Optimal redundancy in computations from random oracles

Abstract

It is a classic result in algorithmic information theory that every infinite binary sequence is computable from an infinite binary sequence which is random in the sense of Martin-Löf. 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. We devise a new coding method that achieves optimal logarithmic redundancy. For any computable non-decreasing function g such that ∑i2−g(i) is bounded we show that there is a coding process that codes any given infinite binary sequence into a Martin-Löf random infinite binary sequence with redundancy g. This redundancy bound is known to be the best possible.