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

Abstract

The Kučera-Gács theorem 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 $\sqrt{n}\log n$. A similar upper bound is implicit in the later proof by Merkle and Mihailović. In this paper we obtain strict optimal lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function $g$ such that ${\sum }_n{2}^{-g(n)}=\infty$ 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 ${\sum }_n{2}^{-g(n)}<\infty$. Moreover, the class of such reals is comeager and includes a ${\Delta }_2^0$ real as well as all weakly 2-generic reals. This excludes many slow growing functions such as $\log n$ from bounding the redundancy in computations from random oracles for a large class of reals. On the other hand it was recently shown that if ${\sum }_n{2}^{-g(n)}<\infty$ then $g$ is a general upper bound for the redundancy in computations of any real from some Martin-Löf random oracle. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.