Solovay functions and their applications in algorithmic randomness

TL;DR

This paper explores Solovay functions, computable functions closely approximating Kolmogorov complexity, and demonstrates their applications in deriving classical results in algorithmic randomness, extending properties from K to these functions.

Contribution

The paper proves that Solovay functions uniquely satisfy key properties of Kolmogorov complexity and extends these characterizations to right-c.e. functions, enriching the understanding of randomness measures.

Findings

Solovay functions approximate Kolmogorov complexity for infinitely many strings.

Classical properties of K hold when replaced by Solovay functions.

Characterizations extend to right-c.e. functions.

Abstract

Classical versions of Kolmogorov complexity are incomputable. Nevertheless, in 1975 Solovay showed that there are computable functions $f > K+O(1)$ such that for infinitely many strings $\sigma$, $f(\sigma)=K(\sigma)+O(1)$, where $K$ denotes prefix-free Kolmogorov complexity (while $C$ denotes plain Kolmogorov complexity). Such an $f$ is now called a Solovay function. We prove that many classical results about $K$ can be obtained by replacing $K$ by a Solovay function. For example, the three following properties of a function $g$ all hold for the function $K$. (i) The sum of the terms $\sum_n 2^{-g(n)}$ is a Martin-L\"of random real. (ii) A sequence A is Martin-L\"of random if and only if $C(A \upharpoonright n) > n -g(n)-O(1)$. (iii) A sequence A is K-trivial if and only if $K(A \upharpoonright n) < g(n) + O(1)$. We show that when fixing any of these three properties, then among all computable functions exactly the Solovay functions possess this property. Furthermore, this characterization extends accordingly to the larger class of right-c.e. functions.