealand and the China Basic Research Program (973) grant No. 2014CB340302. Lewis-Pye was supported by a Royal Society University Research Fellowship. 1When f and g are unbounded, we say g is exponentially smaller than f if there exists a constant c such that 2c·g(n) < f (n) for all n.