The K-Degrees, Low for K Degrees,and Weakly Low for K Sets

Abstract

We call A weakly low for K if there is a c such that $K^A(\sigma )\ge K(\sigma )-c$ for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K if and only if Chaitin's Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely often maximal. This had previously been proved for plain Kolmogorov complexity.