Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

The Kučera-Gács theorem [Kuč85, Gác86] is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first n bits of a sequence requires n + h(n) bits of the random oracle, then h is the redundancy of the computation. Kučera implicitly achieved redundancy n log n while Gács used a more elaborate coding procedure which achieves redundancy √ n log n. A similar bound is implicit in the later proof byMerkle and Mihailović [MM04]. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function g such that ∑ n 2 −g(n) = ∞ is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf random oracle satisfies ∑ n 2 −g(n) < ∞. Moreover, the class of such reals is comeager and includes a ∆ 2 real as well as all weakly 2-generic reals. On the other hand, it has been recently shown in [BLP16] that any real is computable from a Martin-Löf random oracle with redundancy g, provided that g is a computable nondecreasing function such that ∑ n 2 −g(n) < ∞. Hence our lower bound is optimal, and excludes many slow growing functions such as log n from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop. 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]: 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 Jason Teutsch Department of Computer and Information Sciences, University of Alabama at Birmingham. E-mail: [email protected]. Web: http://people.cs.uchicago.edu/∼teutsch ∗Barmpalias was supported by the 1000 Talents Program for Young Scholars from the Chinese Government, grant no. D1101130. 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. We thank the two anonymous referees and Peter Gács for feedback that improved the presentation of this article.